aa r X i v : . [ m a t h . C T ] M a y POLYNOMIALS IN CATEGORIES WITH PULLBACKS
MARK WEBER
Abstract.
The theory developed by Gambino and Kock, of polynomials over a locallycartesian closed category E , is generalised for E just having pullbacks. The 2-categoricalanalogue of the theory of polynomials and polynomial functors is given, and its rela-tionship with Street’s theory of fibrations within 2-categories is explored. Johnstone’snotion of “bagdomain data” is adapted to the present framework to make it easier tocompletely exhibit examples of polynomial monads.
1. Introduction
Thanks to unpublished work of Andr´e Joyal dating back to the 1980’s, polynomials admita beautiful categorical interpretation. Given a multivariable polynomial function p withnatural number coefficients, like say p ( w, x, y, z ) = ( x y + 2 , x z + y ) (1)one may break down its formation as follows. There is a set In = { w, x, y, z } of “inputvariables” and a two element set Out of “output variables”. Rewriting p ( w, x, y, z ) =( x y + 1 + 1 , x z + x z + x z + y ), there is a set MSum = { x y, (1) , (1) , ( x z ) , ( x z ) , ( x z ) , y } of “monomial summands”, and a set UVar = { x , x , x , y , x , x , z , x , x , z , x , x , z , y } of “usages of variables”, informally consisting of no w ’s, nine x ’s, two y ’s and three z ’s.The task of forming the polynomial p can then be done in three steps. First one takesthe input variables and duplicates or ignores them according to how often each variable isused. The book-keeping of this step is by means of the evident function p : UVar → In ,which in our example forgets the subscripts of elements of UVar . In the second step oneperforms all the multiplications, and this is book-kept by taking products over the fibresof the function p : UVar → MSum which sends each usage to the monomial summandin which it occurs, that is x , x , x , y x y x , x , z ( x z ) x , x , z ( x z ) x , x , z ( x z ) y y. (cid:13) Mark Weber, 2015. Permission to copy for private use granted. p : MSum → Out . Thus the polynomial p “is” the diagram In UVar MSum Out o o p p / / p / / (2)in the category Set . A categorical interpretation of the formula (1) from the diagram(2) begins by regarding an n -tuple of variables as (the fibres of) a function into a givenset of cardinality n . Duplication of variables is then interpretted by the functor ∆ p : Set / In → Set / UVar given by pulling back along p , taking products by the functor Π p : Set / UVar → Set / MSum and taking sums by applying the functor Σ p : Set / MSum → Set / Out given by composing with p . Composing these functors gives P ( p ) : Set / In → Set / Out the polynomial functor corresponding to the polynomial p .Functors of the form ∆ p , Π p and Σ p are part of the bread and butter of categorytheory. For any map p in any category, one may define Σ p between the appropriateslices, and one requires only pullbacks in the ambient category to interpret ∆ p moregenerally. The functor Π p is by definition the right adjoint of ∆ p , and its existence is acondition on the map p , called exponentiability . Locally cartesian closed categories are bydefinition categories with finite limits in which all maps are exponentiable. Consequentlya reasonable general categorical definition of polynomial is as a diagram X A B Y o o p p / / p / / (3)in some locally cartesian closed category E . The theory polynomials and polynomialfunctors was developed at this generality in the beautiful paper [11] of Gambino andKock. There the question of what structures polynomials in a locally cartesian closed E form was considered, and it was established in particular that polynomials can be seenas the arrows of certain canonical bicategories, with the process of forming the associatedpolynomial functor giving homomorphisms of bicategories.In this paper we shall focus on the bicategory Poly E of polynomials and cartesian mapsbetween them in the sense of [11]. Our desire to generalise the above setting comes fromthe existence of canonical polynomials and polynomial functors for the case E = Cat andthe wish that they sit properly within an established framework. While local cartesianclosedness is a very natural condition of great importance to categorical logic, enjoyed forexample by any elementary topos, it is not satisfied by
Cat . Avoiding the assumption oflocal cartesian closure may be useful also for applications in categorical logic. For example,the categories of classes considered in Algebraic Set Theory [14] are typically not assumedto be locally cartesian closed, but the small maps are assumed to be exponentiable.The natural remedy of this defect is to define a polynomial p between X and Y in acategory E with pullbacks to be a diagram as in (3) such that p is an exponentiable map.Since exponentiable maps are pullback stable and closed under composition, one obtainsthe bicategory Poly E together with the “associated-polynomial-functor homomorphism”,as before. We describe this in Section 3.The main technical innovation of Sections 2 and 3 is to remove any reliance on typetheory in the proofs, giving a completely categorical account of the theory. In establishingthe bicategory structure on Poly E in Section 2 of [11], the internal language of E is usedin an essential way, especially in the proof of Proposition 2.9. Our development makes nouse of the internal language. Instead we isolate the concept of a distributivity pullbackin Section 2.2 and prove some elementary facts about them. Armed with this technologywe then proceed to give an elementary account of the bicategory of polynomials, andthe homomorphism which encodes the formation of associated polynomial functors. Ourtreatment requires only pullbacks in E .Our second extension to the categorical theory of polynomials is motivated by the factthat Cat is a 2-category. Thus in Section 4.1 we develop the theory of polynomials withina 2-category K with pullbacks, and the polynomial 2-functors that they determine. In thiscontext the structure formed by polynomials is a degenerate kind of tricategory, called a2-bicategory, which roughly speaking is a bicategory whose homs are 2-categories insteadof categories. However except for this change, the theory works in the same way as forcategories. In fact our treatment of the 1-categorical version of the theory in Section 3was tailored in order to make the previous sentence true (in addition to giving the desiredgeneralisation).A first source of examples of 2-categorical polynomials come from the 2-monads consid-ered first by Street [27] whose algebras are fibrations. In Proposition 4.2.3 these 2-monadsare exhibited as being polynomial in general. Fibrations in a 2-category play another rolein this work, because it is often the case that the maps participating in a polynomial maythemselves be fibrations or opfibrations in the sense of Street. This has implications forthe properties that the resulting polynomial 2-functor inherits. To this end, the generaltypes of 2-functor that are compatible with fibrations are recalled from [34] in Section4.3, and the polynomials that give rise to them are identified in Theorem 4.4.5.As explained in [7, 23] certain 2-categorical colimits called codescent objects are im-portant in 2-dimensional monad theory. Theorem 4.4.5 has useful consequences in [32], inwhich certain codescent objects which arise naturally from a morphism of 2-monads areconsidered. When these codescent objects arise from a situation conforming appopriatelyto the hypotheses of Theorem 4.4.5, they acquire extra structure which facilitates theircomputation. Also of relevance to the computation of associated codescent objects, wehave in Theorem 4.5.1 identified sufficient conditions on polynomials in Cat so that theirinduced polynomial 2-functors preserve all sifted colimits.While the bicategorical composition of polynomials has been established in [11], andmore generally in Sections 3 and 4 of this paper, actually exhibiting explicitly a polynomialmonad requires some effort due to the complicated nature of this composition. Howeverone can often avoid the need to check monad axioms by using an alternative approach,based on Johnstone’s notion of “bagdomain data” [13]. The essence of this approach isdescribed in Theorem 5.3.3 and its 2-categorical analogue Theorem 5.4.1. These methodsare then illustrated in Section 5.4, where various fundamental examples of polynomial2-monads on
Cat are exhibited. In particular the 2-monads on
Cat for symmetric andfor braided monoidal categories are polynomial 2-monads.Polynomial functors over some locally cartesian closed category E arise in diverse math-ematical contexts as explained in [11]. They arise in computer science under the nameof containers [1]. Tambara in [30] studied polynomials over categories of finite G -setsmotivated by representation theory and group cohomology. Very interesting applicationsof Tambara’s work were found by Brun in [8] to Witt vectors, and in [9] also to equiv-ariant stable homotopy theory and cobordism. Moreover in [22] one finds applications ofpolynomial functors to higher category theory.Having generalised to the consideration of non-locally cartesian closed categories wehave expanded the possible scope of applications. In this article we have described somebasic examples of polynomial monads over Cat . Further examples for
Cat of relevanceto operads are provided in [3, 32, 33]. The results of Section 3 apply also to polynomialsover
Top which were a part of the basic setting of the work of Joyal and Bisson [4] onDyer-Lashof operations.
Notations . We denote by [ n ] the ordinal { < ... < n } regarded as a category. Thecategory of functors A → B and natural transformations between them is usually denotedas [ A , B ], though in some cases we also use exponential notation B A . For instance E [1] isthe arrow category of a category E , and E [2] is a category whose objects are composablepairs of arrows of E . A 2-monad is a Cat -enriched monad, and given a 2-monad T on a2-category K , we denote by T -Alg s the 2-category of strict T -algebras and strict maps, T -Alg the 2-category of strict algebras and strong maps and Ps- T -Alg for the 2-categoryof pseudo- T -algebras and strong maps, following the usual notations of 2-dimensionalmonad theory [5, 23].
2. Elementary notions
In this section we describe the elementary notions which underpin our categorical treat-ment of the bicategory of polynomials in Section 3. In Section 2.1 we recall basic factsand terminology regarding exponentiable morphisms. In Section 2.2 we introduce dis-tributivity pullbacks, and prove various general facts about them.
Given a morphism f : X → Y in a category E ,we denote by Σ f : E /X → E /Y the functor given by composition with f . When E has pullbacks Σ f has a right adjoint denoted as ∆ f , given by pulling back maps along f .When ∆ f has a right adjoint, denoted as Π f , f is said to be exponentiable . A commutative Which are T -algebra morphisms up to coherent isomorphism square in E as on the left A BDC f / / k (cid:15) (cid:15) / / g (cid:15) (cid:15) h E /A E /B E /D E /C Σ f / / O O ∆ k / / Σ g ∆ h O O α + E /A E /B E /D E /C o o ∆ f Π k (cid:15) (cid:15) ∆ g o o (cid:15) (cid:15) Π h k s β determines a natural transformation α as in the middle, as the mate of the identityΣ k Σ f = Σ g Σ h via the adjunctions Σ h ⊣ ∆ h and Σ k ⊣ ∆ k . We call α a left Beck-Chevalley cell for the original square. There is another left Beck-Chevalley cell for thissquare, namely Σ h ∆ f → ∆ g Σ k , obtained by mating the identity Σ k Σ f = Σ g Σ h with theadjunctions Σ f ⊣ ∆ f and Σ g ⊣ ∆ g . If in addition h and k are exponentiable maps, thentaking right adjoints produces the natural transformation β from α , and we call this a right Beck-Chevalley cell for the original square. There is another right Beck-Chevalleycell ∆ k Π g → Π f ∆ h when f and g are exponentiable. It is well-known that the originalsquare is a pullback if and only if either associated left Beck-Chevalley cell is invertible,and when h and k are exponentiable, these conditions are also equivalent to the rightBeck-Chevalley cell β being an isomorphism. Under these circumstances we shall speakof the left or right Beck-Chevalley isomorphisms.Clearly exponentiable maps are closed under composition and any isomorphism isexponentiable. Moreover, exponentiable maps are pullback stable. For given a pullbacksquare as above in which g is exponentiable, one has Σ h ∆ f ∼ = ∆ g Σ k , and since Σ h iscomonadic, ∆ g has a right adjoint by the Dubuc adjoint triangle theorem [10].When E has a terminal object 1 and f is the unique map X →
1, we denote byΣ X , ∆ X and Π X the functors Σ f , ∆ f and Π f (when it exists) respectively. In fact sinceΣ X : E /X → E takes the domain of a given arrow into X , it makes sense to speak of iteven when E doesn’t have a terminal object. An object X of a finitely complete category E is exponentiable when the unique map X → X exists). A finitely complete category E is cartesian closed when all its objectsare exponentiable, and locally cartesian closed when all its morphisms are exponentiable.Note that as right adjoints the functors ∆ f and Π f preserve terminal objects. Anobject h : A → X of the slice category E /X is terminal if and only if h is an isomorphismin E , but there is also a canonical choice of terminal object for E /X – the identity 1 X .So for the sake of convenience we shall often assume below that ∆ f and Π f are chosen sothat ∆ f (1 Y ) = 1 X and Π f (1 X ) = 1 Y . For f : A → B in E a category with pullbacks,∆ f : E /B → E /A expresses the process of pulling back along f as a functor. One maythen ask: what basic categorical process is expressed by the functor Π f : E /A → E /B ,when f is an exponentiable map?Let us denote by ε (1) f the counit of Σ f ⊣ ∆ f , and when f is exponentiable, by ε (2) f thecounit of ∆ f ⊣ Π f . The components of these counits fit into the following pullbacks: Y XBA ε (1) f,b / / b (cid:15) (cid:15) / / f (cid:15) (cid:15) ∆ f b pb Q P ABR ε (2) f,a / / a / / f (cid:15) (cid:15) / / Π f a (cid:15) (cid:15) ε (1) f, Π f a pb (4)Now the universal property of ε (1) f , as the counit of the adjunction Σ f ⊣ ∆ f , is equivalentto the square on the left being a pullback as indicated. An answer to the above questionis obtained by identifying what is special about the diagram on the right in (4), thatcorresponds to the universal property of ε (2) f as the counit of ∆ f ⊣ Π f . To this end wemake Definition.
Let g : Z → A and f : A → B be a composable pair of morphisms ina category E . Then a pullback around ( f, g ) is a diagram X Z ABY p / / g / / f (cid:15) (cid:15) / / r (cid:15) (cid:15) q pb in which the square with boundary ( gp, f, r, q ) is, as indicated, a pullback. A morphism( p, q, r ) → ( p ′ , q ′ , r ′ ) of pullbacks around ( f, g ) consists of s : X → X ′ and t : Y → Y ′ suchthat p ′ s = p , qs = tq ′ and r = r ′ s . The category of pullbacks around ( f, g ) is denotedPB( f, g ).For example the pullback on the right in (4) exhibits ( ε (2) f,a , ε (1) f, Π f a , Π f a ) as a pullbackaround ( f, a ). One may easily observe directly that the universal property of ε (2) f,a isequivalent to ( ε (2) f,a , ε (1) f, Π f a , Π f a ) being a terminal object of PB( f, a ). Thus we make Definition.
Let g : Z → A and f : A → B be a composable pair of morphisms ina category E . Then a distributivity pullback around ( f, g ) is a terminal object of PB( f, g ).When ( p, q, r ) is a distributivity pullback, we denote this diagramatically as follows: X Z ABY p / / g / / f (cid:15) (cid:15) / / r (cid:15) (cid:15) q dpb and we say that this diagram exhibits r as a distributivity pullback of g along f .Thus the answer to the question posed at the beginning of this section is: when f : A → B is an exponentiable map in E a category with pullbacks, the functor Π f : E /A → E /B encodes the process of taking distributivity pullbacks along f .For any ( p, q, r ) ∈ PB( f, g ) one has a Beck-Chevalley isomorphism as on the leftΠ q ∆ p ∆ g ∼ = ∆ r Π f δ p,q,r : Σ r Π q ∆ p → Π f Σ g which when you mate it by Σ r ⊣ ∆ r and Σ g ⊣ ∆ g , gives a natural transformation δ p,q,r ason the right in the previous display. When this is an isomorphism, it expresses a type ofdistributivity of “sums” over “products”, and so the following proposition explains whywe use the terminology distributivity pullback. Proposition.
Let f be an exponentiable map in a category E with pullbacks. Then ( p, q, r ) is a distributivity pullback around ( f, g ) if and only if δ p,q,r is an isomorphism. Proof.
Since ( ε (2) f,g , ε (1) f, Π f g , Π f g ) is terminal in PB( f, g ), one has unique morphisms d and e fitting into a commutative diagram Z X Y BED w w p ♦♦♦♦♦♦♦♦♦ q / / r ' ' ❖❖❖❖❖❖❖❖❖ g g ε (2) f,g ❖❖❖❖❖❖❖❖❖ ε (1) f, Π f g / / Π f g ♦♦♦♦♦♦♦♦♦ d (cid:15) (cid:15) e (cid:15) (cid:15) pb in which the middle square is a pullback by the elementary properties of pullbacks. Thus( p, q, r ) is a distributivity pullback if and only if e is an isomorphism. Since the adjunctionsΣ r ⊣ ∆ r and Σ g ⊣ ∆ g are cartesian, δ p,q,r is cartesian, and so it is an isomorphism ifand only if its component at 1 Z ∈ E /Z is an isomorphism. Since ∆ p (1 Z ) = 1 X andΠ q (1 X ) = 1 Y one may easily witness directly that ( δ p,q,r ) X = e .When manipulating pullbacks in a general category, one uses the “elementary fact”that given a commutative diagram of the form A B CFED / / / / (cid:15) (cid:15) / / / / (cid:15) (cid:15) (cid:15) (cid:15) pb then the front square is a pullback if and only if the composite square is. In the remainderof this section we identify three elementary facts about distributivity pullbacks. Lemma. (Composition/cancellation) Given a diagram of the form B B BX YB B B Z h / / h / / h (cid:15) (cid:15) / / g / / f (cid:15) (cid:15) h (cid:15) (cid:15) h (cid:15) (cid:15) h / / h (cid:15) (cid:15) h h (cid:15) (cid:15) pbdpb pb in any category with pullbacks, then the right-most pullback is a distributivity pullbackaround ( g, h ) if and only if the composite diagram is a distributivity pullback around ( gf, h ) . Proof.
Let us suppose that right-most pullback is a distributivity pullback, and that C , C , k , k and k as in B B BX YB B B ZC C C h / / h / / h (cid:15) (cid:15) / / g / / f (cid:15) (cid:15) h (cid:15) (cid:15) h (cid:15) (cid:15) h / / h (cid:15) (cid:15) h h (cid:15) (cid:15) & & k k / / k (cid:9) (cid:9) ✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒ k (cid:25) (cid:25) k , , k k $ $ k (cid:22) (cid:22) k (cid:20) (cid:20) k (cid:127) (cid:127) k (cid:31) (cid:31) pbdpb dpb are given such that the square with boundary ( hk , gf, k , k ) is a pullback. Then wemust exhibit r : C → B and s : C → B unique such that h h r = k , h h r = sk and h s = k . Form C , k and k by taking the pullback of k along g , and then k is uniquesuch that k k = k and k k = f hk . Clearly the square with boundary ( hk , f, k , k )is a pullback around ( f, h ). From the universal property of the left-most distributivitypullback, one has k and k as shown unique such that k = h k , h k = k k and h k = k . From the universal property of the right-most distributivity pullback, one has k and k as shown unique such that k = h k , h k = k k and h k = k . Clearly h k k = h k and so by the universal property of the top-left pullback square one has k as shown unique such that h k = k and h k = k k . Clearly h h k = k , h h k = k k and h k = k and so we have established the existence of maps r and s with the required properties.As for uniqueness, let us suppose now that r : C → B and s : C → B are givensuch that h h r = k , h h r = sk and h s = k . We must verify that r = k and s = k . Since the right-most distributivity pullback is in particular a pullback, one has k ′ : C → B unique such that h h k ′ = k and h k ′ = sk . Since ( h h , h ) are jointlymonic, and clearly h h h r = h h k ′ k and h h r = h k ′ k , we have h r = k ′ k . Bythe universal property of the left-most distributivity pullback, it follows that h k ′ = k and h r = k . Thus by the universal property of the left-most distributivity pullback, itfollows that k = k ′ and k = s . Since ( h , h ) are jointly monic, h k = k = h r and h k = k k = h r , we have r = k .Conversely, suppose that the composite diagram is a distributivity pullback around( gf, h ), and that C , C , k , k and k as in B B BX YB B B ZC C C h / / h / / h (cid:15) (cid:15) / / g / / f (cid:15) (cid:15) h (cid:15) (cid:15) h (cid:15) (cid:15) h / / h (cid:15) (cid:15) h h (cid:15) (cid:15) k (cid:9) (cid:9) ✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒✒ k / / / / k k $ $ k (cid:29) (cid:29) k (cid:18) (cid:18) k (cid:127) (cid:127) k (cid:31) (cid:31) pbdpb pb are given such that the square with boundary ( h k , g, k , k ) is a pullback. We must give r : C → B and s : C → B unique such that k = h r , h r = sk and h s = k .Pullback k along h to produce C , k and k . This makes the square with boundary( hh k , gf, k , k k ) a pullback around ( gf, h ). Thus one has k and k as shown uniquesuch that h k = k , h h k = k k k and k h = k . By universal property of the rightpullback and since gh k = h k k , one has k as shown unique such that h k = k and h k = k k . By the uniquness part of the universal property of the left distributivitypullback, it follows that h k = k , and so we have established the existence of maps r and s with the required properties.As for uniqueness let us suppose that we are given r : C → B and s : C → B such that k = h r , h r = sk and h s = k . We must verify that r = k and s = k .By the universal property of the top-left pullback one has k ′ unique such that h k ′ = k and h k ′ = rk . By the uniquness part of the universal property of the left distributivitypullback, it follows that h k = h k ′ and h k = h r . Thus by the uniquness part of theuniversal property of the composite distributivity pullback, it follows that k = k ′ and s = k . Since ( h h , h ) are jointly monic, it follows that r = k . Lemma. (The cube lemma). Given a diagram of the form A A A B B D D C C C f / / k (cid:15) (cid:15) / / g (cid:15) (cid:15) h f / / k (cid:15) (cid:15) / / g (cid:15) (cid:15) h h (cid:15) (cid:15) d ) ) ❘❘❘❘❘❘❘❘ d ) ) ❘❘❘❘❘❘❘❘ d ❧❧❧❧❧❧❧❧ d ❧❧❧❧❧❧❧❧ d u u ❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧ d i i ❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘ pbpb (1) (2)(3) dpb in any category with pullbacks, in which regions (1) and (2) commute, region (3) is apullback around ( f , d ) , the square with boundary ( f , k , g , h ) is a pullback and the bottom distributivity pullback is around ( g , d ) . Then regions (1) and (2) are pullbacks ifand only if region (3) is a distributivity pullback around ( f , d ) . Proof.
Let us suppose that (1) and (2) are pullbacks and p , q and r are given as in A A A B B D D C C C f / / k (cid:15) (cid:15) / / g (cid:15) (cid:15) h f / / k (cid:15) (cid:15) / / g (cid:15) (cid:15) h h (cid:15) (cid:15) d ) ) ❘❘❘❘❘❘❘❘ d ) ) ❘❘❘❘❘❘❘❘ d ❧❧❧❧❧❧❧❧ d ❧❧❧❧❧❧❧❧ d u u ❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧ d i i ❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘ pbpbpb pbpbdpb X Y (cid:5) (cid:5) p ☛☛☛☛☛☛☛ q / / r (cid:8) (cid:8) ✒✒✒✒✒✒✒✒✒✒✒ s (cid:12) (cid:12) t (cid:25) (cid:25) s s s t + + such that the square with boundary ( q, r, f , d p ) is a pullback. Then one can use thebottom distributivity pullback to induce s and t as shown, and then the pullbacks (1)and (2) to induce s and t , and these clearly satisfy d s = p , f s = tq and r = d t . On theother hand given s ′ : X → A and t ′ : Y → B satisfying these equations, define s ′ = h s ′ and t ′ = k t ′ . But then by the uniqueness part of the universal property of the bottomdistributivity pullback it follows that s ′ = s and t ′ = t , and from the uniqueness partsof the universal properties of the pullbacks (1) and (2), it follows that s = s ′ and t = t ′ ,thereby verifying that s and t are unique satisfying the aforementioned equations.For the converse suppose that (3) is a distributivity pullback. Note that (2) being apullback implies that (1) is by elementary properties of pullbacks, so we must show that(2) is a pullback. To that end consider s and t as in A A A B B D D C C C f / / k (cid:15) (cid:15) / / g (cid:15) (cid:15) h f / / k (cid:15) (cid:15) / / g (cid:15) (cid:15) h h (cid:15) (cid:15) d ) ) ❘❘❘❘❘❘❘❘ d ) ) ❘❘❘❘❘❘❘❘ d ❧❧❧❧❧❧❧❧ d ❧❧❧❧❧❧❧❧ d ❧❧❧❧❧❧❧❧❧ u u ❧❧❧❧❧❧❧❧❧ d i i ❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘ pbpb = = dpbdpb P Z s (cid:8) (cid:8) ✒✒✒✒✒✒✒✒✒✒✒ t (cid:25) (cid:25) ✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸ u (cid:22) (cid:22) ✱✱✱✱✱✱✱✱✱✱✱ v / / w (cid:12) (cid:12) x (cid:5) (cid:5) y s s z + + such that k s = d t , and then pullback s and f to produce P , u and v . Using the factthat the bottom distributivity pullback is a mere pullback, one has w unique such that d d w = h u and g w = tv . Using the inner left pullback, one has x unique such that h x = d w and d x = u . Using the distributivity pullback (3), one has y and z unique1such that d y = x , f y = zv and s = d z . By the uniqueness part of the universal propertyof the bottom distributivity pullback, it follows that t = k z . Thus we have constructed z satisfying s = d z and t = k z . On the other hand given z ′ : Z → B such that s = d z ′ and t = k z ′ , one has y ′ : P → A unique such that d d y = u and f y = zv , using thefact that the top distributivity pullback is a mere pullback. Then from the uniquenesspart of the universal property of that distributivity pullback, it follows that y = y ′ and z = z ′ . Thus as required z is unique satisfying s = d z and t = k z . Lemma. (Sections of distributivity pullbacks). Let
D A BCE p / / g / / f (cid:15) (cid:15) / / r (cid:15) (cid:15) q dpb be a distributivity pullback around ( f, g ) in any category with pullbacks. Three maps s : B → A s : B → D s : C → E which are sections of g , gp and r respectively, and are natural in the sense that s = ps and qs = s f , are determined uniquely by the either of the following: (1) the section s ;or (2) the section s . Proof.
Given s a section of g , induce s and s uniquely as shown: D A BCE p / / g / / f (cid:15) (cid:15) / / r (cid:15) (cid:15) q dpb BC s (cid:29) (cid:29) f (cid:15) (cid:15) ; ; s / / s / / using the universal property of the distributivity pullback. On the other hand given thesection s , one induces s using the fact that the distributivity pullback is a mere pullback,and then put s = ps .We often assume that in a given category E with pullbacks, some choice of all pullbacks,and of all existing distributivity pullbacks, has been fixed. Moreover we make the followingharmless assumptions, for the sake of convenience, on these choices once they have beenmade. First we assume that the chosen pullback of an identity along any map is anidentity. This ensures that ∆ X = 1 E /X and that ∆ f (1 B ) = 1 A for any f : A → B .Similarly we assume that all diagrams of the form • • ••• / / / / f (cid:15) (cid:15) / / (cid:15) (cid:15) f dpb • • ••• / / g / / (cid:15) (cid:15) / / g (cid:15) (cid:15) dpb f (1 A ) = 1 B for any exponentiable f : A → B , and that Π X = 1 E /X .
3. Polynomials in categories
This section contains our general theory of polynomials and polynomial functors. InSection 3.1 we give an elementary account of the composition of polynomials, culminatingin Theorem 3.1.10, in which polynomials in a category E with pullbacks are exhibitedas the 1-cells of the bicategory Poly E . Then in Section 3.2, we study the process offorming the associated polynomial functor, exhibiting this as the effect on 1-cells of thehomomorphism P E : Poly E → CAT in Theorem 3.2.6. At this generality, the homs ofthe bicategory
Poly E have pullbacks, and the hom functors of P E preserve them. Thisgives the sense in which the theory of polynomial functors could be iterated, and this isdescribed in Section 3.3. The organisation of this section has been chosen to facilitate itsgeneralisation to the theory of polynomials in 2-categories, in Section 4.1. Let E be a category with pullbacks. In thissection we give a direct description of a bicategory Poly E , whose objects are those of E , and whose one cells are polynomials in E in the following sense. For X , Y in E , a polynomial p from X to Y in E consists of three maps X A B Y o o p p / / p / / such that p is exponentiable. Let p and q be polynomials in E from X to Y . A cartesianmorphism f : p → q is a pair of maps ( f , f ) fitting into a commutative diagram X A B YB ′ A ′ (cid:127) (cid:127) p ⑧⑧⑧⑧⑧⑧⑧ p / / p (cid:31) (cid:31) ❄❄❄❄❄❄❄ _ _ q ❄❄❄❄❄❄❄ q / / q ? ? ⑧⑧⑧⑧⑧⑧⑧ f (cid:15) (cid:15) f (cid:15) (cid:15) pb We call f the 0 -component of f , and f the 1 -component of f . With composition inheritedin the evident way from E , one has a category Poly E ( X, Y ) of polynomials from X to Y and cartesian morphisms between them. These are the homs of our bicategory Poly E .In order to describe the bicategorical composition of polynomials, we introduce theconcept of a subdivided composite of a given composable sequence of polynomials. Thisenables us to give a direct description of n -ary composition for Poly E , and then to describethe sense in which coherence for this bicategory “follows from universal properties”.Consider a composable sequence of polynomials in E of length n , that is to say, poly-nomials X i − A i B i X i o o p i p i / / p i / / in E , where 0 < i ≤ n . We denote such a sequence as ( p i ) ≤ i ≤ n , or more briefly as ( p i ) i .3 Definition.
Let ( p i ) ≤ i ≤ n be a composable sequence of polynomials of length n .A subdivided composite over ( p i ) i consists of objects ( Y , ..., Y n ), morphisms q : Y → X q ,i : Y i − → Y i q : Y n → X n for 0 < i ≤ n , and morphisms r i : Y i − → A i s i : Y i → B i for 0 < i ≤ n , such that p r = q , p n s n = q and Y i A i +1 X i B i r i +1 / / p i +1 , (cid:15) (cid:15) / / p i (cid:15) (cid:15) s i = Y i − Y i B i A i q i / / s i (cid:15) (cid:15) / / p i (cid:15) (cid:15) r i pb For example a subdivided composite over ( p , p , p ), that is when n = 3, assemblesinto a commutative diagram like this: • • • • • • • • • •• • • • o o p p / / p / / o o p p / / p / / o o p p / / p / / q / / q / / q / / r (cid:15) (cid:15) s (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧ r (cid:31) (cid:31) ❄❄❄❄❄❄❄ s (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧ r (cid:31) (cid:31) ❄❄❄❄❄❄❄ s (cid:15) (cid:15) q (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧ q (cid:31) (cid:31) ❄❄❄❄❄❄❄ pb pb pb We denote a general subdivided composite over ( p i ) i simply as ( Y, q, r, s ). Definition.
Let ( p i ) ≤ i ≤ n be a composable sequence of polynomials of length n .A morphism ( Y, q, r, s ) → ( Y ′ , q ′ , r ′ , s ′ ) of subdivided composites consists of morphisms t i : Y i → Y ′ i for 0 ≤ i ≤ n , such that q = q ′ t , q ′ i t i − = t i q i , q = q ′ t n , r i = r ′ i t i − and s i = s ′ i t i . With compositions inherited from E , one has a category SdC( p i ) i of subdividedcomposites over ( p i ) i and morphisms between them.Given a subdivided composite ( Y, q, r, s ) over ( p i ) i , note that the morphisms q i areexponentiable since exponentiable maps are pullback stable, and that the composite q : Y → Y n defined as q = q n ...q is also exponentiable, since exponentiable maps areclosed under composition. Thus we make Definition.
The associated polynomial of a given subdivided composite (
Y, q, r, s )over ( p i ) i is defined to be X Y Y n X n o o q q / / q / / The process of taking associated polynomials is the object map of a functorass : SdC( p i ) i −→ Poly E ( X , X n ) . Having made the necessary definitions, we now describe the canonical operations onsubdivided composites which give rise to the bicategorical composition of polynomials.4Let n > p i ) ≤ i ≤ n be a composable sequence of polynomials in E . One has evidentforgetful functors res and res n as inSdC( p i )
The morphisms ε ( Y,q,r,s ) just described are the components of the counitof an adjunction res n ⊣ p n · ( − ) . Proof.
Let ( Y ′ , q ′ , r ′ , s ′ ) be a subdivided composite over ( p i ) ≤ i ≤ n , then for t as inres n ( p n · ( Y, q, r, s )) (
Y, q, r, s )res n ( Y ′ , q ′ , r ′ , s ′ ) ε ( Y,q,r,s ) / / t ♦♦♦♦♦♦♦♦♦ res n ( t ′ ) g g t ′ unique so that the above triangle commutes. The following commutativediagram assembles this given data in the case n = 4. • • • • • • • • • •• • • • • • •• • • o o p p / / p / / o o p p / / p / / o o p p / / p / / o o p p / / p / / q / / q / / q / / • • • r (cid:15) (cid:15) s ⑧⑧ (cid:127) (cid:127) ⑧⑧ r ❄❄ (cid:31) (cid:31) ❄❄ s ⑧⑧ (cid:127) (cid:127) ⑧⑧ r ❄❄ (cid:31) (cid:31) ❄❄ s (cid:15) (cid:15) q ⑧⑧ (cid:127) (cid:127) ⑧⑧ q ❄❄ (cid:31) (cid:31) ❄❄ w w ♦♦♦♦ (cid:26) (cid:26) ✹✹✹✹✹✹✹ w w ♦♦♦♦ ( p · q ) / / (cid:15) (cid:15) ( p · q ) ✴✴✴✴ (cid:23) (cid:23) ✴✴✴✴ ε ⑧⑧ (cid:127) (cid:127) ⑧⑧ ε ttt z z ttt ε ⑧⑧ (cid:127) (cid:127) ⑧⑧ ( p · q ) / / ( p · q ) / / ( p · q ) / / pb pb pbpb pb pb pb dpb • • • • • q ′ / / q ′ / / q ′ / / q ′ / / t (cid:17) (cid:17) t (cid:13) (cid:13) t (cid:17) (cid:17) t (cid:17) (cid:17) q ′ (cid:13) (cid:13) r ′ (cid:20) (cid:20) s ′ (cid:23) (cid:23) Since q n − t n − = p n r ′ n one induces u : Y ′ n − → C using the defining pullback of C , andthen one induces t ′ n − : Y ′ n − → Y n − and t ′ n : Y ′ n → Y n from the maps u and s ′ n using thedistributivity pullback. The rest of the t ′ i are induced inductively as follows. For 0 < i < n given t ′ i : Y ′ i → Y i , one induces t ′ i − using the maps t i − and t ′ i and the pullback whichdefines ( p n · Y ) i − . By construction the t ′ i are the components of the required unique map t ′ . For ( Y, q, r, s ) a subdivided composite over ( p i )
1, define thecomponents of the morphism ε ′ Y,q,r,s : res (( Y, q, r, s ) · p ) −→ ( Y, q, r, s ) . Lemma.
The morphisms ε ′ Y,q,r,s just described are the components of the counit ofan adjunction res ⊣ ( − ) · p . Proof.
Given (
Y, q, r, s ) in SdC( p i ) ≤ i ≤ n and t as inres (( Y, q, r, s ) · p ) ( Y, q, r, s )res ( Y ′ , q ′ , r ′ , s ′ ) ε ′ Y,q,r,s / / t ❧❧❧❧❧❧❧❧❧❧ res ( t ′ ) i i we must exhibit t ′ as shown unique so that the above diagram commutes. In the case n = 4 the data ( Y ′ , q ′ , r ′ , s ′ ) and t fit into the following diagram: • • • • • • • • • • • • •• • • • o o p p / / p / / o o p p / / p / / o o p p / / p / / o o p p / / p / / q / / q / / q / / r (cid:15) (cid:15) s ☎☎ (cid:2) (cid:2) ☎☎ r ✿✿ (cid:28) (cid:28) ✿✿ s ☎☎ (cid:2) (cid:2) ☎☎ r ✿✿ (cid:28) (cid:28) ✿✿ s (cid:15) (cid:15) q ☎☎ (cid:2) (cid:2) ☎☎ q ✿✿ (cid:28) (cid:28) ✿✿ pb pb pb ••• ••• •• •• (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) ( q · p ) / / ( q · p ) / / ( q · p ) / / ( q · p ) / / / / / / (cid:0) (cid:0) ✁✁✁✁✁✁✁✁✁✁✁✘✘✘✘✘✘✘✘✘✘✘ (cid:12) (cid:12) ✘✘✘✘✘✘✘✘✘✘✘ ✫✫✫✫✫✫✫✫✫✫✫ (cid:18) (cid:18) ✫✫✫✫✫✫✫✫✫✫✫ pbpb dpb dpb dpbpb pb • • • • • (cid:17) (cid:17) q ′ q ′ / / q ′ / / q ′ / / q ′ / / q ′ (cid:13) (cid:13) t (cid:20) (cid:20) t (cid:20) (cid:20) t (cid:20) (cid:20) t (cid:20) (cid:20) r ′ (cid:10) (cid:10) s ′ (cid:15) (cid:15) C and the maps s ′ and t , one induces Y ′ → C .Using the distributivity pullbacks one induces successively the morphisms Y ′ i → C i and Y ′ i +1 → C i for 0 < i < n . In the case i = n − t ′ n − and t ′ n respectively. The components t ′ i for 0 ≤ i < n − Y · p ) i . By construction the t ′ i are the componentsof the required unique map t ′ . Proposition.
For any composable sequence ( p i ) ≤ i ≤ n of polynomials in a category E with pullbacks, the category SdC( p i ) i has a terminal object. Proof.
We proceed by induction on n . In the case n = 0, observe that a subdividedcomposite consists just of the data Y , q : Y → X and q : Y → X , and that SdC() isthe category Span E ( X , X ) of endospans of X . The identity endospan is terminal. Forthe inductive step apply either of the functors p n · ( − ) or ( − ) · p which as right adjoints,preserve terminal objects. Definition.
Let E be a category with pullbacks. A composite of a composablesequence ( p i ) ≤ i ≤ n of polynomials in E , is defined to be the associated polynomial of aterminal object in the category SdC( p i ) i . When such a composite has been chosen, it isdenoted as p n ◦ ... ◦ p .Let us consider now some degenerate cases of Definition 3.1.7. • n = 0: Choosing identity spans as terminal nullary subdivided composites (see theproof of Proposition 3.1.6), nullary composition of polynomials gives polynomialswhose constituent maps are all identities. That is, X X X X o o X X / / X / / is the “identity polynomial on X ” as one would hope. • n = 1: One may identify SdC( p ) as the slice Poly E ( X , X ) /p , and thus choose 1 p as the terminal unary subdivided composite over ( p ). Thus the unary composite ofa given polynomial p is just p . • n = 2: applying p · ( − ) to p , or ( − ) · p to p , gives the same subdivided composite,namely • • • • • • ••• •• o o p p / / p / / o o p p / / p / / (cid:15) (cid:15) { { ✇✇✇✇✇✇ ●●●●●● (cid:10) (cid:10) ✔✔✔✔✔✔✔✔✔✔ / / / / (cid:20) (cid:20) ✯✯✯✯✯✯✯✯✯✯ (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ (cid:31) (cid:31) ❄❄❄❄❄❄❄❄❄❄❄❄❄ pb dpbpb which is terminal by the case n = 1 and since the functors p · ( − ) and ( − ) · p ,as right adjoints, preserve terminal objects. Thus the associated composite of theabove is the binary composite p ◦ p , and this agrees with the binary compositionof polynomials given in [11].8 Lemma.
Let n > and ( p i ) ≤ i ≤ n be a composable sequence of polynomials in acategory E with pullbacks. Then one has canonical isomorphisms SdC( p i ) ≤ i The canonical isomorphism on the left follows from the definitions and the el-ementary properties of pullbacks. The canonical isomorphism on the right follows fromthe definitions, and iterated application of Lemma 2.2.4.In order to make explicit the horizontal composition of 2-cells in Poly E we consider ahorizontally composable sequence of morphisms of polynomials of length n , that is to saydiagrams X i − A i B i X i B ′ i A ′ i w w p i ♦♦♦♦♦♦♦ p i / / p i ' ' ❖❖❖❖❖❖❖❖ q i ♦♦♦♦♦♦♦♦ / / q i q i g g ❖❖❖❖❖❖❖ f i (cid:15) (cid:15) f i (cid:15) (cid:15) pb in E , for 0 < i ≤ n . We denote such a sequence as ( f i , f i ) i : ( p i ) i → ( q i ) i since itis a morphism of the category Q ni =1 Poly E ( X i − , X i ). The process of vertically stackingsubdivided composites and their morphisms on top of ( f i , f i ) i gives a functorSdC( f i , f i ) i : SdC( p i ) i −→ SdC( q i ) i . The assignation ( f i , f i ) i SdC( f i , f i ) i is functorial, and natural in the evident sensewith respect to the restriction and associated polynomial functors defined above. Forany choice t and t of terminal object of SdC( p i ) i and SdC( q i ) i respectively, one hascomposites p n ◦ ... ◦ p = ass( t ) q n ◦ ... ◦ q = ass( t )by Definition 3.1.7, and a unique morphism u t ,t : SdC( f i , f i ) i ( t ) → t . Definition. Let E be a category with pullbacks and ( f i , f i ) i : ( p i ) i → ( q i ) i bea horizontally composable sequence of polynomial morphisms of length n . Then in thecontext just described, the 2-cell f n ◦ ... ◦ f : p n ◦ ... ◦ p −→ q n ◦ ... ◦ q is defined to be ass( u t ,t ).9In the case n = 2 the original data and the chosen terminal subdivided compositescomprise the solid parts of the diagram, • • • • • • •••••••• w w ♦♦♦♦♦♦ / / ' ' ❖❖❖❖❖❖ w w ♦♦♦♦♦♦ / / ' ' ❖❖❖❖❖❖ ♦♦♦♦♦♦ / / g g ❖❖❖❖❖❖ ♦♦♦♦♦♦ / / g g ❖❖❖❖❖❖ f (cid:15) (cid:15) f (cid:15) (cid:15) f (cid:15) (cid:15) f (cid:15) (cid:15) pb pb •• •• (cid:15) (cid:15) w w ♦♦♦♦♦♦ ' ' ❖❖❖❖❖❖ / / (cid:22) (cid:22) ✲✲✲✲✲✲✲ (cid:30) (cid:30) ❂❂❂❂❂❂❂❂❂❂❂❂❂ (cid:0) (cid:0) ✁✁✁✁✁✁✁✁✁✁✁✁✁ (cid:8) (cid:8) ✑✑✑✑✑✑✑ / / pbpb dpb •• •• O O g g ❖❖❖❖❖❖ ♦♦♦♦♦♦ / / H H ✑✑✑✑✑✑✑ @ @ ✁✁✁✁✁✁✁✁✁✁✁✁✁ ^ ^ ❂❂❂❂❂❂❂❂❂❂❂❂❂ V V ✲✲✲✲✲✲✲ / / pbpb dpb φ (cid:22) (cid:22) φ (cid:9) (cid:9) φ (cid:15) (cid:15) φ (cid:15) (cid:15) and one then induces φ using the pullback defining its codomain, φ and φ are theninduced by the universal property of the bottom distributivity pullback, and finally φ is induced by the bottom pullback. One verifies easily that ( φ , φ , φ ) a morphism ofsubdivided composites, thus it is u t ,t , and so by Definition 3.1.9 the composite f ◦ f isgiven by ( φ , φ ). Theorem. Let E be a category with pullbacks. One has a bicategory Poly E , whoseobjects are those of E , whose hom from X to Y is Poly E ( X, Y ) , horizontal compositionof 1-cells is given by Definition 3.1.7, and horizontal composition of 2-cells is given byDefinition 3.1.9. Proof. By induction on n , using the fact that the functors p n · ( − ) and ( − ) · p pre-serve terminal objects, and Lemma 3.1.8, it follows that any iterated binary compositeof polynomials of length n , is a composite in the sense of Definition 3.1.7. That is, suchan iterated composite is the associated polynomial of a terminal subdivided polynomial,which arises from the composable sequence of polynomials that participates in the giveniterated binary composite. Hence between any two alternative brackettings of a givencomposite, there is a unique isomorphism of their underlying subdivided composites, giv-ing rise to a “coherence” isomorphism of the composites themselves upon application of“ass”. Any diagram of such coherence isomorphisms must commute, since it is the imageby the appropriate “ass” functor, of a diagram whose vertices are all terminal subdividedcomposites. Thanks to our conventions regarding chosen pullbacks and chosen distribu-tivity pullbacks of identities described in Section 2.2, the unit coherence isomorphismshere turn out to be identities.The functoriality of horizontal composition comes from the functoriality of ( f i , f i ) i SdC( f i , f i ) i and the naturality of SdC( f i , f i ) i with respect to the “ass” functors. Itremains to verify the naturality of the coherence isomorphisms identified in the previousparagraph. To this end we suppose that a horizontally composable sequence ( f i , f i ) i :( p i ) i → ( q i ) i of morphisms of polynomials of length n , and binary brackettings β and β of n things is given. Let us denote by t β ( p i ) i , t β ( p i ) i , t β ( q i ) i and t β ( q i ) i the terminalsubdivided composites witnessing the iterated binary composites of ( p i ) i and ( q i ) i via the0given brackettings. In SdC( q i ) i one has the diagramSdC( f i , f i ) i ( t β ( p i ) i ) SdC( f i , f i ) i ( t β ( p i ) i ) t β ( q i ) i t β ( q i ) i / / (cid:15) (cid:15) / / (cid:15) (cid:15) o o o o in which the top horizontal arrows are the effect of applying SdC( f i , f i ) i to the uniquemorphisms, and the other morphisms are determined uniquely and both squares commutebecause t β ( q i ) i and t β ( q i ) i are terminal. Applying ass : SdC( q i ) i → Poly E ( X , X n ) tothis diagram gives the squares witnessing the naturality of the coherence morphisms.A span in E as on the left X Z Y o o s t / / X Z Z Y o o s Z / / t / / may be identified as a polynomial in which the middle map is an identity as on the right.Polynomial composition of spans coincides exactly with span composition, giving us astrict inclusion Span E ֒ → Poly E of bicategories which is the identity on objects and locally fully faithful. For a given map f : X → Y in E , we denote by f • : X → Y and f • : Y → X the polynomials X X X Y o o / / f / / Y X X X o o f / / / / respectively. These are spans, it is well known that one has f • ⊣ f • and that this is partof the basic data of the proarrow equipment ( E , Span E ) [36, 37]. By the above strictinclusion, this extends to another proarrow equipment ( E , Poly E ), and all this at thegenerality of a category E with pullbacks. It is worth noting that polynomial compositesof the form f • ◦ p and q ◦ g • are particularly easy, these being • • • • o o p p / / fp / / • • • • o o gq q / / q / / respectively.The homs of Poly E interact well with the slices of E . For all X and Y one has obviousforgetful functors E /X Poly E ( X, Y ) E /Y o o l X,Y r X,Y / / and we refer to these as the left and right projections of the homs of Poly E . From theabove descriptions of composites of the form f • ◦ p and q ◦ g • , one obtains immediatelythe sense in which these forgetful functors are natural. Lemma. For all f : Y → Z and g : X → W one has Σ g l X,Y = l W,Y (( − ) ◦ g • ) Σ f r X,Y = r X,Z ( f • ◦ ( − )) l X,Y = l X,Z ( f • ◦ ( − )) r X,Y = r W,Y (( − ) ◦ g • )1 Let E be a category with pullbacks. In this section wedefine a homomorphism of bicategories P E : Poly E −→ CAT X 7→ E /X with object map as indicated, in Theorem 3.2.6. Given a polynomial p : X → Y in E ,the functor P E ( p ) : E /X → E /Y is defined to be the composite Σ p Π p ∆ p , which for thesake of brevity, will also be denoted as p ( − ) : E /X → E /Y . In more elementary terms theeffect of p ( − ) on an object x : C → X of E /X is described by the following commutativediagram: X A B YC C C C o o p p / / p / / o o x (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / (cid:15) (cid:15) p ( x ) ❄❄❄❄❄❄ (cid:31) (cid:31) ❄❄❄❄❄❄ dpbpb Similarly one may, by exploiting the universal property of the pullback and distributivitypullback in this description, induce the maps which provide the arrow map of p ( − ). Theseexplicit descriptions together with Lemma 3.1.11 enables us to catalogue all the ways onecan use the composition of Poly E to describe the functor p ( − ), and we record this in Lemma. Let p : X → Y be a polynomial in E .1. Given x : C → X in E /X , one has p ( x ) = r Z,Y ( p ◦ x • ◦ g • ) for all Z and g : C → Z .2. Given x : C → X , x : C → X and h : C → C over X , one has p ( h ) = r Z,Y ( p ◦ h ′ ◦ g • ) for all Z and g : C → Z , where h ′ : x • → x • h • is the mate of the identity x • h • = x • via h • ⊣ h • . To prove Theorem 3.2.6 we exhibit an analogous result, Lemma 3.2.5, giving all theways of expressing P E ’s 2-cell map in terms of composition in Poly E . Preliminary tothis result we reconcile two ways of describing P E ’s 2-cell map – that given in [11] versusa direct description in terms of morphisms of induced from pullbacks and distributivitypullbacks, in Lemma 3.2.4. Lemmas 3.2.2 and 3.2.3 are preliminary to Lemma 3.2.4. Thereader not interested in such technical details is encouraged to skip ahead to the statementof Theorem 3.2.6 below.2The description of P E ’s 2-cell map given in [11], is to associate to a given cartesianmorphism f : p → q between polynomials from X to Y , the following natural transfor-mation E /X E /A E /B E /Y E /B ′ E /A ′ ∆ p < < ②②②②②② Π p / / ❊❊ Σ p " " ❊❊❊❊❊ ∆ q " " ❊❊❊❊❊❊ Π q / / ②② Σ q < < ②②②②② ∆ f O O ∆ f O O ∼ = ∼ = (cid:11) (cid:19) (5)in which the isomorphism in the middle is a Beck-Chevalley isomorphism, and Σ p ∆ f → Σ q is the mate of the identity via Σ f ⊣ ∆ f . The advantage of this description is thatthe functoriality of the resulting hom functor( P E ) X,Y : Poly E ( X, Y ) −→ CAT ( E /X, E /Y )is evident. We will show that the component at x : C → X of ( P E ) X,Y ( f ) is given by themap f ,x , constructed in X A B YA ′ B ′ C C C C C ′ C ′ C ′ u u p ❦❦❦❦❦❦❦❦❦ p / / p ) ) ❙❙❙❙❙❙❙❙❙ i i q ❙❙❙❙❙❙❙❙ q / / q ❦❦❦❦❦❦❦❦ { { ✇✇✇✇✇✇✇✇✇✇ u u ❦❦❦❦❦❦❦❦ / / p ( x ) (cid:29) (cid:29) ❀❀❀❀❀❀❀❀❀❀❀❀❀ c c ●●●●●●●●●● i i ❙❙❙❙❙❙❙❙ / / q ( x ) A A ✄✄✄✄✄✄✄✄✄✄✄✄ x / / ) ) ❙❙❙❙❙❙❙❙ ❦❦❦❦❦❦❦❦ (cid:15) (cid:15) O O f (cid:15) (cid:15) f (cid:15) (cid:15) f ,x (cid:26) (cid:26) f ,x (cid:23) (cid:23) f ,x (cid:23) (cid:23) pbpb dpbdpb pb (6)To construct this diagram one induces f ,x using f and the bottom pullback, and thenit follows that the square ( C , A, A ′ , C ′ ) is a pullback. One can then induce f ,x and f ,x using the bottom distributivity pullback. Lemma 2.2.5 ensures that ( C , B, B ′ , C ′ )is a pullback, and elementary properties of pullbacks ensure that ( C , C , C ′ , C ′ ) and( C , C , C ′ , C ′ ) are also pullbacks.Our next task is to explain why (6) really does describe the components of (5). Thisverification begins by unpacking, for a given commuting square as shown on the right A BDC f / / k (cid:15) (cid:15) / / g (cid:15) (cid:15) h E /A E /B E /D E /C Σ f / / O O ∆ k / / Σ g ∆ h O O α + E /A E /B E /D E /C o o ∆ f Π k (cid:15) (cid:15) ∆ g o o (cid:15) (cid:15) Π h k s β α , and when h and k are exponentiable, the right Beck-Chevalley cell β , in elementary terms. Since α is obtained by taking the mate of theidentity Σ g Σ h = Σ k Σ f via the adjunctions Σ h ⊣ ∆ h and Σ k ⊣ ∆ k , it follows that α isuniquely determined by the equation Σ k ε (1) f = ( ε (1) g Σ k )(Σ g α ). On the other hand one hasthe commutative diagram A A A BDCC h x / / f / / k (cid:15) (cid:15) / / g / / x (cid:15) (cid:15) ε (1) h,x α x \ \ ∆ k ( gx ) ! ! ε (1) k,gx (cid:31) (cid:31) h (cid:15) (cid:15) pb (7)and the above equation, for the component x , is witnessed by the commutativity of thebottom triangle. Thus Lemma. The components of α are induced as in (7). Moreover we can see directly from (7) that if the original square is a pullback, then α is invertible, and the converse follows by considering the case x = 1 C . The right Beck-Chevalley cell β may be obtained by taking the mate of ∆ f ∆ k ∼ = ∆ h ∆ g via ∆ h ⊣ Π h and∆ k ⊣ Π k . Thus it is uniquely determined by the commutativity of∆ h ∆ g Π k ∆ h Π h ∆ f ∆ f ∆ f ∆ k Π k ∆ h β / / ε (2) h ∆ f (cid:15) (cid:15) / / ∆ f ε (2) k (cid:15) (cid:15) coh.Π k (8)whereas inside E for all B and x : B → B we have A BDCC A A B B D C A f / / k (cid:15) (cid:15) / / g (cid:15) (cid:15) h ' ' ❖❖❖❖❖ ♦♦♦♦ (cid:15) (cid:15) / / ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ (cid:15) (cid:15) + + ❲❲❲❲❲❲❲❲❲❲❲ / / / / w w ♦♦♦♦ x w w ♦♦♦♦♦ (cid:15) (cid:15) g g ❖❖❖❖❖❖❖❖❖❖❖❖❖ / / (cid:31) (cid:31) β x ? ? pbpbpbdpb dpb (9)constructed as follows. Take the distributivity pullback of x along k and then pullbackthe result along g . Pullback x along f and then take the distributivity pullback of theresult along h . Then form the top pullback and induce the morphism A → C . Bythe elementary properties of pullbacks, it follows that the squares ( A , B , D , C ) and( A , A, C, C ) are pullbacks. From this last we induce the dotted arrows using the leftdistributivity pullback.4 Lemma. The components of β are induced as in (9). Proof. Note that the square ( A , A , C , C ) is also a pullback, and so one can identifythe commuting triangle ( A , A , A ) with (8), once one has understood that A → A isthe (appropriate component of) the lower composite in (8).From this construction and Lemma 2.2.5 one may witness directly that if the originalsquare is a pullback, then β is invertible, and the converse is easily witnessed by consid-ering the case x = 1 B . With these details sorted out we can now proceed to the proofof Lemma. The component at x : C → X of the natural transformation described in(5) is the morphism f ,x described in (6). Proof. The proof consists of unpacking the definition of ( P E ) X,Y ( f ) x with reference tothe diagram (6), keeping track of the canonical isomorphisms that participate in thedefinition. All of this may be witnessed in X A B YA ′ B ′ C C C C C ′ C ′ C ′ w w ♦♦♦♦♦♦♦♦ ♦♦♦♦ / / ' ' ❖❖❖❖❖❖❖❖❖❖❖❖❖ g g ❖❖❖❖❖❖❖❖ ❖❖❖❖ / / ♦♦♦♦♦♦♦♦♦♦♦♦♦ (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ w w ♦♦♦♦♦♦♦♦♦♦♦♦ / / (cid:17) (cid:17) _ _ ❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄ g g ❖❖❖❖❖❖❖❖❖❖❖❖ / / M M / / ❖❖❖❖❖❖❖❖ ' ' ❖❖❖♦♦♦♦♦♦♦♦ ♦♦♦ (cid:15) (cid:15) O O (cid:15) (cid:15) (cid:15) (cid:15) (cid:27) (cid:27) f ,x (cid:23) (cid:23) f ,x (cid:23) (cid:23) D (cid:13) (cid:13) , , w w D D x x / / (cid:25) (cid:25) φ (cid:26) (cid:26) (cid:20) (cid:20) } } D (cid:127) (cid:127) α (cid:12) (cid:12) D r r (cid:13) (cid:13) γ Y Y β k k in which the solid arrows appeared already in (6), and the dotted arrows are constructedas follows. Form D by pulling back f and C ′ → A ′ , then D → C is the composite forthe triangle ( D , C ′ , C ). The form D , D and D by taking the distributivity pullbackof D → A along p . Form D by pulling back f and C ′ → B ′ . The construction of the5rest of the data proceeds in the same way as for (9) as shown in A A ′ B ′ BD D D C ′ C ′ C ′ D D / / (cid:15) (cid:15) / / (cid:15) (cid:15) ) ) ❘❘❘❘ ❧❧❧❧ (cid:15) (cid:15) / / ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ (cid:15) (cid:15) , , ❨❨❨❨❨❨❨❨❨❨❨ / / / / u u ❧❧❧❧ u u ❧❧❧❧ (cid:15) (cid:15) i i ❘❘❘❘❘❘❘❘❘❘❘❘ / / γ " " β < < pbpbpbdpb dpb Thus the arrow labelled as β is by Lemma 9 the right Beck-Chevalley isomorphism, and γ is also invertible. Clearly φ is an isomorphism witnessing the pseudo-functoriality of∆ ( − ) , and considering (7) for the square B YYB ′ p / / Y (cid:15) (cid:15) / / q (cid:15) (cid:15) f the arrow labelled α is evidently the appropriate component of the left Beck-Chevalleycell by Lemma 3.2.2. Thus ( P E ) X,Y ( f ) x is by definition the composite C D D C ′ / / β − / / α / / and to finish the proof we must show that this composite is f ,x . Provisionally let usdenote by ξ this composite, and by ζ the composite C D D C ′ . / / γ − / / / / Observe that the squares C C C ′ C ′ o o ζ (cid:15) (cid:15) o o (cid:15) (cid:15) C C C ′ C ′ / / ξ (cid:15) (cid:15) / / (cid:15) (cid:15) ζ C BB ′ C ′ / / f (cid:15) (cid:15) / / (cid:15) (cid:15) ξ are commutative, and so by the uniqueness aspect of the universal property of the bottomdistributivity pullback, it follows that ζ = f ,x and ξ = f ,x .The importance of this alternative description is that it can, in various ways, be writtenin terms of composition in the bicategory Poly E whose composition and coherence weunderstand. These ways are described in the following result, which follows immediatelyfrom Lemma 3.1.11 and Lemma 3.2.4.6 Lemma. Let p and q : X → Y be polynomials in E and f : p → q be a cartesianmorphism between them. Then given x : C → X in E /X , one has P E ( f ) x = f ,x = r Z,Y ( p ◦ x • ◦ g • ) for all Z and g : C → Z . The fact that the one and 2-cell maps of P E have, by Lemmas 3.2.1 and 3.2.5, beendescribed in terms of the bicategory structure of Poly E , is the reason why they give ahomomorphism of bicategories. We expand on this further in the proof of Theorem. Let E be a category with pullbacks. With the object map X 7→ E /X ,arrow map p Σ p Π p ∆ p , and 2-cell map depicted in (5), one has a homomorphism P E : Poly E −→ CAT of bicategories. Proof. It remains to exhibit the coherence isomorphisms and verify the coherence axioms.We assume a canonical choice of all pullbacks and existing distributivity pullbacks asexplained at the end of Section 2.2. In particular this implies that identities in Poly E are strict, making P E (1 X ) = 1 E /X for all X ∈ E by Lemma 3.2.1. Let p : X → Y and q : Y → Z be polynomials. For x : C → X in E /X one has the associativity isomorphism α − q,p,x • : q ◦ ( p ◦ x • ) ∼ = ( q ◦ p ) ◦ x • and so the component of the coherence isomorphism π q,p,x : P E ( q ) P E ( p )( x ) ∼ = P E ( q ◦ p )( x )is defined to be l C,Z ( α − q,p,x • ). Naturality in q , p and x is clear by definition. Note that byLemma 3.1.11 there are many other descriptions of this same component, namely π q,p,x = r D,Z ( α − q,p,x • ◦ g • )for any D and g : C → D . Using this and Lemmas 3.2.1 and 3.2.5, one can exhibitany component of any bicategorical homomorphism coherence diagram, as the image ofa diagram of coherence isomorphisms in Poly E , by a right projection of one of Poly E ’shoms. By Theorem 3.1.10 all such diagrams commute. Definition. A polynomial functor over E is a functor which is isomorphic to a com-posite of functors of the form Σ f , ∆ g and Π h , where f and g can be arbitrary morphismsof E , and h can be an exponentiable morphism of E .It follows from Theorem 3.2.6 that a functor between slices of E is polynomial if andonly if it is in the essential image of P E .7 Definition. Let X ∈ E . A monad T on E /X is a polynomial monad if it isisomorphic to a monad of the form P E ( p ), where p is a monad on X in Poly E . Amorphism φ : S → T of monads on E /X is a polynomial monad morphism if φ factors as S P E ( q ) P E ( p ) T ι / / P E ( φ ′ ) / / ι / / where ι and ι are isomorphisms of monads, and φ ′ : q → p is a morphism of monads on X in Poly E .We conclude this section by observing that the hom functors of P E are faithful andconservative. Proposition. For any category E with pullbacks and objects X, Y ∈ E , the homfunctor ( P E ) X,Y is faithful and conservative. Proof. Considering the instance of (6) in which x = 1 X , it is clear that f , X = f ,and so by Lemma 3.2.4 ( P E ) X,Y ( f ) uniquely determines f . Let us now consider the case x = q . In that case C = A ′ and the morphisms C ′ → C and C ′ → A ′ , namely theprojections of the pullback defining C ′ , have a common section s : A ′ → C ′ . ApplyingLemma 2.2.6 to the bottom distributivity pullback, one obtains the sections s : A ′ → C ′ and s : B ′ → C ′ satisfying the naturality conditions of that lemma. Since the square( C , B, B ′ , C ′ ) is a pullback, one induces unique s : B → C which is a section of thegiven map C → B and satisfies f ,x s = s f . Applying Lemma 2.2.6, this time to thetop distributivity pullback, one induces the natural sections s : A → C and s : A → C .From the naturality conditions of the sections so constructed and the commutativities inthe original general diagram (6), it follows easily that f is equal to the composite A C C = A ′ , s / / / / which by construction and Lemma 3.2.4, is determined uniquely by ( P E ) X,Y ( f ), and so( P E ) X,Y is faithful. If ( P E ) X,Y ( f ) is invertible, then f , which we saw is the componentat 1 X of this natural transformation, must also be invertible. Since f is a pullback along q of f , f is also invertible, and so ( P E ) X,Y is conservative.Example 2.10 of [11] shows that ( P E ) X,Y is not full in general, and this is discussedfurther in Remark 3.3.4. CAT pb . Recall from [6] that the category CAT pb of cate-gories with pullbacks and pullback preserving functors is cartesian closed. The product in CAT pb is as in CAT , and the internal hom [ X, Y ] is the category of pullback preservingfunctors X → Y and cartesian transformations between them. A CAT pb -bicategory is abicategory B whose homs have pullbacks and whose compositionscomp X,Y,Z : B ( Y, Z ) × B ( X, Y ) → B ( X, Z )8preserve them. The basic example is CAT pb itself. A homomorphism F : B → C of CAT pb -bicategories is a homomorphism of their underlying bicategories whose homfunctors preserve pullbacks. The point of this section is to show that for any category E with pullbacks, the homomorphism P E is in fact a homomorphism of CAT pb -bicategories.For all f : A → B in a category E with pullbacks, it is easy to witness directly thatthe adjunction Σ f ⊣ ∆ f lives in CAT pb . So polynomial functors preserve pullbacks, andthe diagram (5) may be regarded as living in CAT pb . In other words, P E sends 2-cellsin Poly E to cartesian transformations. Thus the hom maps of P E may be regarded aslanding in the homs of CAT pb , that is, one can write( P E ) X,Y : Poly E ( X, Y ) → CAT pb ( E /X, E /Y ) . In fact these functors themselves live in CAT pb . To see this we first we note that Lemma. For any category E with pullbacks and objects X, Y ∈ E , the category Poly E ( X, Y ) has pullbacks, and a commutative square in Poly E ( X, Y ) is a pullback ifand only if its -component is a pullback in E . Proof. One has a canonical inclusion Poly E ( X, Y ) −→ E ←→→ of Poly E ( X, Y ) into the functor category. In general, given a category C , an arrow α in C , and a square P BCA q / / g (cid:15) (cid:15) / / f (cid:15) (cid:15) p in [ C , E ], then if the naturality squares of f and g at α are pullbacks, then so are those for p and q , by the elementary properties of pullback squares. Since exponentiable maps arepullback stable, and one may choose pullbacks in E so that identity arrows are pullbackstable, it follows that pullbacks in Poly E ( X, Y ) exist and are formed as in E ←→→ . Thusit follows in particular that a commutative square in Poly E ( X, Y ) as in the statementis a pullback if and only if its 0 and 1-components are pullbacks in E . But from theelementary properties of pullbacks, if the 1-component square is a pullback then so is the0-component.and so one has Proposition. For any category E with pullbacks and objects X, Y ∈ E , the functor ( P E ) X,Y preserves and reflects pullbacks. Proof. Since by Proposition 3.2.9 ( P E ) X,Y is conservative, it suffices to show that it pre-serves pullbacks. But by the elementary properties of pullbacks, a square in CAT pb ( E /X, E /Y )is a pullback if and only if its component at 1 X is a pullback. Since by Lemma 3.2.4 thecomponent at 1 X of ( P E ) X,Y ( f ) is just f , the result follows from Lemma 3.3.1.9 Theorem. Let E be a category with pullbacks. Then Poly E is a CAT pb -bicategoryand P E : Poly E −→ CAT pb is a homomorphism of CAT pb -bicategories. Proof. By Proposition 3.3.2 it suffices to show that the composition functors of Poly E preserve pullbacks. One has for each X, Y, Z ∈ E , an isomorphism Poly E ( Y, Z ) × Poly E ( X, Y ) Poly E ( X, Z ) CAT pb ( E /X, E /Z ) CAT pb ( E /Y, E /Z ) × CAT pb ( E /X, E /Y ) ◦ / / ( P E ) X,Z (cid:15) (cid:15) / / ◦ (cid:15) (cid:15) ( P E ) Y,Z × ( P E ) X,Y ∼ =The vertical functors preserve and reflect pullbacks by Proposition 3.3.2, the bottom onepreserves pullbacks since CAT pb is a CAT pb -bicategory by cartesian closedness, and sothe composition functor for Poly E preserves pullbacks as required. Remark. In the case where E is locally cartesian closed, the work of Gambinoand Kock [11] tells us more. In that case E is in particular a monoidal category via itscartesian product, and it acts as a monoidal category on its slices. Moreover polynomialfunctors over such E acquire a canonical strength. Then by Proposition 2.9 of [11], theimage of P E consists of the slices of E , polynomial functors over E and strong cartesiantransformations between them. Remark. Since for any category E with pullbacks the homs of Poly E also havepullbacks, the above result can be applied to any of those homs in place of E , givinga sense in which the theory of polynomials may be iterated. Such iteration is implicitin unpublished work of Martin Hyland on fibrations, type theory and the Dialecticainterpretation. Building on this, the thesis [31] of von Glehn provides polynomial modelsof type theory, and shows the possibility of a more sophisticated kind of iteration.We conclude this section by describing the sense in which the hom functors of P E arefibrations. First we require some preliminary definitions. Given a functor F : A → B , amorphism f : X → Y of A is F -cartesian when for all g : Z → X and h : F Z → F X suchthat F ( g ) h = F f , there exists a unique k : Z → X such that F k = h and f k = g . When F is a fibration in the bicategorical sense of Street [29], we say that it is a bi-fibration .This property on F can be formulated in elementary terms as follows: every f : B → F A factors as B F C F B g / / F h / / where h is F -cartesian and g is an isomorphism. Proposition. Suppose that E is a category with finite limits, and let X and Y ∈ E .Then ( P E ) X,Y : Poly E ( X, Y ) −→ CAT pb ( E /X, E /Y ) is a bi-fibration, and every morphism of Poly E ( X, Y ) is ( P E ) X,Y -cartesian. Proof. We begin by verifying that ( φ , φ ) : ( p , p , p ) → ( q , q , q ) as in X E B YB E w w p ♦♦♦♦♦♦♦♦♦ p / / p ' ' ❖❖❖❖❖❖❖❖❖ q ♦♦♦♦♦♦♦♦♦ / / q q g g ❖❖❖❖❖❖❖❖❖ φ (cid:15) (cid:15) φ (cid:15) (cid:15) pb is ( P E ) X,Y -cartesian. Denoting by 1 the terminal object of E , note that P ( φ , φ ) = φ . Given ( ψ , ψ ) : ( r , r , r ) → ( q , q , q ), and a cartesian natural transformation α : P E ( r , r , r ) → P E ( q , q , q ) such that P E ( φ , φ ) α = P E ( ψ , ψ ), we must exhibit( β , β ) : ( r , r , r ) → ( p , p , p ) unique such that P E ( β , β ) = α and ( φ , φ )( β , β ) =( ψ , ψ ) in Poly E . But the first of these equations forces β = α , and the second equationforces β to be induced as in X E B YB E w w ♦♦♦♦♦♦♦♦♦ / / ❖❖❖ ' ' ❖❖❖❖❖❖ q ♦♦♦♦♦♦♦♦♦ / / q q g g ❖❖❖❖❖❖❖❖❖ φ (cid:15) (cid:15) φ (cid:15) (cid:15) pb E B (cid:6) (cid:6) r r / / r (cid:21) (cid:21) ✱✱✱✱✱✱✱✱✱✱✱✱ β (cid:4) (cid:4) ψ (cid:5) (cid:5) α (cid:4) (cid:4) ✡✡✡✡✡✡ ψ (cid:5) (cid:5) The statement that ( P E ) X,Y is a bi-fibration is a reformulation of the fact that if P : E /X → E /Y is a polynomial functor and φ : Q → P is a cartesian transformation, then Q is also polynomial. This appeared as Lemma 2.10 of [11], and the proof given thereworks at the present generality. Corollary. Let E be a category with finite limits, X ∈ E , P and Q be monads on E /X , and φ : Q → P be a monad morphism. If P is a polynomial monad and φ is acartesian monad morphism, then Q is a polynomial monad and φ is a polynomial monadmorphism. Proof. It suffices to show that if p is a monad on X in Poly E and φ : Qto P E ( p ) is acartesian monad morphism, then φ factors as Q P E ( q ) P E ( p ) ι / / P E ( ψ ) / / where ι is an isomorphism of monads on E /X and ψ : q → p is a morphism of monads on X in Poly E . At the level of endomorphisms this follows from Proposition 3.3.6, so it sufficesto exhibit a monad structure on q making ι and ψ into monad morphisms. The unit η q : 1 → q is defined to be the unique 2-cell such that ψη = η p and P E ( η q ) = η P E ( q ) = ιη Q ,by the ( P E ) X,X -cartesianness of ψ . The multiplication µ q : q ◦ q → q is defined similarly1as the unique 2-cell such that ψµ q = µ p ( ψ ◦ ψ ) and P E ( µ q ) = ιµ Q ( ι ◦ ι ) − . The monadaxioms for ( q, η q , µ q ) are deduced from those of p and the uniqueness aspects of ψ ’s( P E ) X,X -cartesianness, and with respect to this structure, ι and ψ are monad morphismsessentially by definition. 4. Polynomials in 2-categories We now develop the 2-categorical aspects of the theory of polynomials. In Section 4.1 wedirectly generalise Section 3 to the setting of 2-categories. One sense in which the studyof polynomial 2-functors is richer than its 1-dimensional counterpart, is that one canconsider whether such 2-functors are compatible with the theory of fibrations internal tothe 2-category in which the corresponding polynomial lives. We review briefly the theoryof fibrations in a 2-category in Section 4.2, and in Section 4.3 recall from [34] the gen-eral theory of familial 2-functors, which are those 2-functors (not necessarily polynomial)which are compatible with the 2-categorical theory of fibrations. Then in Section 4.4 wegive conditions on polynomials and morphisms thereof, so that the resulting polynomial2-functors and morphisms thereof are familial. Since 2-dimensional monad theory [5] ismost usefully applied to sifted colimit preserving 2-monads as explained at the begin-ning of Section 4.5, we give conditions on a polynomial in Cat so that ensure that itscorresponding polynomial 2-functor preserves sifted colimits in Theorem 4.5.1. In this section we extend the developments of Section 3to the setting of 2-categories. Let K be a 2-category with pullbacks. Recall that when onespeaks of pullbacks, or more generally any weighted limit in a 2-category, the universalproperty has a 2-dimensional aspect. That is, a square S in K is by definition a pullbackin K if and only if for all X ∈ K , the square K ( X, S ) in CAT is a pullback in CAT . Onobjects this is the usual universal property of a pullback as in ordinary category theory,and on arrows this is the “2-dimensional aspect”. Recall also [17] that if K admits tensorswith [1], then the usual universal property implies this 2-dimensional aspect, but in theabsence of tensors, one must verify the 2-dimensional aspect separately.Similarly when we speak of distributivity pullbacks in K we will also demand that thesesatisfy a 2-dimensional universal property. Let g : Z → A and f : A → B be in K . Wedescribe first the 2-category PB( f, g ) of pullbacks around ( f, g ). The underlying categoryof PB( f, g ) is described as in Definition 2.2.1. Let ( s, t ) and ( s ′ , t ′ ) : ( p, q, r ) → ( p ′ , q ′ , r ′ )be morphisms in PB( f, g ). Then a 2-cell between them consists of 2-cells σ : s → s ′ and τ : t → t ′ of K , such that p ′ σ = 1 p , qσ = τ q ′ and 1 r = r ′ σ . Compositions for PB( f, g )are inherited from K . One thus defines a distributivity pullback around ( f, g ) in K to bea terminal object of the 2-category PB( f, g ).The meaning of distributivity pullbacks in this 2-categorical environment is the sameas in the discussion of Section 2.2. First note that Σ f : K /A → K /B is a 2-functor, andthat by virtue of the 2-dimensional universal property of pullbacks in K , ∆ f : K /B → K /A is a 2-functor and Σ f ⊣ ∆ f is a 2-adjunction. To say that all distributivity pullbacks along2 f exist in K is to say that ∆ f has a right 2-adjoint, denoted Π f as before, and this rightadjoint encodes the process of taking distributivity pullbacks along f . Such morphisms f in K are said to be exponentiable, and as in the 1-dimensional case, exponentiablemaps are closed under composition and are stable by pullback along arbitrary maps.Morever Lemmas 2.2.4, 2.2.5 and 2.2.6 remain valid in our 2-categorical environment.The verification of this is just a matter of using the 2-dimensional aspects of pullbacksand distributivity pullbacks to induce the necessary 2-cells, in exact imitation of how oneinduced the arrows during these proofs in Section 2.2.Polynomials in K and cartesian morphisms between them are defined as in Section3.1. Given polynomials p and q : X → Y , and cartesian morphisms f and g : p → q , a2-cell φ : p → q consists of 2-cells φ : f → g and φ : f → g such that p = q φ , q φ = φ p and q φ = p . With compositions inherited from K one has a 2-category Poly K ( X, Y ) together with left and right projections K /X Poly K ( X, Y ) K /Y. o o l X,Y r X,Y / / For a composable sequence ( p i ) i of polynomials as in X i − A i B i X i o o p i p i / / p i / / one defines the 2-category SdC( p i ) i of subdivided composites over ( p i ) i as follows. Theobjects and arrows are defined as in Definitions 3.1.1 and 3.1.2. Given morphisms t and t ′ : ( Y, q, r, s ) → ( Y ′ , q ′ , r ′ , s ′ ) of subdivided composites, a 2-cell τ : t → t ′ consists of 2-cells τ i : t i → t ′ i in K for 0 ≤ i ≤ n , such that q = q ′ τ , q ′ i τ i = τ i q i , q = q ′ τ n , r i = r ′ i τ i and s i = s ′ i τ i . Compositions in SdC( p i ) i are inherited from K . The process of taking theassociated polynomial of a subdivided composite, as described in Definition 3.1.3, is 2-functorial. The forgetful functors res n and res become 2-functors. The fact that Lemmas3.1.4 and 3.1.5 remain valid in our 2-categorical environment, is once again a matter ofusing the 2-dimensional aspects of pullbacks and distributivity pullbacks to induce thenecessary 2-cells in the same way that the arrows during these proofs were induced inthe 1-dimensional case. Thus these 2-categories of subdivided composites admit terminalobjects, and so one may define the composition of polynomials as in Definition 3.1.7.Moreover composition is 2-functorial.Lemma 3.2.4 gives a direct description of the arrow map of the hom functors of P E asbeing induced by the universal properties of pullbacks and distributivity pullbacks. Thusin our 2-categorical setting, with the 2-dimensional aspects of these universal propertiesavailable, we can do the same one dimension higher and induce directly the componentsof the modification induced by a 2-cell between maps of polynomials. Thus we have2-functors ( P K ) X,Y : Poly K ( X, Y ) → ( K /X, K /Y )for all objects X and Y of a 2-category K with pullbacks. We now describe the structurethat polynomials in a 2-category form.3 Definition. A consists of a bicategory B whose hom categories areendowed with 2-cells making them 2-categories and the composition functorscomp X,Y,Z : B ( Y, Z ) × B ( X, Y ) → B ( X, Z )are endowed with 2-cell maps making them into 2-functors. In addition we ask that thecoherence isomorphisms of B be natural with respect to the 3-cells.Since a 2-bicategory B is a degenerate sort of tricategory, we shall call the 2-cells inits homs of B . Examples. 1. Given a 2-category K with pullbacks, the bicategory Span K has the additionalstructure of a 2-bicategory in which a 3-cell f → g consists of a 2-cell φ as in X A YB u u s ❧❧❧❧❧❧❧❧❧❧❧❧❧ t ) ) ❘❘❘❘❘❘❘❘❘❘❘❘❘ t ❧❧❧❧❧❧❧❧❧❧❧❧❧ s i i ❘❘❘❘❘❘❘❘❘❘❘❘❘ f (cid:25) (cid:25) g (cid:5) (cid:5) φ + such that s φ = id and t φ = id.2. Dually one has a 2-bicategory Cospan K of cospans in any 2-category K withpushouts.3. Any strict 3-category such as is a 2-bicategory. Definition. A homomorphism F : B → C of 2-bicategories is a homomorphism oftheir underlying bicategories whose hom functors are endowed with 2-cell maps makingthem into 2-functors, and whose coherence data is natural with respect to 3-cells. Theorem. Let K be a 2-category with pullbacks. One has a 2-bicategory Poly K whose objects are those of K , whose hom between X and Y ∈ K is Poly K ( X, Y ) , and whosecompositions are defined as above. Moreover with object, arrow and 2-cell maps definedas in the categorical case, and 3-cell map as defined above, one has a homomorphism P K : Poly K −→ of 2-bicategories. Proof. By the way that things have been set up, Lemma 3.1.8 and Theorem 3.1.10lift to our 2-categorical setting, with the extra naturality of the coherences coming fromthe 2-dimensional aspect of all the universal properties being used. Thus Poly K is a2-bicategory. Since in the proof of Theorem 3.2.6 the coherences of P E were obtainedfrom associativity coherences in Poly E , the extra naturality enjoyed by the associativitiesin Poly K gives the extra naturality required for the coherences of P K . Thus P K is ahomomorphism of 2-bicategories.4The notions polynomial 2-functor , polynomial 2-monad and morphism of polynomial2-monads are defined as in Definitions 3.2.7 and 3.2.8. Similarly one may speak of poly-nomial pseudo monads when the coherences are themselves also in the essential image of P K . Example. Suppose that a morphism U : E → B in a 2-category K with finitelimits is both exponentiable and a classifying discrete opfibration in the sense of [35].One can then define a discrete opfibration in K to be U -small when it arises by pullingback U . By definition U -small discrete opfibrations are pullback stable in K . Often theyare also closed under composition, and when this is the case, one can consider the fullsub-2-bicategory S U of Poly K consisting of those polynomials I A C J o o s p / / t / / such that p is a U -small discrete opfibration. The condition of being a classifying discreteopfibration then implies that the polynomial1 E B o o U / / / / (10)is a biterminal object in S U (1 , K = Cat and U is the forgetful functor Set f , • → Set f from the category of finitepointed sets to that of finite sets, a U -small discrete opfibration is one with finite fibres, andthese are evidently closed under composition. By the finitary analogue of [34] Corollary5.12, the endofunctor associated to (10) is the finite “Fam” construction, which associatesto a category its finite coproduct completion. Replacing U by U op gives a polynomialpseudo monad on Cat whose underlying endofunctor gives the finite product completionof a category. To summarise, the finite coproduct and finite product completion pseudomonads on Cat are polynomial pseudo monads.The 2-categorical analogues of Proposition 3.3.6 and Corollary 3.3.7 are also valid,with proofs adapted from the 1-dimensional case in the same way as above. We recordthese results as follows. Proposition. Let K be a 2-category with finite limits and let X and Y ∈ K . Let P be a polynomial 2-functor K /X → K /Y , and T be a polynomial 2-monad on K /X . Let φ : Q → P be a 2-natural transformation, and ψ : S → T be a morphism of 2-monads.1. If φ is cartesian then Q is a polynomial 2-functor.2. If ψ is cartesian then S is a polynomial 2-monad and ψ is a morphism of polynomial2-monads. The 2-monads whose algebras are fibrations and opfibra-tions to be recalled here, play two roles in this work: (1) as part of the background to thediscussion of familial 2-functors below in Section 4.3, and (2) as examples of polynomial52-monads in Proposition 4.2.3. Fibrations internal to a 2-category were introduced byStreet in [27].Elementary descriptions of fibrations were given in Section 2 of [35] in terms of carte-sian 2-cells. A similar 2-categorical reformulation of split fibrations, in which cleavageswere expressed 2-categorically via the notion of chosen cartesian 2-cell, was given in Sec-tion 3 of [34]. Thus for any 2-category K , one can define fibrations and split fibrations in K . Moreover, given fibrations f : A → B and g : C → D in K , a morphism of fibrations( u, v ) : f → g is a commutative square A CDB u / / g (cid:15) (cid:15) / / v (cid:15) (cid:15) f such that post-composition with u sends f -cartesian 2-cells to g -cartesian 2-cells. When f and g are split fibrations, ( u, v ) is a strict morphism when post-composing with u preserves chosen cartesian 2-cells.Thus one has 2-categories Fib( K ) and SFib( K ) of fibrations, morphisms thereof and2-cells; and of split fibrations and strict morphisms respectively, coming with forgetful2-functors into K [1] . Dually an opfibration in a general 2-category K is a fibration in the2-category K co obtained by reversing 2-cells. We freely use the associated dual notions,such as (chosen) f -opcartesian 2-cells below, and we defineOpFib( K ) = Fib( K co ) co SOpFib( K ) = SFib( K co ) co . When K has comma objects, Street [27] observed that fibrations can be regarded aspseudo algebras for certain easy to define 2-monads. The comma squares1 B ↓ f ABB q f / / f (cid:15) (cid:15) / / B (cid:15) (cid:15) Φ K ( f ) λ f + f ↓ B BBA Ψ K ( f ) / / B (cid:15) (cid:15) / / f (cid:15) (cid:15) p f λ ′ f + describe the effect on objects of the underlying endofunctors of 2-monads Φ K and Ψ K on K [1] . For f : A → B the component of the unit of Φ K is of the form ( η f , B ) where η f isunique such that Φ K ( f ) η f = f q f η f = 1 A λ f η f = idand the f -component of the multiplication is of the form ( µ f , B ) where µ f is unique suchthat Φ K ( f ) µ f = Φ K ( f ) q f µ f = q f q Φ K ( f ) λ f µ f = ( λ f q Φ K ( f ) ) λ Φ K ( f ) . The unit and multiplication of Ψ K are described dually.6We follow the established notation of 2-dimensional monad theory by denoting, for a2-monad T on a 2-category K , T -Alg s , Ps- T -Alg and Kl( T ) the 2-categories of strict T -algebras and strict maps, pseudo T -algebras and strong maps and the Kleisli 2-categoryof T respectively. Moreover one has U T : T -Alg s −→ K F T : K −→ Kl( T )the right adjoint part of the Eilenberg-Moore adjunction for T , and the left adjoint part ofthe Kleisli adjunction for T respectively. We proceed now to exhibit explicit descriptionsof these for the 2-monads Φ K and Ψ K .Denote by K [1]colax the 2-category of functors [1] → K , colax natural transformationsbetween them, and modifications; and by K [1]lax the 2-category of functors [1] → K , laxnatural transformations between them, and modifications. Thus a morphism f → g of K [1]colax is the unlabelled data in A CDB / / g (cid:15) (cid:15) / / (cid:15) (cid:15) f + and for K [1]lax the 2-cell is in the opposite direction. Since strict natural transformationsare degenerate instances of lax and colax ones, one has canonical inclusions K [1] ֒ → K [1]colax and K [1] ֒ → K [1]lax . The following result is known, though perhaps formulated slightly moreglobally than usual. Proposition. [20, 27] Let K be a 2-category with comma objects.1. One has isomorphisms Ps-Φ K -Alg ∼ = Fib( K ) Φ K -Alg s ∼ = SFib( K )Ps-Ψ K -Alg ∼ = OpFib( K ) Ψ K -Alg s ∼ = SOpFib( K ) commuting with the evident 2-functors into K [1] .2. One has isomorphisms Kl(Φ K ) ∼ = K [1]colax Kl(Ψ K ) ∼ = K [1]lax commuting with the evident 2-functors out of K [1] . Proof. In the case K = Cat of (1) is completely standard. See [20] for a recent discussion.The result for general K follows by a representable argument since the 2-monads Φ K andΨ K are described in terms of limits, and the notions of cartesian and opcartesian 2-cell arerepresentable. The isomorphisms (2) are easily exhibited by using the universal propertyof comma objects and the definition of Kl(Φ K ) and Kl(Ψ K ). Meaning that the coherence data consists of invertible 2-cells K [1] → K over B ∈ K is exactly the slice 2-category K /B , and these 2-monads restrict to the 2-monads Φ K ,B and Ψ K ,B on K /B defined originally by Street in [27]. Using Proposition 4.2.1 one then has an explicitdescription of the algebras of Φ K ,B (resp. Ψ K ,B ) as fibrations (resp. opfibrations) withcodomain B . Similarly 1-cells x → y of Kl(Φ K ,B ) (resp. Kl(Ψ K ,B )) may be identified withlax triangles X YB / / y (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧ (cid:31) (cid:31) x ❄❄❄❄❄❄❄ + X YB. / / y (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧ (cid:31) (cid:31) x ❄❄❄❄❄❄❄ k s For any nice symmetric monoidal category V over which one may wish to enrich, onehas a notion of monoidal V -category, and so in particular taking V = Cat (with thecartesian tensor product), one has a canonical notion of monoidal 2-category . In thissense, for any 2-bicategory B and object B therein, the hom B ( B, B ) is a monoidal 2-category. Following [18, 21] we define a pseudo monoid in a monoidal 2-category K , withunit and multiplication denoted u : I → M and m : M ⊗ M → M , to be colax idempotent (resp. lax idempotent ) when u ⊗ M ⊣ m ⊣ M ⊗ u (resp. M ⊗ u ⊣ m ⊣ u ⊗ M ). Definition. Let B be a 2-bicategory and B ∈ B . Then a pseudo monad (resp. ) on B in B is a pseudo monoid (resp. monoid) in the monoidal 2-category B ( B, B ). A pseudo monad on B in B is colax idempotent (resp. lax idempotent ) when itscorresponding pseudo monoid in B ( B, B ) is so.Writing [ n ] for the ordinal { < ... < n } one has a cospan in Cat as on the left[0] [1] [0] δ / / o o δ B B [1] B o o d d / / B [1] BBB d / / B (cid:15) (cid:15) / / B (cid:15) (cid:15) d + and cotensoring this with B gives the span in K in the middle, where Z A denotes thecotensor of Z ∈ K with the category A . This span fits into comma square as depicted onthe right in the previous display. To some extent the following result is implicit in the workof Street [27, 28, 29], and all that we have done is to observe that Street’s approach to thefibrations 2-monads exhibits them as polynomial 2-monads. In fact, since the underlyingendofunctors come from spans, they are examples of “linear” polynomial 2-functors. Proposition. Let K be a 2-category with comma objects and pullbacks and let B ∈ K . Monoidal bicategories in the most general sense whose underlying bicategory is a 2-category areweaker than this. Φ K ,B is the result of applying P K to a colax idempotent 2-monad in Poly K whoseunderlying endomorphism is B B [1] B [1] B. o o d / / d / / Ψ K ,B is the result of applying P K to a lax idempotent 2-monad in Poly K whoseunderlying endomorphism is B B [1] B [1] B. o o d / / d / / Proof. The two statements are dual, so we consider just the monad Φ K ,B . By theelementary properties of comma squares and pullbacks, one can factor the defining commasquare of Φ K ,B ( f ) = Φ K ( f ) as 1 K ,B ↓ f B [1] BBBA / / d / / K ,B (cid:15) (cid:15) / / K ,B / / f (cid:15) (cid:15) d (cid:15) (cid:15) Φ K ,B ( f ) pb k s and so one has Φ K ,B ( f ) = Σ d ∆ d ( f ). This expresses on objects that Φ K ,B is the resultof applying P K to the endopolynomial of the statement.Recall that the inclusion ∆ ֒ → Cat is a cocategory object, and as described in [24, 29],the standard presentation of (topologists’) ∆ as a subcategory of Cat has the 2-categoricalfeature that the successive generating coface and codegeneracy maps are adjoint. Thusthe diagram [0] [1] [2] [3] ...... [0][0] σ o o σ o o / / σ o o ⊥⊥ σ o o / / o o / / σ o o ⊥⊥⊥⊥ t (cid:4) (cid:4) δ = t (cid:8) (cid:8) t (cid:15) (cid:15) t (cid:19) (cid:19) ... b Z Z δ = b V V b O O b K K ... (11)in which each functor labelled as “ t ” picks out the top element of its codomain, and eachfunctor labelled as “ b ” picks out a bottom element, is a colax idempotent 2-monad in Cospan Cat . Cotensoring it with an object B in any finitely complete 2-category K , givesa colax idempotent 2-monad in Span K and by means of the inclusion Span K ֒ → Poly K ,one has a polynomial colax idempotent 2-monad on B . The observation that associated2-monad on K /B , obtained via application of P K , is the 2-monad Φ K ,B is easily verified,and was implicit in Section 2 of [29].9 Remark. In general d is a split fibration and d is a split opfibration. In 2-categories K , such as the case K = Cat , in which fibrations and opfibrations are expo-nentiable, it follows that Φ K ,B and Ψ K ,B are themselves left adjoints. For such situationsfibrations and opfibrations are thus also coalgebras for 2-comonads (ie those obtainedfrom Φ K ,B and Ψ K ,B by taking right adjoints), and the forgetful 2-functors U Φ K ,B and U Ψ K ,B create all colimits. We now recall, and to some extent update, the theory offamilial 2-functors from [34]. Intuitvely, a familial 2-functor is one that is compatible inan appropriate sense with the theory of fibrations recalled in the previous section. InSection 4.4 we will identify conditions on polynomials in a 2-category which ensure thatthe corresponding 2-functor is familial.The compatibility of familial 2-functors with the theory of fibrations is expressedformally by the formal theory of monads [26], in which monads in a 2-category A wereorganised in various useful ways into 2-categories. Since this material is used so extensivelyin this section, we recall it briefly now.We denote a monad in a 2-category A as a pair ( A, t ) where A ∈ A , t is the underlyingone-cell of the monad on A in A , and we denote the unit and multiplication 2-cells as η t and µ t respectively. A lax morphism f : ( A, t ) → ( B, s ) in A consists of an arrow f : A → B in A , and a “coherence” 2-cell f l : sf → f t which satisfies f l ( η s f ) = f η t and f l ( µ s f ) = ( f µ t )( f l t )( sf l ). A colax morphism f : ( A, t ) → ( B, s ) in A consists ofan arrow f : A → B in A , and f c : f t → sf satisfying f c ( f η t ) = η s f and f c ( f µ t ) =( µ s f )( sf c )( f c t ). Lax and colax morphisms of monads were called “monad functors” and“monad opfunctors” in [26] respectively.Given a monad ( A, t ) in A one may have an associated Eilenberg-Moore object, whichis a type of 2-categorical limit, whose universal 0 and 1-cell data is denoted as A t and u t : A t → A respectively. The corresponding colimit notion is that of a Kleisli object,the universal 0 and 1-cell data of which is denoted A t and f t : A → A t respectively. See[26] for the precise definitions. When A = Cat , A t is the category of algebras and A t isthe Kleisli category, and in the general situation u t has a left adjoint and f t has a rightadjoint. When A is the 2-category of 2-categories, a monad in A is a 2-monad ( K , T ),and the 0 and 1-cell data of the Eilenberg-Moore and Kleisli objects are denoted as U T : T -Alg s −→ K F T : K −→ Kl( T )respectively.In general, when the 2-category A admits Eilenberg-Moore objects, given monads( A, t ) and ( B, s ) in A , and an arrow f : A → B , then 2-cells f l : sf → f t providingcoherence data of a lax monad morphism are in bijection with liftings f to the level of0algebras as on the left in A t B s BA f / / u s (cid:15) (cid:15) / / f (cid:15) (cid:15) u t = A s B s BA f / / O O f s / / ff t O O = and dually, when A admits Kleisli objects, colax coherence data f c : f t → sf is inbijection with extensions of f to f as on the right in the previous display. Remark. Let K be a 2-category with comma objects and pullbacks and f : A → B be a morphism therein. The 2-functor Σ f : K /A → K /B clearly extends to laxtriangles, and so one has Σ f : Kl(Φ K ,A ) → Kl(Φ K ,B ) such that Σ f F Φ K ,A = F Φ K ,B Σ f . Suchan extension is equivalent to the data of a 2-natural transformation σ f providing thecoherence datum for a colax morphism of 2-monads(Σ f , σ f ) : ( K /A, Φ K ,A ) −→ ( K /B, Φ K ,B )and so by taking mates [19] with respect to Σ f ⊣ ∆ f , also to the coherence datum δ f fora lax morphism of 2-monads(∆ f , δ f ) : ( K /B, Φ K ,B ) −→ ( K /A, Φ K ,A ) . This last is in turn equivalent to lifting ∆ f to the level of split fibrations, that is to say,to giving a 2-functor ∆ f : Φ K ,B -Alg s → Φ K ,A -Alg s such that U Φ A ∆ f = ∆ f U Φ B . The2-functor ∆ f witnesses the pullback stability of split fibrations.We recall now the familial 2-functors of [34]. For a 2-functor T : K → L and an object X ∈ K , we denote by T X : K /X → L /T X the 2-functor given on objects by applying T to morphisms into X . A local right adjoint K → L is a 2-functor T : K → L equippedwith a left adjoint to T X for all X ∈ K . Given an adjunction as on the left A B R / / L o o ⊥ A /X B /RX R X / / o o ⊥ and X ∈ A , one always has an adjunction as on the right in the previous display. Thuswhen K has a terminal object 1, to exhibit T as a local right adjoint, it suffices to give aleft adjoint to T . Definition. Let K and L be 2-categories with comma objects, and suppose that K has a terminal object 1. Then a familial 2-functor K → L consists of local right adjoint T : K → L together with T : K → Φ T -Alg s such that U Φ T T = T .Familial 2-functors are exactly those 2-functors which are compatible with fibrationsbecause of1 Proposition. Let K and L be 2-categories with comma objects, suppose that K has a terminal object and let T be a 2-functor. To give T the structure of a familial2-functor is to give the coherence data φ T for a lax morphism ( T [1] , φ T ) : ( K [1] , Φ K ) −→ ( L [1] , Φ L ) of 2-monads. Proof. It is straight forward to adapt Theorem 6.6 of [34] to provide, given T , the liftingof T [1] to a 2-functor Φ K -Alg s → Φ L -Alg s . Conversely, given such a lifting one recovers T by restricting to split fibrations in K into 1.Implicit in Definition 5.2 of [34] is the idea that morphisms of familial 2-functorsare simply cartesian 2-natural transformations. In this work we wish to adopt a morespecialised notion which is compatible with Propostion 4.3.3. Definition. Let S and T be familial 2-functors K → L . Then a 2-natural trans-formation α : S → T is familial when it is cartesian and for all X ∈ K , α ’s naturalitysquare with respect to the unique map t X : X → α X , α ) : St X → T t X . Lemma. Let α : S → T be a familial natural transformation between familial2-functors K → L , and f : A → B be a split fibration in K . Then the naturality square SA T AT BSB α A / / T f (cid:15) (cid:15) / / α B (cid:15) (cid:15) Sf is a morphism ( α A , α B ) : Sf → T f of split fibrations in L . Proof. Recall from Lemma 6.3 of [34] that a 2-cell ψ as indicated on the left in X SA x ! ! y = = ψ (cid:11) (cid:19) X SDSASC g / / Sh (cid:15) (cid:15) / / Sh (cid:15) (cid:15) g Sδ ♦♦♦♦♦ ♦♦♦♦♦ ψ + Sψ + is chosen Sf -cartesian if and only if ψ is chosen f -cartesian, where x = S ( h ) g and y = S ( h ) g are S -generic factorisations in the sense of [34], and ψ and ψ are uniquesuch that the composite on the right in the previous display equal to ψ and ψ is chosen St D -cartesian. We must show that if ψ is chosen Sf -cartesian, then α A ψ is chosen T f -cartesian. In X SDSASC g / / Sh (cid:15) (cid:15) / / Sh (cid:15) (cid:15) g Sδ ♦♦♦♦♦ ♦♦♦♦♦ ψ + Sψ + T DT AT C T h (cid:15) (cid:15) / / T h T δ ♦♦♦♦♦♦♦♦♦♦♦ T ψ + α D % % ❑❑❑❑❑❑❑❑❑ ❑❑❑❑ ❑❑ % % α C % % ❑❑❑❑❑❑❑❑❑ α D ψ is chosen T t D -cartesian since ( α D , α ) is a morphism of split fibrations St D → T t D ,and so the result follows by Lemma 6.3 of [34] applied to T .When T : K → L is familial we denote by T : Φ K -Alg s → Φ L -Alg s the lifting of T [1] corresponding to φ T . Then one obtains the following result immediately from Lemma4.3.5. Proposition. Let S and T be familial 2-functors and α : S → T be a cartesiannatural transformation between them. Then α is familial if and only if it lifts to a 2-naturaltransformation α : S → T . We freely use and apply the dual notions and results, an opfamilial 2-functor being onewhich is compatible in the same way with opfibrations. A familial (resp. opfamilial) 2-monad is one whose underlying endo-2-functor, unit and multiplication are familial (resp.familial). Remark. All the examples familial and opfamilial 2-monads exhibited in [34] turnout to be familial and opfamilial 2-monads in the more restricted sense of this article. Forthose that arise from polynomials, we shall establish this from general results below. In this section we identify conditionson polynomials in a 2-category and their morphisms, ensuring that the associated 2-functors and 2-natural transformations are familial. The central result to this effectappears in this section as Theorem 4.4.5. The task of proving this result breaks up intothat of identifying conditions ensuring that 2-functors of the form ∆ f and Π f , and the“Beck-Chevalley 2-cells” lift to the level of split fibrations. These conditions are presentedin turn in Lemmas 4.4.1-4.4.4 below. Lemma. [12] Let f : A → B be a split opfibration in a 2-category K with pullbacksand comma objects.1. One has ∆ f : Kl(Φ K ,B ) → Kl(Φ K ,A ) such that ∆ f F Φ K ,B = F Φ K ,A ∆ f .2. If f is a discrete opfibration then Σ f ⊣ ∆ f . Proof. (1): Given ( k, γ ) : x → y in Kl(Φ K ,B ), we pullback to obtain X and Y in X Y A X YB q / / ✿✿✿✿✿ x (cid:28) (cid:28) ✿✿✿✿✿ / / f (cid:28) (cid:28) ✿✿✿✿✿ ∆ f x ✿✿✿✿✿ h (cid:13) (cid:13) q / / ✖✖✖✖✖✖✖ y (cid:11) (cid:11) ✖✖✖✖✖✖✖ / / (cid:11) (cid:11) ✖✖✖✖✖✖✖ ∆ f y ✖✖✖ ✖✖✖✖ k ❧❧❧❧❧❧❧❧❧❧❧ k γ ❧❧❧❧ γ @ ②②②② γ is the chosen f -opcartesian lift of γq , and then k is unique such that ∆ f ( y ) k = h and q k = kq . One defines ∆ f ( k, γ ) = ( k , γ ). Given a 2-cell κ : ( k, γ ) → ( k ′ , γ ′ ), then3by the opcartesianness of γ one has κ : ∆ f ( y ) k → ∆ f ( y ) k ′ unique such that κ γ = γ ′ and f κ = yκq . By the 2-dimensional universal property of the pullback defining Y ,one has κ : k → k ′ unique such that ∆ f ( y ) κ = κ and q κ = κq . We then define∆ f ( κ ) = κ . The 2-functoriality of these assignations follows easily from the uniquenessof the various lifts in these constructions and the 2-dimensional universal property ofpullbacks in K .(2): Note that in the above diagram q and q are the components of the counit ofΣ f ⊣ ∆ f at x and y respectively. Thus the commutativity of the diagram ensures thatthe counit is natural with respect to lax triangles, and 2-naturality follows similarly. Itremains to verify the 2-naturality of the unit of Σ f ⊣ ∆ f with respect to lax triangleswhen f is a discrete opfibration. Denoting the defining pullback of ∆ f Σ f x as on the leftin X XBA q x / / fx (cid:15) (cid:15) / / f (cid:15) (cid:15) ∆ f Σ f x pb X YA k / / y (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧ (cid:31) (cid:31) x ❄❄❄❄❄❄❄ γ + η x : X → X is unique such that ∆ f Σ f ( x ) η x = x and q x η x = 1 X . Naturality with respectto ( k, γ ) amounts to checking that η y k = k η x and γ = γ η x in X YA k ♦♦♦♦♦♦♦ y ✔✔✔✔ (cid:10) (cid:10) ✔✔✔✔ x ●●●● γ ; ♣♣♣♣ B f (cid:15) (cid:15) A / / f X / / q x Y / / q y k ♦♦♦♦♦♦ ✙✙✙ ∆ f Σ f y ✙✙✙✙ (cid:12) (cid:12) ✙✙✙✙✙✙✙ (cid:24) (cid:24) ✵✵✵✵✵✵✵✵✵✵ γ ; ♦♦♦♦ X / / η x Y / / η y k ♦♦♦♦♦♦♦ ❂❂❂❂ ❂❂❂❂❂ (cid:30) (cid:30) ❂❂❂❂❂❂❂❂❂ ( ( x PPPPPPPPPPPPPPPPPPPPPPPP γ ; ♥♥♥♥ in which γ is unique such that f γ = f γq x . Thus f γ η x = f γ and so since f is a discreteopfibration, the uniqueness of lifts implies γ η x = γ and ∆ f Σ f ( y ) k η x = ky . From thislast one obtains the first equation of∆ f Σ f ( y ) η y k = ∆ f Σ f ( y ) k η x q y k η x = q y η y k and the second equation is also easily verified, so that by the joint monicness of (∆ f Σ f ( y ) , q y ),one has η y k = k η x . The verification of the 2-dimensional part of naturality proceedsstraight forwardly along similar lines, and so is left to the reader. Lemma. [12] Let f : A → B be an exponentiable split opfibration in a 2-category K with pullbacks and comma objects.1. One has Π f : Φ K ,A -Alg s → Φ K ,B -Alg s such that U Φ K ,B Π f = Π f U Φ K ,A .2. If f is a discrete opfibration then ∆ f ⊣ Π f . Proof. (1): By the formal theory of monads the data of the extension ∆ f of Lemma4.4.1(1) is equivalent to that of a 2-natural transformation ∂ f providing the coherencedatum for an oplax morphism(∆ f , ∂ f ) : ( K /B, Φ K ,B ) −→ ( K /A, Φ K ,A ) . Since f is exponentiable, this is in turn equivalent, by taking mates with respect to theadjunction ∆ f ⊣ Π f , to the data of a 2-natural transformation π f providing the coherencedatum of a lax morphism(Π f , π f ) : ( K /A, Φ K ,A ) −→ ( K /B, Φ K ,B ) . This last is in turn equivalent to the lifting Π f of Π f to the level of split fibrations.(2): The data of the adjunction Σ f ⊣ ∆ f established in Lemma 4.4.1(2) correspondsby the formal theory of monads to the data of an adjunction of oplax morphisms of 2-monads whose underlying 2-adjunction is Σ f ⊣ ∆ f . This means that in addition to havingthe oplax coherence data σ f and ∂ f described above, one has the compatibility of the unit η f and counit ε f of the adjunction Σ f ⊣ ∆ f with respect to these coherences, which inexplicit terms is the commutativity ofΦ K ,A ∆ f Σ f Φ K ,A ∆ f Φ K ,B Σ f Φ K ,A ∆ f Σ fη f Φ K ,A / / ∆ f σ f (cid:15) (cid:15) ∂ f Σ f o o (cid:15) (cid:15) Φ K ,A η f Σ f ∆ f Φ K ,B Σ f Φ K ,A ∆ f Φ K ,B Σ f ∆ f . Φ K ,B Σ f ∂ f / / σ f ∆ f (cid:15) (cid:15) Φ K ,B ε f o o (cid:15) (cid:15) ε f Φ K ,B In terms of string diagrams (which go from top to bottom) in the 2-category of 2-categories(see [15]), the above commutative diagrams are expressed as η f σ f ∂ f ✿✿✿✿☎☎☎☎ Φ K ,A ✒✒✒✒✒✒✒ Σ f ✱✱✱✱✱✱✱ Φ K ,A ☛☛ ∆ f ✸✸ = η f ∆ f ✕✕✕✕✕✕✕ Σ f ✮✮✮✮✮✮✮ Φ K ,A Φ K ,A ε f σ f ∂ f ✿✿✿✿☎☎☎☎ Φ K ,B ✒✒✒✒✒✒✒ Σ f ✱✱✱✱✱✱✱ Φ K ,B ☛☛ ∆ f ✸✸ = ε f ∆ f ✕✕✕✕✕✕✕ Σ f ✮✮✮✮✮✮✮ Φ K ,B Φ K ,B (12)It suffices now to verify that the lax coherence data δ f and π f are compatible with theunit η ′ f and counit ε ′ f of ∆ f ⊣ Σ f in the same way, which is to say that η ′ f π f δ f ☎☎☎☎✿✿✿✿✱✱✱✱✱✱✱✱✒✒✒✒✒✒✒✒ ✸✸✸✸☛☛☛☛ = η ′ f ✮✮✮✮✮✮✮✮✕✕✕✕✕✕✕✕ ε ′ f π f δ f ☎☎☎☎✿✿✿✿ ✱✱✱✱✱✱✱✱ ✒✒✒✒✒✒✒✒✸✸✸✸ ☛☛☛☛ = ε ′ f ✮✮✮✮✮✮✮✮ ✕✕✕✕✕✕✕✕ (13)5because then by the formal theory of monads this corresponds to giving the rest of thedata of the required adjunction ∆ f ⊣ Π f . The definition of δ f and π f as mates of σ f and ∂ f in explicit terms says that η f σ f ε f ✿✿✿✿ ✿✿✿✿✖✖✖✖✖✖✖✖✖✖✖✖✖ ✕✕✕✕✕✕✕✕✕ ✕✕✕✕✕✕✕✕✕ ✖✖✖✖✖✖✖✖✖✖✖✖✖ = δ f ✕✕✕✕✕✕✕✕✕ ✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮ ✕✕✕✕✕✕✕✕✕ η ′ f ∂ f ε ′ f ✿✿✿✿ ✿✿✿✿✖✖✖✖✖✖✖✖✖✖✖✖✖ ✕✕✕✕✕✕✕✕✕ ✕✕✕✕✕✕✕✕✕ ✖✖✖✖✖✖✖✖✖✖✖✖✖ = π f ✕✕✕✕✕✕✕✕✕ ✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮✮ ✕✕✕✕✕✕✕✕✕ (14)and so the left equation of (13) is established by the calculation η ′ π δ ☎☎☎✿✿✿✱✱✱✱✱✱✒✒✒✒✒✒ ✸✸✸☛☛☛ = η ′ ∂ ε ′ η ′ η σ ε ✿✿✿ ✿✿✿ ☎☎☎✿✿✿ ✿✿✿ ✒✒✒✒✒✒✢✢✢✢✢✢✢✢✢ ✘✘✘✘✘✘ = η ′ ∂η σ ε ▲▲▲▲▲✿✿✿ ✿✿✿ ✘✘✘✘✘✘☎☎☎✢✢✢✢✢✢✢✢✢ ✘✘✘✘✘✘ = η ′ η ε ✘✘✘✘✘✘ ✿✿✿ = η ′ ✮✮✮✮✮✕✕✕✕✕ which uses (12), (14) and the adjunction triangle equations. The right equation in (13)follows from a similar calculation.Recall that given a commutative square in K as on the left A CDB h / / g (cid:15) (cid:15) / / k (cid:15) (cid:15) f K /A K /C K /D K /B Σ h / / O O ∆ g / / Σ k ∆ f O O α + K /A K /C K /D K /B o o ∆ h Π g (cid:15) (cid:15) ∆ k o o (cid:15) (cid:15) Π f k s β (15)one has a canonical 2-natural transformation α is obtained from Σ k Σ f = Σ g Σ h and theadjunctions Σ f ⊣ ∆ f and Σ g ⊣ ∆ g . If f and g are exponentiable, then by taking rightadjoints one has a canonical 2-natural transformation β as on the right. Lemma. If in a 2-category K with comma objects and pullbacks, the left-mostsquare in (15) underlies a morphism of split opfibrations f → g , then α extends to a2-natural transformation α as on the left Kl(Φ K ,A ) Kl(Φ K ,C )Kl(Φ K ,D )Kl(Φ K ,B ) Σ h / / O O ∆ g / / Σ k ∆ f O O α + Φ K ,A -Alg s Φ K ,C -Alg s Φ K ,D -Alg s Φ K ,B -Alg s o o ∆ h Π g (cid:15) (cid:15) ∆ k o o (cid:15) (cid:15) Π f k s β and if f and g are exponentiable, then β lifts to a 2-natural transformation β as on theright. Proof. By the formal theory of monads α extends to α if and only if α underlies a monad2-cell, that is to say, satisfies axioms of compatibility with the colax monad morphismstructures on Σ h ∆ f and ∆ g Σ k . When f and g are exponentiable this in turn is equivalent,by the calculus of mates, to asking that β satisfy compatibility with respect to the laxmonad morphism structures on Π f ∆ h and ∆ k Π g , which is moreover equivalent to theexistence of the lifting β . Thus it suffices to exhibit α , in other words, that α is 2-naturalwith respect to lax triangles.For the 1-dimensional part of this naturality note that the component of α at x : X → B is as shown on the right X XBD.CA x / / x (cid:15) (cid:15) k (cid:15) (cid:15) / / g (cid:15) (cid:15) h (cid:15) (cid:15) x f / / pb X XBDCA X x / / x (cid:15) (cid:15) k (cid:15) (cid:15) / / g (cid:15) (cid:15) h (cid:15) (cid:15) x (cid:30) (cid:30) α x x ✁✁ @ @ ✁✁ x ☛☛☛ (cid:5) (cid:5) ☛☛☛ pb Denoting by ( k , ψ ) the result of pulling back ( k, ψ ) along fX Y A X YB / / ✾✾✾ x (cid:28) (cid:28) ✾✾✾ / / f (cid:28) (cid:28) ✾✾✾ x ✾✾✾ / / ✖✖✖✖✖ y (cid:11) (cid:11) ✖✖✖✖✖ / / (cid:11) (cid:11) ✖✖✖✖✖ y ✖✖ ✖✖ k ❧❧❧❧❧❧ k ❧❧❧❧❧❧ ψ : ❧❧❧❧ ψ : ♠♠♠♠ as in Lemma 4.4.1, the desired naturality comes down to the commutativity of the prism( X , Y , Y , X , C, A ) in X Y C X YD x / / ✾✾✾ kx (cid:28) (cid:28) ✾✾✾ / / g (cid:28) (cid:28) ✾✾✾ x ✾✾✾ y / / ✖✖✖✖✖ ky (cid:11) (cid:11) ✖✖✖✖✖ / / (cid:11) (cid:11) ✖✖✖✖✖ y ✖✖ ✖✖ k ❧❧❧❧❧❧ k ❧❧❧❧❧❧ kψ : ❧❧❧❧ ψ : ♠♠♠♠ AX Y ✾✾✾ (cid:28) (cid:28) ✾✾✾ k ❧❧❧❧❧❧ ✖✖✖✖ (cid:11) (cid:11) ✖✖✖✖✖ α x / / α y / / h / / ψ : ♠♠♠♠ in which ( k , ψ ) is the result of pulling back ( k, kψ ) along g . Since gψ α x = ghψ ,and both ψ α x and hψ are chosen g -opcartesian, it follows that ψ α x = hψ . Takingcodomains of this gives y k α x = y α y k , and y k α x = y β y k , so k α x = α y k . Theuniqueness part of opcartesianness can be used to verify 2-naturality.Given a commutative triangle as on the left A BC h / / g (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧ (cid:31) (cid:31) f ❄❄❄❄❄❄❄ K /A K /B K /C o o ∆ h Σ g (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧ (cid:31) (cid:31) Σ f ❄❄❄❄❄❄ κ + (16)7one has a canonical 2-natural transformation as on the right obtained from Σ g Σ h = Σ f and Σ h ⊣ ∆ h . When moreover f and g are split fibrations, since split fibrations compose,the 2-functors Σ f and Σ g lift to 2-functorsΣ f : Φ K ,A -Alg s −→ Φ K ,C -Alg s Σ g : Φ K ,B -Alg s −→ Φ K ,C -Alg s respectively. Lemma. The 2-natural transformation (16) lifts to a 2-natural transformation Φ K ,A -Alg s Φ K ,B -Alg s Φ K ,C -Alg s o o ∆ h Σ g | | ②②②②②② " " Σ f ❊❊❊❊❊❊ κ + when f and g each have the structure of a split fibration. Proof. Let p : E → B be a split fibration. We have to show that the component κ p is a morphism of split fibrations, that is to say, that post-composition with κ p preserveschosen cartesian 2-cells. So we consider a chosen f p -cartesian 2-cell ψ as in X E EBCA x ' ' h / / x κ p / / p (cid:15) (cid:15) g (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧ (cid:31) (cid:31) f ❄❄❄❄❄❄❄ (cid:15) (cid:15) p ψ (cid:11) (cid:19) pb and we must show that κ p ψ is chosen gp -cartesian. By the way in which one describes thecleavage for a composite of split fibrations, to say that κ p ψ is chosen gp -cartesian is to saythat κ p ψ is chosen p -cartesian and pκ p ψ is chosen g -cartesian; and to say that ψ is chosen f p -cartesian is to say that ψ is chosen p -cartesian and p ψ is chosen f -cartesian. Bythe way in which one describes the cleavage of a morphism resulting from pulling back asplit fibration, to say that ψ is chosen p -cartesian is to say that κ p ψ is chosen p -cartesian.Since h is a morphism of split fibrations and p ψ is chosen f -cartesian, hp ψ is chosen g -cartesian, and so the result follows. Theorem. Let K be a 2-category with pullbacks and comma objects. Denote by P : K /I → K /J the polynomial 2-functor associated to the polynomial on the left I E B J o o s p / / t / / I E B JB E w w s ♦♦♦♦♦♦♦♦♦ p / / t ' ' ❖❖❖❖❖❖❖❖❖ t ♦♦♦♦♦♦♦♦♦ / / p s g g ❖❖❖❖❖❖❖❖❖ f (cid:15) (cid:15) f (cid:15) (cid:15) pb= = and by φ : P → Q the 2-natural transformation associated to the morphism of polynomialsindicated on the right in the previous display. Then1. If p has the structure of a split opfibration and t has the structure of a split fibration,then P lifts to a 2-functor P : Φ K ,I -Alg s → Φ K ,J -Alg s such that U Φ K ,J P = P U Φ K ,I .2. If in the context of (1) I is discrete, then P is a familial 2-functor.3. If ( f , f ) : p → p is a morphism of split opfibrations and ( f , J ) : t → t is amorphism of split fibrations, then φ lifts to a 2-natural transformation φ : P → Q such that U Φ K ,J φ = φU Φ K ,I .4. If in the context of (3) I is discrete, then φ is a familial 2-natural transformation. Proof. When I is discrete Φ K ,I is the identity 2-monad, and so (2) follows from (1) bythe definition of familial 2-functor, and (4) follows from (3) by the definition of familial2-natural transformation. To obtain (1), define P = Σ t Π p ∆ p , where Π p exists by Lemma4.4.1 and the remarks immediately following that result, and Σ t exists by the compos-ability of split fibrations. To obtain (3), note that φ may be obtained as the composite K /I K /E K /B K /J. K /B K /E s ❧❧❧❧❧❧❧❧❧ Π p / / Σ t ) ) ❘❘❘❘❘❘❘❘ Σ t ❧❧❧❧❧❧❧❧ / / Π p ) ) ∆ s ❘❘❘❘❘❘❘❘❘ O O ∆ f O O ∆ f ∼ = β (cid:11) (cid:19) κ (cid:11) (cid:19) The unnamed isomorphism is obtained by adjunction from the identity Σ f Σ s = Σ s which clearly extends to the level of lax slices, and so this isomorphism lifts to the level ofsplit fibrations by the same argument as that given in the first paragraph of the proof ofLemma 4.4.3. By Lemma 4.4.3 β also lifts to the level of split fibrations, and by Lemma4.4.4 κ does too, and so (3) follows. Remark. In subsequent work we shall also use the dual of the above result, inwhich K is replaced by K co . For instance the dual version of (2) says that if I is discrete, p has the structure of a split fibration and t has the structure of a split opfibration, then P is opfamilial. Note also that when J is discrete, t , t and t are automatically split(op)fibrations in a unique way and f is a morphism thereof. Two dimensional monad theory becomes aparticularly powerful framework when applied to codescent object preserving 2-monads.For instance, as explained in [7, 23], a codescent object preserving 2-monad T on a 2-category of the form Cat ( E ) automatically satisfies the conditions of Power’s generalcoherence theorem [23, 25]. Thus knowing T preserves codescent objects implies a coher-ence theorem for T -algebras in this case. In particular one recovers the usual coherencetheorems for monoidal, braided monoidal and symmetric monoidal categories in this way,9without any combinatorial analysis. Moreover as discussed in [32], knowing that a 2-monad preserves codescent objects is one desirable condition leading to the ability tocompute internal algebra classifiers involving T .Codescent objects are particular instances of a class of 2-categorical colimits calledsifted colimits, which include also reflexive coequalisers and filtered colimits. Recall thatany weight J : C op → Cat (with C a small 2-category) determines, by virtue of thecocompleteness of Cat as a 2-category, a functor J ∗ − : [ C , Cat ] −→ Cat given on objects by taking colimits in Cat weighted by J , and J is a sifted weight when J ∗ − preserves finite products. A sifted colimit is a weighted colimit whose weight issifted. In other words, sifted colimits are exactly those colimits which in Cat commutewith finite products. Sufficient conditions on a polynomial in Cat so that its correspondingpolynomial 2-functor preserves sifted colimits is provided by Theorem. The polynomial 2-functor associated to a polynomial I A B J o o s p / / t / / in Cat such that I is discrete and p is a discrete fibration or a discrete opfibration withfinite fibres, preserves sifted colimits. Proof. It suffices to show that Π p ∆ s preserves sifted colimits since Σ t as a left adjointpreserves all colimits. We consider the case where p is a discrete fibration; the proof forthe case where p is a discrete opfibration is similar. By Theorem 4.4.5(1) applied in thecase K = Cat co , one has the commutative diagram Cat /I Ψ K ,E -Alg s Ψ K ,B -Alg s Cat /B. Cat /E ∆ s ❥❥❥❥❥❥❥❥❥❥❥❥❥❥ Π p / / U Ψ K ,B (cid:15) (cid:15) / / Π p / / ∆ s U Ψ K ,E (cid:15) (cid:15) By Remark 4.2.4 U Ψ K ,E and U Ψ K ,B create all colimits, and so it suffices to show that Π p preserves sifted colimits.Since Ψ K ,E -Alg s and Ψ K ,B -Alg s are 2-equivalent to the functor 2-categories [ E, Cat ]and [ B, Cat ], and in these terms ∆ p corresponds to the process of precomposition with p , by Lemma 4.4.2(2) one may identify Π p with the process of right Kan extension along p . But since p is a discrete fibration, such right Kan extensions are computed simply bytaking products over the fibres of p . Since these products are finite, and limits and colimitsin [ B, Cat ] are componentwise, the result follows by the definition of “sifted colimit”.0 Examples. By Theorem 4.5.1 the finite product completion and finite coproductcompletion endofunctors of Cat described in Example 4.1.5 preserve sifted colimits. Examples. In part 3 of [2] Batanin and Berger exhibit various flavours of “operad”as algebras of polynomial monads over Set . The underlying endo-polynomial in all of theirexamples is of the form I E B I o o s p / / t / / (17)where p is a function with finite fibres. The algebras of the associated monad on Set /I arethe particular flavour of operad under consideration in Set , however for the homotopicalaspects of that work, it is also important to regard (17) as a componentwise discretepolynomial in Cat . The corresponding 2-monad on Cat /I is familial and opfamilial byTheorem 4.4.5 and preserves sifted colimits by Theorem 4.5.1. 5. Examples of polynomial 2-monads on Cat In this section we exhibit the 2-monads1. M for monoidal categories,2. S for symmetric monoidal categories,3. B for braided monoidal categories,4. C fin for categories with finite coproducts, and5. P fin for categories with finite productson Cat , as polynomial 2-monads. These are all well-known cartesian 2-monads whichwere, for instance, basic examples for Kelly and his collaborators [5, 16, 18], in the es-tablishment of 2-dimensional monad theory. We recall them as such in Section 5.1, andthen in Section 5.2 exhibit their underlying endofunctors as polynomial. In Section 5.3we reexpress Johnstone’s idea of bagdomain data, as a way of indirectly exhibiting theunit and multiplication of a polynomial monad. We apply this method in Section 5.4 toour examples. Many other examples are exhibited in [33]. Let X be a category. Then the objectsof M ( X ), S ( X ), B ( X ), C fin ( X ) and P fin ( X ) are the same, namely they are finite sequences( x , ..., x n ) of objects of X . Denoting n as the discrete category { , ..., n } with n objects,we regard a sequence ( x , ..., x n ) as a functor x : n → X .A morphism x → y in C fin ( X ) consists of a function φ and a natural transformation φ as on the left in m nX φ / / y (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧ (cid:31) (cid:31) x ❄❄❄❄❄❄❄ φ + x x x x y y y (cid:31) (cid:31) ❄❄❄ φ ❄❄❄❄❄❄❄❄❄ (cid:10) (cid:10) ✔✔✔ φ ✔✔✔✔✔✔ (cid:10) (cid:10) ✔✔✔✔✔✔ φ ✔✔✔✔ (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧ φ ⑧⑧⑧⑧⑧⑧ φ ) decorated by themorphisms of X (the components of φ ) as indicated in the example on the right. Sinceit is useful to be able to reason precisely from this latter point of view, we make a briefnotational digression.We denote by S the category whose objects are natural numbers, and whose morphisms m → n are functions m → n . The category S is a skeleton of the category of finite sets andfunctions. We denote a sequence ( x , ..., x n ) of objects of X alternatively as ( x k ) ≤ k ≤ n ,or as ( x k ) k , as convenience dictates. The result of concatenating a sequence of sequences(( x k,l ) ≤ l ≤ n k ) ≤ k ≤ m , is denoted ( x k,l ) k,l . Implicit in this notation is the identification { ( k, l ) : 1 ≤ k ≤ m, ≤ l ≤ n k } = { , ..., n + ... + n m } via the lexicographic ordering of the former. Given morphisms φ : m → m and φ k : n ,k → n ,φ ( k ) in S for 1 ≤ k ≤ m , we denote by φ ( φ k ) k the morphism Σ k n ,k → Σ k n ,k given by ( φ ( φ k ) k )( k, l ) = ( φ ( k ) , φ k ( l )). In these terms a general morphism of C fin ( X ), isof the form ( φ, ( φ k ) ≤ k ≤ m ) : ( x k ) ≤ k ≤ m −→ ( y k ) ≤ k ≤ n where φ : m → n is in S , and φ k : x k → y φk is in X for 1 ≤ k ≤ m . We refer to the datum φ of such a morphism as the indexing function .The unit for C fin is given by the full inclusion of sequences of length 1. On objectsthe components µ X : C ( X ) → C fin ( X ) are given by concatenation of sequences, and onmorphisms by µ X ( φ, ( φ k , ( φ k,l ) l ) k ) = ( φ ( φ k ) k , ( φ k,l ) k,l ) , which in intuitive terms, is just substitution of decorated functions. With the notationprovided it is straight forward to verify that ( C fin , η, µ ) is a cartesian 2-monad on Cat .We have already considered a version of this 2-monad in Example 4.1.5. The differenceis that here, we have taken a skeleton of the category of finite sets to index our families,and have more carefully book-kept the combinatorics. It is straight forward to check that C fin is lax idempotent and thus to exhibit the pseudo algebras of either version as beingthe same, namely, categories with finite coproducts. The virtue of the more pain-stakingapproach taken here, is that C fin is a 2-monad rather than just a pseudo monad.One defines the 2-monad P fin as P fin ( X ) = C fin ( X op ) op . In direct terms, a morphismof P fin ( X ) is of the form( φ, ( φ k ) ≤ k ≤ m ) : ( x k ) ≤ k ≤ m −→ ( y k ) ≤ k ≤ n where φ : n → m is in S , and φ k : x φk → y k is in X for 1 ≤ k ≤ n . This is acolax idempotent cartesian 2-monad, and its pseudo algebras are categories with finiteproducts.The morphisms of the category S ( X ) are defined to be those of C fin ( X ) (or equallywell, of P fin ( X )) whose underlying indexing functions are bijections. It is clear that S is a Or in older language, C fin is a Kock-Z¨oberlein 2-monad of colimit like variance. C fin , and that the inclusion ι SC : S ֒ → C fin is a cartesian monad morphism.It follows that S is also a cartesian 2-monad. It is straight forward to verify directlythat pseudo S -algebras are exactly unbiased symmetric monoidal categories, and strictalgebras are exactly symmetric strict monoidal categories. Moreover the various notionsof S -morphism – lax, colax, pseudo and strict – correspond to symmetric lax, colax, strongand strict monoidal functors respectively. Similarly one defines the cartesian 2-monad M for monoidal categories, by defining the morphisms of M ( X ) as those of C fin ( X ) whoseindexing functions are identities.As explained in [15] the 2-monad B for braided monoidal categories, there denoted B ≀ ( − ), is given similarly as for S , except that the indexing bijections are replaced byindexing braids, in the definition of the morphisms of B ( X ). The process of taking theunderlying permutation of a braid, gives a cartesian monad morphism π : B → S , andso B is also a cartesian 2-monad. Once again the different types of algebras and algebramorphisms of B reconcile in the expected way with braided (strict) monoidal categoriesand braided (lax, colax, strong or strict) monoidal functors. One also has a cartesianmonad morphism ι MB : M → B , whose components can be regarded as the identity onobjects inclusion which regards morphisms of M ( X ) as identity braids whose strings aredecorated by the morphisms of X . The composite πι MB is denoted ι MS , and its componentsregards morphisms of M ( X ) as identity permutations whose strings are decorated by themorphisms of X .To summarise, one has a diagram of cartesian 2-monads and cartesian monad mor-phisms as on the left M B S C fin P fin ι MB / / π / / ι SC ♦♦♦♦ ι SP ' ' ❖❖❖❖ ι MS " " N B P SS op / / / / ♦♦♦♦♦♦ ' ' ❖❖❖❖❖ " " (18)and the result of evaluating at 1 is denoted as on the right. Each monad morphism hereis componentwise an identity on objects, and so in particular each functor on the right isthe identity on objects. On the right one has M (1) = N the natural numbers, B (1) = B the braid category, S (1) = P the permutation category, and so on. Recall, the non-emptyhom-sets of P are P ( n, n ) = Σ n , where Σ n is the group of permutations of n elements, andsimilarly, B ( n, n ) is Br n , the n -th braid group. In this section we shall establish that theleft diagram of (18) is a diagram of polynomial 2-monads and morphisms thereof. We denote by S ∗ thecoslice 1 / S , whose objects may be regarded as pairs ( i, n ) where n ∈ N and 1 ≤ i ≤ n ,with i regarded as a “chosen basepoint”. A morphism ( i, m ) → ( j, n ) of S ∗ is thus a This means that an n -ary tensor product is defined for all n , and invertible coherence maps aregiven. The coherence theorems one has available enable one to identify these with symmetric monoidalcategories defined in the usual biased way, with a unit and binary product, and so we regard pseudo S -algebras as symmetric monoidal categories, disregarding the biased/unbiased distinction. f : m → n such that f i = j . By definition, S ∗ comes with a forgetful functor U S : S ∗ → S , and U S is a discrete opfibration with finite fibres. The fibre over n ∈ S maybe identified with the set n = { , ..., n } . Since U S is a discrete opfibration, and ( U S ) op isa discrete fibration, U S and ( U S ) op are exponentiable functors. Lemma. The underlying endofunctors of C fin and P fin are the result of applying P Cat to S ∗ S o o U S / / / / S op ∗ S op o o ( U S ) op / / / / respectively. Proof. We give the proof for C fin , the case of P fin follows similarly. Provisionally wewrite P for the 2-functor corresponding to the polynomial on the left in the above display.By definition one has 1 S ∗ S X X × S ∗ P ∗ X P X o o U S / / / / o o (cid:15) (cid:15) p X (cid:15) (cid:15) (cid:15) (cid:15) / / q X (cid:15) (cid:15) ❄❄❄❄❄❄❄❄ (cid:31) (cid:31) ❄❄❄❄❄❄❄❄ dpbpb (19)and we now proceed to identify P X with C fin ( X ). An object of P X may be regarded asa functor h : [0] → P X , and thus also as a pair n : [0] → S together with h : [0] → P X such that qh = n . By ∆ U S ⊣ Π U S and since q X = Π U S ( p X ), such an h is in bijection with k : ( U S ) − { n } → X × S ∗ over S ∗ , but this in turn is just an n -tuple ( x , ..., x n ) of objectsof X . Thus an object of P X is a pair ( n, x ) where n ∈ N and x : n → X . Similarlyregarding arrows of P X as functors [1] → P X and using the adjointness ∆ U S ⊣ Π U S inthe same way, one finds that a morphism ( m, x ) → ( n, y ) of P X consists of φ : m → n in S together with φ as in m nX φ / / y (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧ (cid:31) (cid:31) x ❄❄❄❄❄❄❄ φ + and so P and C fin agree on objects. Note that P ∗ X also has an easy explicit description.Namely, an object is a triple ( i, n, x ) where ( n, x ) ∈ P X and 1 ≤ i ≤ n , and a morphism( i, m, x ) → ( j, n, y ) consists of φ and φ as above making ( φ, φ ) : ( m, x ) → ( n, y ) amorphism of P X , such that φi = j . Moreover, all the functors participating in (19) alsoadmit straight forward direct descriptions in these terms.We must now verify that P f = C fin ( f ) for f : X → Y . Note that P f is inducedfrom f by the pullbacks and distributivity pullbacks that go into defining P X and P Y ,whereas C fin ( f ) is the functor described by composing with f , that is on objects one has C fin ( f )( n, x ) = ( n, f x ). Let us define C fin ∗ ( f ) : P ∗ X → P ∗ Y to be the functor given on4objects by ( i, n, x ) ( i, n, f y ). By the uniqueness aspects of the universal properties ofthe pullbacks and distributivity pullbacks involved in defining P Y , it suffices to show that X × S ∗ P ∗ X P X S P YP ∗ YY × S ∗ o o / / q X ' ' ❖❖❖❖❖❖❖❖ q Y ♦♦♦♦♦♦♦♦ / / o o (cid:15) (cid:15) f × S ∗ C fin ∗ ( f ) (cid:15) (cid:15) C fin ( f ) (cid:15) (cid:15) commutes. Given that everything in this diagram has been made so explicit, this is astraight forward calculation.The functor ( ι SC ) : P → S is the inclusion of the maximal subgroupoid of S , andsimilarly ( ι SP ) is also a maximal subgroupoid inclusion. We define P ∗ as the maximalsubgroupoid of S ∗ . It comes with a forgetful functor U P : P ∗ → P . Equivalently, P ∗ = 1 / P .The functor U P fits into pullback squares in P ∗ PSS ∗ U P / / ( ι SC ) (cid:15) (cid:15) / / U S (cid:15) (cid:15) pb w w ♦♦♦♦♦♦♦ g g ❖❖❖❖❖❖❖ ' ' ❖❖❖❖❖❖❖❖ ♦♦♦♦♦♦♦♦ P ∗ PS op S op ∗ U P / / ( ι SP ) (cid:15) (cid:15) / / ( U S ) op (cid:15) (cid:15) pb w w ♦♦♦♦♦♦♦ g g ❖❖❖❖❖❖❖ ' ' ❖❖❖❖❖❖❖❖ ♦♦♦♦♦♦♦ By Propositions 3.3.6 and 4.1.6 the underlying endofunctor of S is a polynomial 2-functor,and one may regard these morphisms of polynomials as corresponding to the 2-naturaltransformations ι SC and ι SP . Similarly by pulling back along π and ( ι MS ) , one defines U B : B ∗ → B and U N : N ∗ → N , exhibits B and M as polynomial 2-functors, and π , ι MS and ι MB as arising from morphisms of polynomials. To summarise, we have Corollary. At the level of the underlying endofunctors, the left diagram of (18) isin the essential image of P Cat . To establish that (18) is a diagram of polynomial 2- monads and morphisms thereof, oneway to proceed would be to write down the polynomial monad structures for C fin and P fin explicitly, and then use Proposition 4.1.6. To avoid computations involving compositesof such polynomials, for instance when verifying monad axioms, we use an alternativeapproach. This approach is described in general in the next section, and applied to ourexamples in Section 5.4. The construction of thefinite coproduct and finite product completion were seen to underlie polynomial pseudo monads in Cat in Example 4.1.5. However as explained in Section 5.1 these constructionsunderlie 2-monads, and a framework for these is described in this section. It is little morethan an explicit encoding of some of the developments of Section 2 of [13], in the languageof polynomials.Recall that when a category E has pullbacks the codomain functor cod : E [1] → E is afibration, and that a morphism of E [1] is cod-cartesian if and only if its underlying squarein E is a pullback square.5 Definition. Let E be a category with pullbacks and p : E → B be a morphismtherein. Then a p -fibration is a cod-cartesian arrow into p , and a morphism of p -fibrations is a morphism in E [1] ↓ p between p -fibrations.As such a p -fibration consists of an arrow f : X → Y of E together with ( u, v ) fittinginto a pullback square X YBE f / / v (cid:15) (cid:15) / / p (cid:15) (cid:15) u pb X Y Y X f / / v (cid:15) (cid:15) / / f (cid:15) (cid:15) u pb as on the left. In this context we say that ( u, v ) is a p -fibration structure on f . For amorphism of p -fibrations ( f , u , v ) → ( f , u , v ) one has morphisms u and v as on theright in the previous display such that u u = u and v v = v . Clearly, u is uniquelydetermined by v and the universal property of the pullback containing ( f , u , v ), andso in minimalistic terms, ( u , v ) amounts to v : Y → Y over B . Example. Let E = Set and regard U N : N ∗ → N ∈ Set . Explicitly, N ∗ = { ( i, j ) : i ∈ N , ≤ j ≤ i } and U N ( i, j ) = i . Observe that for n ∈ N , | ( U N ) − { n }| = n . To givea function f : X → Y the structure of a U N -fibration is by definition to give functions u and v fitting into a pullback square X Y NN ∗ f / / v (cid:15) (cid:15) / / U N (cid:15) (cid:15) v pb and so f must have finite fibres since U N does. Moreover for y ∈ Y such that f − { y } = n , u restricts to a bijection u y between f − { y } and ( U N ) − { n } , and so amounts to a linearorder on f − { y } . Conversely given a function f : X → Y with finite fibres and a linearorder on each fibre, one determines v by the formula vy = | f − { y }| , and then u by taking ux = ( n, i ), where n = | f − { f x }| and i is x ’s position in the linear order on f − { f x } .Thus a U N -fibration is a function with finite fibres together with a linear order on eachfibre.Recall that a double category (resp. double functor ) is a category (resp. functor)internal to CAT . Given a category E we denote by D ( E ) the double category as on theleft E [2] E [1] E id E o o dom E / / cod E / / / / comp E / / / / X X d / / d / / whose objects are those of E , vertical arrows and horizontal arrows are morphisms of E ,and squares are commutative squares in E . Any double category X has, by forgettinghorizontal identities and compositions, an underlying graph internal to CAT , consisting6of the source and target functors as on the right in the previous display. Our conventionsare that X is the category of objects and vertical arrows, and X is the category ofhorizontal arrows and squares between them.In the context of Definition 5.3.1 we define U p : D ( p ) −→ D ( E )a graph morphism internal to CAT , as follows. The category D ( p ) of objects and verticalarrows of D ( p ) is E , and ( U p ) = 1 E . The category D ( p ) of horizontal arrows and squaresof D ( p ), is the category of p -fibrations and their morphisms. Every p -fibration has anunderlying arrow of E , and every morphism of p -fibrations has an underlying square, theassignations of which provide ( U p ) . Theorem. Let E be a category with finite limits and p : E → B an exponentiablemorphism therein. There is a bijection between the following types of data:1. Unit and multiplication 2-cells in Poly E making E B o o p / / / / the underlying endoarrow of a monad in Poly E .2. Double category structures on D ( p ) making U p a double functor. Proof. We denote by P : 1 → u : 1 → P in Poly E is to give ( u , u ) fitting into a pullback square1 1 BE / / u (cid:15) (cid:15) / / p (cid:15) (cid:15) u pb X X X / / (cid:15) (cid:15) / / (cid:15) (cid:15) pb X XYY X / / f (cid:15) (cid:15) / / Y (cid:15) (cid:15) f pb as on the left. Given this data any identity arrow 1 X : X → X acquires the structure ofa p -fibration, namely that which makes the square on the middle in the previous displaya morphism 1 X → of p -fibrations. Given a morphism f : X → Y of E , with respectto these p -fibration structures on identity arrows, the square on the right in the previousdisplay is clearly a morphism of p -fibrations. Thus one has a functor id p : E → D ( p ) such that ( U p ) id p = id E . Conversely, such a functor amounts to assigning a p -fibrationstructure to each identity map in such a way that for all f : X → Y the right squareabove is a morphism of p -fibrations. Thus in particular there is a p -fibration structure on1 , which amounts to ( u , u ) as on the left in the previous display. The processes justdescribed are easily verified to exhibit a bijection between 2-cells u : 1 → P of Poly E and functors id p as above.To give a functor comp p : D ( p ) × E D ( p ) → D ( p ) such that ( U p ) comp p = comp E ( U p ) ,where ( U p ) is induced in the evident way using ( U p ) , is to give, for each composable pair7 f : X → Y , g : Y → Z in E together with p -fibration structures on f and g , a p -fibrationstructure on the composite gf , and that this assignation be functorial. This functorialitymeans that given morphisms of p -fibrations ( u, v ) : f → f ′ and ( v, w ) : g → g ′ as in X Y ZZ ′ Y ′ X ′ f / / g / / w (cid:15) (cid:15) / / g ′ / / f ′ (cid:15) (cid:15) u v (cid:15) (cid:15) the composite square underlies a p -fibration morphism ( u, w ) : gf → g ′ f ′ .Suppose that a 2-cell m : P ◦ P → P in Poly E is given. Then given also a composablepair f : X → Y , g : Y → Z in E together with p -fibration structures on f and g , one has1 E B E B B × EF B (2) E (2) X Y Z o o p / / / / o o p / / / / (cid:15) (cid:15) (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧ π E (cid:31) (cid:31) ❄❄❄❄❄❄❄ (cid:11) (cid:11) ✗✗✗✗✗✗✗✗✗✗✗✗ / / / / (cid:19) (cid:19) ✬✬✬✬✬✬✬✬✬✬✬✬ (cid:3) (cid:3) ✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞ (cid:27) (cid:27) ✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼ f / / g / / W W ✴✴✴✴✴ _ _ ❄❄❄❄❄❄❄ ? ? ⑧⑧⑧⑧⑧⑧⑧ G G ✎✎✎✎✎ pb dpbpbpb pb W W I I O O O O in which the lower pullback squares witness the p -fibration structures on f and g . Bythe universal property of B × E one induces the unique arrow Y → B × E commutingwith the morphisms into B and E . Thus the bottom right pullback is around ( p, π E ),and so by the distributivity pullback one induces the morphisms Y → F and Z → B (2) .Finally using the top right pullback one induces X → E (2) . The squares ( X, Y, F, E (2) )and ( Y, Z, B (2) , F ) so arising are pullbacks, and so the composite pullback X Y ZB (2) FE (2) f / / g / / (cid:15) (cid:15) / / / / (cid:15) (cid:15) (cid:15) (cid:15) pb pb E B m (cid:15) (cid:15) / / p (cid:15) (cid:15) m pb exhibits a p -fibration structure on gf , and so we have described the object map of thefunctor comp p .The composite p (2) : E (2) → B (2) is the middle map of the composite polynomial P ◦ P .When f and g are the morphisms E (2) → F and F → B (2) respectively, the top row ofvertical arrows in the previous display are identities, and so one has a p -fibration structureon p (2) . Moreover for general f and g , the composite of the top pullbacks of the previous8display exhibits a morphism gf → p (2) of p -fibrations. From1 E B E B B × EF B (2) E (2) X ′ Y ′ Z ′ o o p / / / / o o p / / / / (cid:15) (cid:15) (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧ π E (cid:31) (cid:31) ❄❄❄❄❄❄❄ (cid:11) (cid:11) ✗✗✗✗✗✗✗✗✗✗✗✗ / / / / (cid:19) (cid:19) ✬✬✬✬✬✬✬✬✬✬✬✬ (cid:3) (cid:3) ✞✞✞✞✞✞✞✞✞✞✞✞✞✞✞ (cid:27) (cid:27) ✼✼✼✼✼✼✼✼✼✼✼✼✼✼✼ f ′ / / g ′ / / W W ✴✴✴✴✴ _ _ ❄❄❄❄❄❄❄ ? ? ⑧⑧⑧⑧⑧⑧⑧ G G ✎✎✎✎✎ pb dpbpbpb pb W W I I O O O O X Y Z f / / g / / w O O v O O u O O pb pb in which ( u, v ) and ( v, w ) are morphisms of p -fibrations, ( u, w ) is a morphism gf → g ′ f ′ of p -fibrations, thus giving the arrow map for comp p . In this way from a 2-cell m : P ◦ P → P in Poly E , one obtains the functor comp p such that ( U p ) comp p = comp E ( U p ) .Conversely given comp p , the pullbacks appearing in the formation of P ◦ P exhibit p -fibration structures on E (2) → F and F → B (2) , and applying comp p to this composablepair gives a p -fibration structure to p (2) , which amounts to the components ( m , m ) of a2-cell m : P ◦ P → P . It is straight forward to verify that the processes described hereexhibit a bijection between such 2-cells and such functors comp p . The straight forwardverification that the unit and associative laws for u and m correspond with the unit andassociative laws for horizontal composition in the double category D ( p ) is left to thereader.It was the data of Theorem 5.3.3(1) that Johnstone named bagdomain data in Defini-tion 2.1 of [13]. Remark. In the context of Theorem 5.3.3, one can consider polynomials as onthe left I X Y J o o s f / / t / / I X Y JY X w w s ♦♦♦♦♦♦♦♦♦ f / / t ' ' ❖❖❖❖❖❖❖❖❖ t ♦♦♦♦♦♦♦♦♦ / / f s g g ❖❖❖❖❖❖❖❖❖ u (cid:15) (cid:15) v (cid:15) (cid:15) pb= = in which the middle map f has the structure of a p -fibration, morphisms thereof as on theright in which the middle pullback square is a morphism of p -fibrations. We call such apolynomial a p -structured polynomial from I to J . By definition the process of pulling backin E carries along p -fibration structures, and by Theorem 5.3.3 one has a composition of p -fibrations. Thus composition in Poly E can be extended to a composition of p -structuredpolynomials. In this way one has a bicategory Poly p , whose 1-cells are p -structured9polynomials, together with a strict homomorphism of bicategories Poly p → Poly E , andthe polynomial 1 E B o o p / / / / is by definition terminal in the hom category Poly p (1 , E with finite limits, anymonad in Poly E on 1 arises in a manner analogous to the process described in Example4.1.5. Example. By concatenating linear orders on fibres, U N -fibrations as characterisedin Example 5.3.2 can be composed, and this composition is functorial with respect tomorphisms of U N -fibrations. The polynomial monad one gets by Theorem 5.3.3 is themonoid monad on Set . Regarding U N as a functor between discrete categories and arguingthe same way, one recovers the monad M on Cat . The foregoing discussion is adapted to the 2-categoricalcontext in the following way. For an exponentiable morphism p : E → B in a 2-category K with pullbacks, we define p -fibrations and morphisms thereof as in Definition 5.3.1.Given p -fibrations ( f , u , v ) and ( f , u , v ) and morphisms thereof as in X Y BE f / / v (cid:15) (cid:15) / / p (cid:15) (cid:15) u pb X Y BE f / / v (cid:15) (cid:15) / / p (cid:15) (cid:15) u pb X Y Y X f / / v (cid:15) (cid:15) / / f (cid:15) (cid:15) u pb X Y Y X f / / v (cid:15) (cid:15) / / f (cid:15) (cid:15) u pb X Y Y X f / / v (cid:3) (cid:3) / / f (cid:3) (cid:3) u v (cid:27) (cid:27) (cid:27) (cid:27) u α ✐✐✐✐ β ✐✐✐✐ a 2-cell ( u , v ) → ( u , v ) is a pair ( α, β ) such that the cylinder on the right commutes,and u α = id and v β = id. In minimalistic terms using the 2-dimensional universalproperty of pullbacks, ( α, β ) is determined by the 2-cell β over B .We call a category (resp. functor) internal to a double 2-category (resp. double2-functor ). For any 2-category K one has the double 2-category D ( K ) K [2] K [1] K . id K o o dom K / / cod K / / / / comp K / / / / In the situation where K has pullbacks and p is as above, the graph morphism U p : D ( p ) → D ( K ) internal to is defined analogously to the definition given abovewith D ( p ) = K , ( U p ) = 1 K , D ( p ) is the 2-category of p -fibrations as just defined and( U p ) is the evident forgetful 2-functor. The following result is proved in the same wayas Theorem 5.3.3 with the 2-dimensional universal properties one now has providing theadditional required information. Theorem. Let K be a 2-category with finite limits and p : E → B be an exponen-tiable morphism therein. There is a bijection between the following types of data: 1. Unit and multiplication 2-cells in Poly K making E B o o p / / / / the underlying endoarrow of a 2-monad in Poly K .2. Double 2-category structures on D ( p ) making U p a double 2-functor. There is also a version of this result useful for exhibiting morphisms of polynomial2-monads. Given a pullback square E B B E p / / v (cid:15) (cid:15) / / p (cid:15) (cid:15) u pb in K in which p and p are exponentiable, composing with this pullback square is theeffect on objects of a 2-functor D ( u, v ) : D ( p ) → D ( p ). Defining D ( u, v ) : K → K tobe the identity, one has a graph morphism D ( u, v ) : D ( p ) → D ( p ) internal to over D ( K ). It is straight forward to extend the proof of Theorem 5.4.1 to a proof of Theorem. In the context just described, there is a bijection between the followingtypes of data:1. Unit and multiplication 2-cells on in Poly K making E B E B w w ♦♦♦♦♦♦♦♦♦ p / / ' ' ❖❖❖❖❖❖❖❖❖ ♦♦♦♦♦♦♦♦♦ / / p g g ❖❖❖❖❖❖❖❖❖ u (cid:15) (cid:15) v (cid:15) (cid:15) pb the underlying endoarrow of a morphism of 2-monads in Poly K .2. Double 2-category structures on D ( p ) and D ( p ) making U p , U p and D ( u, v ) double2-functors. We now have sufficiently many tools to enable us to witness the diagram (18) as beinga diagram of polynomial 2-monads. First we shall understand some of the relevant double2-categories of p -fibrations for appropriate p . Recall from the definition of U P : P ∗ → P given in Section 5.2, that P is the permutation category, and that P ∗ has objects those of N ∗ as described explicitly in Example 5.3.2. An arrow ( m, i ) → ( n, j ) of P ∗ can only existwhen m = n , in which case it is a permutation ρ ∈ Σ n such that ρi = j . The forgetfulfunctor U P is a discrete fibration of groupoids with finite fibres.1 Lemma. 1. To give a U P -fibration is to give a functor f : X → Y which is a discrete fibrationand a discrete opfibration with finite fibres, together with a linear order on each fibre.2. Given U P -fibrations f : X → Y and f : X → Y , a to give a morphism f → f of U P -fibrations is to give a pair ( h, k ) of functors fitting into a pullback square ason the left X Y Y X f / / k (cid:15) (cid:15) / / f (cid:15) (cid:15) h pb X Y Y .X f / / k (cid:3) (cid:3) / / f (cid:3) (cid:3) h k (cid:27) (cid:27) (cid:27) (cid:27) h α ✐✐✐✐ β ✐✐✐✐ such that for all y ∈ Y , h | f − { y } : f − { y } → f − { ky } is order preserving.3. Given f and f as in (2), and morphisms ( h , k ) and ( h , k ) of U P -fibrations,to give a 2-cell ( h , k ) → ( h , k ) in D ( U P ) is to give a pair ( α, β ) fitting into acommutative cylinder as on the right in the previous display. Proof. In each case we shall explain how to go from U P -fibrations, morphisms or 2-cellsthereof to the data described in the statement, and how to go back, leaving to the readerthe straight forward task of showing that these processes give the required bijections.(1): To give a functor f : X → Y the structure of a U P -fibration is by definition togive functors u and v fitting into a pullback square X Y P . P ∗ f / / v (cid:15) (cid:15) / / U P (cid:15) (cid:15) u pb (20)Thus f is a discrete fibration and a discrete opfibration with finite fibres since U P is, andthese properties on a functor are pullback stable. For y ∈ Y , writing n = vy , because ofthe pullback (20) on objects, u restricts to a bijection between f − { y } and ( U P ) − { n } ,this latter set being { ( n, , ..., ( n, n ) } in explicit terms. But to give such a bijection isto give a linear order on f − { y } , by taking the i -th element of f − { y } to be the elementsent to ( n, i ) by the bijection.Conversely suppose one has a functor f : X → Y which is a discrete fibration anda discrete opfibration with finite fibres, together with a linear order on each fibre. For y ∈ Y one can define vy = | f − { y }| . Putting n = vy and denoting the linearly orderedset f − { y } as { x < ... < x n } , for 1 ≤ i ≤ n we define ux i = ( n, i ). For β : y → y ′ in Y ,the unique lifting property f enjoys by being a discrete opfibration provides a function f − { y } → f − { y ′ } , the unique lifting property f enjoys by being a discrete fibrationprovides a function f − { y ′ } → f − { y } , and the uniqueness of these lifting properties2ensures that these functions are mutually inverse. Let n = vy = vy ′ and denote by { x < ... < x n } and { x ′ < ... < x ′ n } the linearly ordered sets f − { y } and f − { y ′ } respectively. The lifting properties just described give us, for 1 ≤ i ≤ n , a morphism α i : x i → x ′ ρi of X , and the above bijection f − { y } → f − { y ′ } is given by x i x ′ ρi . Thus ρ ∈ Σ n , and we define vβ = ρ and uα i to be ρ : ( n, i ) → ( n, ρi ).(2): We write ( u , v ) and ( u , v ) for the morphisms which exhibit the U P -fibrationstructures of f and f respectively. By definition a morphism f → f of U P -fibrationsdetermines ( h, k ) fitting into a pullback square as in the statement, and the equation u h = u restricted to f − { y } , implies that h | f − { y } is order preserving.Conversely suppose that one has ( h, k ) as in the statement. We must verify that u h = u and v k = v . For y ∈ Y one has v y = | f − { y }| = | f − { ky }| = v ky in which the first and third equalities follow from the definitions of v and v , and thesecond equality follows by the pullback of the statement on objects. Writing x ,i (resp. x ,i ) for the i -th element of f − { y } (resp. f − { y } ), one has u x ,i = ( n, i ) = u x ,i = u hx i where n = v y = v ky and x ,i is the i -th element of f − { ky } , in which the first andsecond equalities follow by the definition of u and u , and the third equality follows since f h = kf and h is order preserving on the fibres of f .For β : y → y ′ in Y , we denote by x ,i , x ,i , x ′ ,i and x ′ ,i the i -th element of f − { y } , f − { ky } , f − { y ′ } and f − { ky ′ } respectively. As in (1) we have α ,i : x ,i → x ′ ,ρ i suchthat f α ,i = β , and α ,i : x ,i → x ′ ,ρ i such that f α ,i = kβ , where n = v y = v y ′ and ρ , ρ ∈ Σ n . We must show that v kβ = v β , that is that ρ = ρ , and that u hα ,i = u α ,i for all 1 ≤ i ≤ n . But since hα ,i : hx ,i → hx ′ ,ρ i , hx ,i = x ,i , hx ′ ,ρ i = x ′ ,ρ i , and f hα ,i = kβ , these equations follow from the uniqueness of lifts for f .(3): By definition a 2-cell ( h , k ) → ( h , k ) of D ( U P ) consists of ( α, β ) of thestatement making the cylinder commute, and moreover verifying u α = id and v β = id.It suffices to show that these last two equations are automatic.Let y ∈ Y and denote by { x < ... < x n } the linearly ordered fibre f − { y } . We have β y : k y → k y in Y , and for 1 ≤ i ≤ n , we have α x i : h x i → h x i in X . By thecommutativity of the cylinder f α x i = β y . By the definition of v on arrows, one has v β y = id, and so by the definition of u on arrows, one has u α x i = id as required.Thanks to Lemma 5.4.3 we have an explicit description of graph D ( U P ) internalto and of the internal graph morphism U U P , which involves forgetting the U P -fibration structures. Similarly one can understand U U S : D ( U S ) −→ D ( Cat ) U U S op : D ( U S op ) −→ D ( Cat )by analysing what U S -fibrations (resp. U S op -fibrations) and their morphisms amount to.To give a functor f : X → Y a U S -fibration (resp. U S op -fibration) structure, is to exhibit3it as a discrete opfibration (resp. discrete fibration) with finite fibres, and to give a linearorder on each fibre. Via the pullback squares relating U P , U S and U S op , one has internalgraph morphisms D ( ι SC ) : D ( U P ) → D ( U S ) and D ( ι SP ) : D ( U P ) → D ( U S op ) over D ( Cat ). Theorem. The diagram M B S C fin P fin ι MB / / π / / ι SC ♦♦♦♦ ι SP ' ' ❖❖❖❖ ι MS " " described in (18) Section 5.1, is a diagram of polynomial 2-monads and morphisms thereof. Proof. Any identity functor has a unique U P -fibration structure. Given U P -fibrations f : X → Y and g : Y → Z , the composite gf is a discrete fibration and discrete opfibrationwhose fibres are finite. For any z ∈ Z , an element of the fibre ( gf ) − { z } may be identifiedas a pair ( x, y ), where y ∈ g − { z } and x ∈ f − { y } . Defining ( x , y ) ≤ ( x , y ) if and onlyif y < y or y = y and x ≤ x , provides ( gf ) − { z } with a linear order. Thus thereis an evident composition of U P -fibrations. Thus D ( U P ) acquires a double 2-categorystructure making U U P : D ( U P ) → D ( K ) a double 2-functor. Moreover U U S , U U S op , D ( ι SC )and D ( ι SP ) are easily witnessed as double 2-functors, thanks to our explicit understandingof U P -fibrations, U S -fibrations and U S op -fibrations. Thus C fin , P fin and S are polynomial2-monads and ι SC and ι SP are morphisms thereof, by Theorems 5.4.1 and 5.4.2. To exhibit B , M , π and ι MB as polynomial, we appeal to Proposition 4.1.6 and Corollary 5.2.2, usingthe fact that S is a polynomial 2-monad.Having exhibited these examples as polynomial, Theorems 4.4.5 and 4.5.1 enable usto read off some of their categorical properties. 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