Monads in Double Categories
aa r X i v : . [ m a t h . C T ] J u l MONADS IN DOUBLE CATEGORIES
THOMAS M. FIORE, NICOLA GAMBINO, AND JOACHIM KOCK
Abstract.
We extend the basic concepts of Street’s formal theory of monadsfrom the setting of 2-categories to that of double categories. In particular,we introduce the double category Mnd( C ) of monads in a double category C and define what it means for a double category to admit the constructionof free monads. Our main theorem shows that, under some mild conditions,a double category that is a framed bicategory admits the construction of freemonads if its horizontal 2-category does. We apply this result to obtain doubleadjunctions which extend the adjunction between graphs and categories andthe adjunction between polynomial endofunctors and polynomial monads. Introduction
The development of the formal theory of monads, begun in [23] and contin-ued in [15], shows that much of the theory of monads [1] can be generalized fromthe setting of the 2-category
Cat of small categories, functors and natural trans-formations to that of a general 2-category. The generalization, which involvesdefining the 2-category Mnd( K ) of monads, monad maps and monad 2-cells in a 2-category K , is useful to study homogeneously a variety of important mathematicalstructures. For example, as explained in [17], categories, operads, multicategoriesand T -multicategories can all be seen as monads in appropriate bicategories. How-ever, the most natural notions of a morphism between these mathematical struc-tures do not appear as instances of the notion of a monad map. For example, itis well known that, while categories can be viewed as monads in the bicategory ofspans [2], functors are not monad maps therein.To address this issue, we define the double category Mnd( C ) of monads, horizon-tal monad maps, vertical monad maps and monad squares in a double category C .Monads and horizontal monad maps in C are exactly monads and monad maps inthe horizontal 2-category of C , while the definitions of vertical monad maps andmonad squares in C involve vertical arrows of C that are not necessarily identities.This combination of horizontal and vertical arrows of C in the definition of Mnd( C )allows us to describe mathematical structures and morphisms between them asmonads and vertical monad maps in appropriate double categories. For example,small categories and functors can be viewed as monads and vertical monad mapsin the the double category of spans.For a double category C , we define also the double category End( C ) of endo-morphisms, horizontal endomorphism maps, vertical endomorphism maps and en-domorphism squares. The double categories Mnd( C ) and End( C ) are related by aforgetful double functor U : Mnd( C ) → End( C ), mapping a monad to its underlyingendomorphism. By definition, a double category C is said to admit the construc-tion of free monads if U has a vertical left adjoint. In view of our applications,we consider the construction of free monads in double categories that satisfy theadditional assumption of being framed bicategories, in the sense of [21]. Our main Date : July 11th, 2010.2000
Mathematics Subject Classification.
Key words and phrases.
Double categories, fibred bicategories, monads. result shows that a framed bicategory satisfying some mild assumptions admits theconstruction of free monads if its horizontal 2-category does. Here, the notion ofa 2-category admitting the construction of free monads is obtained by generalizingthe characterization of the free monads in the 2-category
Cat obtained in [22, § E is a pretopos with parametrizedlist objects, then the double category of spans in E admits the construction of freemonads. Secondly, we consider the construction of the free monad on a polynomialendofunctor (relatively to a locally cartesian closed category, which is always as-sumed here to have a terminal object), which contributes to the category-theoreticanalysis of Martin-L¨of’s types of wellfounded trees, begun in [19] and continuedin [7, 8]. We show that if E is a locally cartesian closed category with finite disjointcoproducts and W-types, then the double category of polynomials in E admits theconstruction of free monads. Both of these results are obtained by application ofour main result, which is possible since the double categories of interest are framedbicategories. Examples of categories E satisfying the hypotheses above abound: forexample, every elementary topos with a natural numbers object is both a pretoposwith parametrized list objects and a locally cartesian closed category with finitedisjoint coproducts and W-types [19]. Thus, our theory applies in particular to thecategory Set of sets and functions and to categories of sheaves.The double categories of spans and of polynomials are defined so that if weconsider the vertical part of the free monad double adjunction, we recover exactlythe adjunction between graphs and categories [16, § II.7] and the adjunction be-tween polynomial endofunctors and polynomial monads [8, § § Plan of the paper.
Section 1 discusses monads in a 2-category, recalling somebasic notions from [23] and giving a characterization of the free monads in a 2-category. Section 2 introduces the double category Mnd( C ) of monads in a doublecategory C and illustrates its definition with examples. Section 3 establishes somespecial properties of Mnd( C ) under the assumption that C is a framed bicategory.In particular, we state our main result, Theorem 3.7, and apply it to our examples.Finally, Section 4 contains the proof of Theorem 3.7.1. Monads in a 2-category
Preliminaries.
We recall some definitions concerning endomorphisms, monadsand their algebras in a 2-category. Let K be a 2-category. An endomorphism in K is a pair ( X, P ) consisting of an object X and a 1-cell P : X → X . An ONADS IN DOUBLE CATEGORIES 3 endomorphism map ( F, φ ) : (
X, P ) → ( Y, Q ) consists of a 1-cell F : X → Y and a 2-cell φ : QF → F P , which is not required to satisfy any condition. An endomorphism 2-cell α : ( F, φ ) → ( F ′ , φ ′ ) is a 2-cell α : F → F ′ making thefollowing diagram commute QF φ / / Qα (cid:15) (cid:15) F P αP (cid:15) (cid:15) QF ′ φ ′ / / F ′ P. We write End( K ) or End K for the 2-category of endomorphisms, endomorphismmaps and endomorphism 2-cells in K . There is a 2-functor Inc : K →
End( K )which sends an object X ∈ K to the identity endomorphism ( X, X ) on X . Let usnow consider a fixed endomorphism ( Y, Q ) in K . For X ∈ K , the category Q -alg X of X -indexed Q -algebras, in the sense of Lambek, is defined by letting Q -alg X = def End K (( X, X ) , ( Y, Q )) . Explicitly, an X -indexed Q -algebra consists of a 1-cell F : X → Y , called theunderlying 1-cell of the algebra, and a 2-cell f : QF → F , called the structuremap of the algebra. Note that the structure map is not required to satisfy anyconditions. These definitions extend to a 2-functor Q -alg ( − ) : K →
Cat . We write U ( − ) : Q -alg ( − ) → K ( − , Y ) for the 2-natural transformations whose com-ponents are the forgetful functors U X : Q -alg X → K ( X, Y ) mapping an X -indexed Q -algebra to its underlying 1-cell.We write Mnd( K ) or Mnd K for the 2-category of monads, monad maps andmonad 2-cells in K , as defined in [23]. As usual, we refer to a monad by mention-ing only its underlying endomorphism, leaving implicit its multiplication and unit.With a minor abuse of notation, we write Inc : K →
Mnd( K ) for the 2-functormapping an object X to the monad ( X, X ). If ( Y, Q ) is a monad, for every X ∈ K we may consider not only the category Q -alg X of Lambek algebras for its under-lying endomorphism, but also the category Q -Alg X of X -indexed Eilenberg-Moore Q -algebras, which is defined by letting Q -Alg X = def Mnd K (( X, X ) , ( Y, Q )) . Note that we write Q -alg X for the category of algebras for the endomorphismand Q -Alg X for the category of Eilenberg-Moore algebras for the monad. Explicitly,an X -indexed Eilenberg-Moore Q -algebra consists of a 1-cell F : X → Y and a 2-cell f : QF → F satisfying the axioms QQF Qf / / µ Q F (cid:15) (cid:15) QF f (cid:15) (cid:15) QF f / / F, F η Q F / / F - - QF f (cid:15) (cid:15) F. Again, these definitions extend to a 2-functor Q -Alg ( − ) : K →
Cat and there isa 2-natural transformation U ( − ) : Q -Alg ( − ) → K ( − , Y ), with components given bythe evident forgetful functors. Since ( Y, Q ) is assumed to be a monad, for every X ∈ K the forgetful functor U X : Q -Alg X → K ( X, Y ) has a left adjoint, defined bycomposition with Q : Y → Y . T. M. FIORE, N. GAMBINO, AND J. KOCK
A characterization of free monads.
We generalize the characterization of thefree monad on an endomorphism given by Staton in [22, Theorem 6.1.5] from the2-category
Cat to an arbitrary 2-category K . The generalization is essentiallystraightforward, but we indicate the main steps of the proof. See [1, § Theorem 1.1.
Let ( Y, Q ) be an endomorphism in a 2-category K . For a monad ( Y, Q ∗ ) and a 2-cell ι Q : Q → Q ∗ , the following conditions are equivalent.(i) The endomorphism map (1 Y , ι Q ) : ( Y, Q ∗ ) → ( Y, Q ) is universal, in the sensethat for every monad ( X, P ) , composition with (1 Y , ι Q ) induces an isomor-phism fitting in the diagram (1) Mnd K (( X, P ) , ( Y, Q ∗ )) ∼ = / / ) ) RRRRRRRRRRRRR
End K (( X, P ) , ( Y, Q )) v v lllllllllllll K ( X, Y ) , where the downward arrows are the evident forgetful functors.(ii) The 2-cell ν Q ∗ : QQ ∗ → Q ∗ , defined as the composite (2) QQ ∗ ι Q Q ∗ / / Q ∗ Q ∗ µ Q ∗ / / Q ∗ , equips Q ∗ with a universal Q -algebra structure, in the sense that for every X ∈ K , the functor K ( X, Y ) → Q - alg X defined by mapping F : X → Y to the Q -algebra with underlying 1-cell Q ∗ F and structure map the 2-cell ν Q ∗ F : QQ ∗ F → Q ∗ F , is left adjoint to the forgetful functor U X : Q - alg X →K ( X, Y ) .Proof. To see that (i) implies (ii), consider the following diagram: Q ∗ -Alg X ∼ = / / Q -alg X Mnd K (( X, X ) , ( Y, Q ∗ )) ) ) RRRRRRRRRRRRRR ∼ = / / End K (( X, X ) , ( Y, Q )) v v lllllllllllll K ( X, Y ) , where the bottom triangular diagram is an instance of the diagram in (1). Thefunctor defined in (ii) is left adjoint to the forgetful functor U X : Q -alg X → K ( X, Y )since it is exactly the composite of the left adjoint K ( X, Y ) → Q ∗ -Alg X , whichis given by composition with Q ∗ (since Q ∗ is a monad), with the isomorphism Q ∗ -Alg X → Q -alg X , which is defined by composition with ι Q .For the proof that (ii) implies (i), we need to define an isomorphism as in (1).Given an endomorphism map ( F, φ ) : (
X, P ) → ( Y, Q ), where φ : QF → F P , weneed to define a monad map (
F, φ ♯ ) : ( X, P ) → ( Y, Q ∗ ), where φ ♯ : Q ∗ F → F P .For this, we exploit the adjointness in (ii). Note that the left adjoint to Q -alg X →K ( X, Y ) sends F to the Q -algebra with underlying 1-cell Q ∗ F and structure map ν Q ∗ F : QQ ∗ F → Q ∗ F . Now, observe that the map QF P φP / / F P P
F µ P / / F P equips
F P with a Q -algebra structure. By adjointness, the map φ ♯ : Q ∗ F → F P is defined as the unique Q -algebra morphism such that the following diagram ONADS IN DOUBLE CATEGORIES 5 commutes F η Q ∗ F / / F η P , , Q ∗ F φ ♯ (cid:15) (cid:15) F P.
Note that saying that φ ♯ is a Q -algebra morphism amounts to saying that thefollowing diagram commutes QQ ∗ F Qφ ♯ / / ν Q ∗ F (cid:15) (cid:15) QF P φP (cid:15) (cid:15) F P P
F µ P (cid:15) (cid:15) Q ∗ F φ ♯ / / F P.
The isomorphism is defined as the identity on 2-cells. It remains to check thatwhat we have defined is indeed an inverse to the functor defined by compositionwith (1 Y , ι Q ), but the verification is essentially identical to the one given in detailin the proof of [22, Theorem 6.1.5] and hence we omit it. (cid:3) Definition 1.2.
A 2-category K is said to admit the construction of free monads iffor every endomorphism ( Y, Q ) there exists a monad (
Y, Q ∗ ) and a 2-cell ι Q : Q → Q ∗ satisfying the equivalent conditions of Theorem 1.1. Remark 1.3.
Let us point out that the universal property of the free monad(
Y, Q ∗ ) on an endomorphism ( Y, Q ) stated in item (i) of Theorem 1.1 includesthe assertion that for every monad (
X, P ) and every endomorphism map (
F, φ ) :(
X, P ) → ( Y, Q ), there exists a unique 2-cell φ ♯ : Q ∗ F → F P such that (
F, φ ♯ ) :( X, P ) → ( Y, Q ∗ ) is a monad map and the diagram QF ι Q F / / φ , , Q ∗ F φ ♯ (cid:15) (cid:15) F P commutes. From the statement in item (ii) of Theorem 1.1, it also follows thatif K is a 2-category that admits the construction of free monads and has localcoproducts, i.e. coproducts in its hom-categories, then for every F : X → Y , theinitial algebra for the endofunctor K ( X, Y ) → K ( X, Y )( − ) F + Q ( − )has Q ∗ F as its underlying object and the copair of the 2-cells η Q ∗ F : F → Q ∗ F and ν Q ∗ F : QQ ∗ F → Q ∗ F as its structure map.By the bicategorical Yoneda lemma [24], every bicategory is biequivalent to a2-category [9, Theorem 1.4]. Hence, the remarks and the results above can beapplied also to bicategories. We now introduce our two main classes of examples:bicategories of spans and bicategories of polynomials. T. M. FIORE, N. GAMBINO, AND J. KOCK
Example 1.4.
Let E be a category with finite limits. Recall that a span in E is adiagram of the form(3) F τ (cid:31) (cid:31) @@@@@@@ σ (cid:127) (cid:127) ~~~~~~~~ X Y, and that a span morphism is a commutative diagram of the form(4)
X F τ / / σ o o φ (cid:15) (cid:15) YX F ′ τ ′ / / σ ′ o o Y. We write
Span E for the bicategory of spans in E , originally defined in [2], which hasthe objects of E as 0-cells, spans as 1-cells and span morphisms as 2-cells. It is well-known that graphs and categories in E can be identified with endomorphisms andmonads in Span E [2, 3]. For our purposes, it is convenient to recall the definitionof the 2-category of linear functors over E , which is biequivalent to the bicategory Span E . Given a span as in (3), we define its associated linear functor to be thecomposite E /X σ ∗ / / E /F τ ! / / E /Y , where σ ∗ acts by pullback along σ and τ ! acts by composition with τ . In general,a functor between slices of E is said to be linear if it is naturally isomorphic to afunctor of this form. Now, recall from [8, § E are ten-sored over E and that linear functors have a canonical strength. The 2-category oflinear functors is then defined as the sub-2-category of Cat having slice categoriesof E as 0-cells, linear functors between them as 1-cells, and strong natural trans-formations as 2-cells, i.e. natural transformations compatible with the canonicalstrength on linear functors. Let us also recall that a strong natural transformationbetween linear functors is cartesian, i.e. its naturality squares are pullbacks. By thebiequivalence, graphs in E can be thought of as linear endofunctors and categoriesin E can be thought of as linear monads, i.e. monads whose underlying functor islinear and whose multiplication and unit are strong natural transformations. Example 1.5.
Let E be a locally cartesian closed category. Recall from [8, § E is a diagram of the form(5) ¯ F σ (cid:127) (cid:127) ~~~~~~~ θ / / F τ (cid:31) (cid:31) ???????? X Y and a cartesian morphism of polynomials is a diagram of the form X ¯ F (cid:15) (cid:15) / / o o _(cid:31) F / / YX ¯ F ′ / / o o F ′ / / Y, where the central square is a pullback. We write Poly E for the bicategory ofpolynomials over E , as defined in [8, § E as 0-cells,polynomials as 1-cells, and cartesian morphisms of polynomials as 2-cells. Working ONADS IN DOUBLE CATEGORIES 7 in the internal logic of E , for a polynomial as in (5) we may represent an element f ∈ F as an arrow f : ( x i | i ∈ I ) → y , where I = def θ − ( f ), the family ( x i | i ∈ I ) is defined by letting x i = def σ ( i ),for i ∈ I , and y = def τ ( f ). Thus, we think of the set I as the arity of the arrow f .The biequivalence between the bicategory of spans and the 2-category of linearfunctors extends to a biequivalence between the bicategory of polynomials and the2-category of polynomial functors [8, Theorem 2.17], as we now proceed to recall.For a polynomial as in (5), the polynomial functor associated to it is defined as thecomposite E /X σ ∗ / / E / ¯ F θ ∗ / / E /F τ ! / / E /Y , where θ ∗ is the right adjoint to the pullback functor θ ∗ . A functor between slicesof E is said to be polynomial if it is naturally isomorphic to a functor of thisform. Like linear functors, polynomial functors have a canonical strength and sowe can define the 2-category of polynomial functors as the sub-2-category of Cat having slices of E as 0-cells, polynomial functors as 1-cells and cartesian strongnatural transformations as 2-cells. The biequivalence between Poly E and the 2-category of polynomial functors allows us to identify endomorphisms and monadsin Poly E with polynomial endofunctors and polynomial monads on slices of E ,respectively, where by a polynomial monad we mean a monad whose underlyingendofunctor is polynomial and whose multiplication and unit are cartesian strongnatural transformations.Let us also recall from [19] that a locally cartesian closed category E is said tohave W-types if every polynomial endofunctor P : E → E has an initial algebra,called the W-type of the functor. Note that a polynomial functor P : E → E hasto be represented by a diagram as in (5) in which both X and Y are the terminalobject of E and hence is competely determined by the map θ . The category-theoreticnotion of a W-type is a counterpart of the notion of a type of wellfounded trees,originally introduced by Martin-L¨of within his dependent type theory [20]. Asshown in [7, Theorem 12], if E has disjoint coproducts, the assumption of W-typesis sufficient to show that, for all X ∈ E , every polynomial endofunctor P : E /X →E /X has an initial algebra. For further material and references on polynomialfunctors, see [8] and its bibliography.Proposition 1.6 provides the horizontal part of Proposition 3.8. Item (i) in itsstatement refers to the notion of a pretopos with parametrized list objects, forwhich we invite the reader to refer to [18]. Proposition 1.6. (i) If E is a pretopos with parametrized list objects, the bicategory Span E admitsthe construction of free monads.(ii) If E is a locally cartesian closed category with disjoint coproducts and W-types,the bicategory Poly E admits the construction of free monads.Proof. We begin by proving (ii). We exploit the biequivalence between the bicat-egory of polynomials and the 2-category of polynomial functors. Let Q : E /Y →E /Y be a polynomial endofunctor. We show that there is a polynomial monad Q ∗ : E /Y → E /Y and a cartesian strong natural transformation ι : Q → Q ∗ thatsatisfy the universal property in item (ii) of Theorem 1.1. By [7, Theorem 12], theassumption that E has W-types implies that the forgetful functor U : Q -alg → E /Y has a left adjoint. We let Q ∗ : E /Y → E /Y be the monad resulting from theadjunction. The monad Q ∗ : E /Y → E /Y is polynomial by [8, Theorem 4.5]. T. M. FIORE, N. GAMBINO, AND J. KOCK If Q : E /Y → E /Y is represented by the polynomial(6) ¯ Q (cid:127) (cid:127) (cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127) θ Q / / Q (cid:31) (cid:31) ???????? Y Y then Q ∗ : E /Y → E /Y is represented by the polynomial(7) ¯ Q ∗ (cid:127) (cid:127) ~~~~~~~~ θ Q ∗ / / Q ∗ @@@@@@@@ Y Y, where the object Q ∗ in (7) is described in the internal logic of E as the set of well-founded trees of profile Q , i.e. trees built up from identities and formal compositesof the arrows in Q . The map θ Q ∗ in (7) describes the arities of the arrows in Q ∗ in the evident way. The inclusion of the arrows in Q into those in Q ∗ is part of adiagram(8) Y ¯ Q θ Q / / o o (cid:15) (cid:15) _(cid:31) Q (cid:15) (cid:15) / / YY ¯ Q ∗ θ Q ∗ / / o o Q ∗ / / Y, which represents the required cartesian strong natural transformation ι : Q → Q ∗ .A direct verification shows that the left adjoint to U : Q -alg → E /Y maps anobject A to the Q -algebra with underlying object Q ∗ A and structure map ν A : QQ ∗ A → Q ∗ A , where ν Q ∗ : QQ ∗ → Q ∗ is defined as in (2). To conclude theproof of item (ii) it is sufficient to observe that, for X ∈ E , the category Q -alg X isequivalent to the category of polynomial functors F : E /X → E /Y equipped witha cartesian strong natural transformation φ : QF → F .The proof of item (i) is similar, except that polynomial functors are replacedby linear functors. In this case, the assumption of W-types can be replaced bythat of parametrized list objects, which suffice to prove the existence of the leftadjoint to the forgetful functor U : Q -alg → E /Y and that the resulting monad Q ∗ : E /Y → E /Y is linear. This is because linear endofunctors (respectively, linearmonads) are just graphs (respectively, categories) internal to E , and, as shownin [18, Proposition 7.3], the assumption of parametrized list objects guarantees theexistence of the free category on a graph in E . (cid:3) If E is a locally cartesian closed pretopos with W-types, then it has list objectsand these are parametrized since we are in a cartesian closed category. Hence, such acategory satisfies the hypotheses of both item (i) and item (ii) of Proposition 1.6. Inthis case, the construction of the free monad for polynomial endofunctors generalizesthe construction of the free monad for linear endofunctors.2. Monads in a double category
Notation and preliminaries.
We assume readers to be familiar with the basicconcepts of the theory of double categories (see [4] for the original reference and [6,10, 11] for modern accounts) and limit ourselves to introducing some notation andrecalling some basic notions. For a double category C , we write Obj C for its classof objects, Hor C for its class of horizontal arrows, Ver C for its class of vertical ONADS IN DOUBLE CATEGORIES 9 arrows and Sq C for its class of squares. We write C for the category of objectsand vertical arrows and C for the category of horizontal arrows and squares. Weallow horizontal composition to be associative and unital up to coherent invertiblesquares rather than strictly. For the sake of readability, however, we shall work asif horizontal composition were strict, as allowed by [10, Theorem 7.5]. Typically, asquare will be written as follows:(9) X F / / u (cid:15) (cid:15) α Y v (cid:15) (cid:15) X ′ F ′ / / Y ′ . Identity squares will be written without a label, as follows: X F / / YX F / / Y, X u (cid:15) (cid:15) X u (cid:15) (cid:15) X ′ X ′ . For a double category C , its horizontal 2-category H C is defined as follows: the0-cells are the objects of C , the 1-cells are the horizontal arrows of C and the 2-cellsare the squares of the form X F / / α YX F ′ / / Y. The notions of horizontal adjunction and vertical adjunction between double cate-gories can be defined using the general notion of an adjunction in a 2-category [14].A horizontal adjunction is an adjunction in the 2-category of double categories, dou-ble functors and horizontal natural transformations; vertical adjunctions are definedanalogously, replacing horizontal natural transformations with vertical ones [10].
Example 2.1.
Let E be a category with finite limits. With a minor abuse ofnotation, we write Span E also for the double category of spans in E , which hasobjects of E as objects, spans as horizontal arrows, maps of E as vertical arrowsand diagrams of the form X u (cid:15) (cid:15) F τ / / φ (cid:15) (cid:15) σ o o Y v (cid:15) (cid:15) X ′ F ′ σ ′ o o τ ′ / / Y ′ as squares. Note that the horizontal bicategory of this double category is exactlythe bicategory of spans in E defined in Example 1.4. Example 2.2.
Let E be a locally cartesian closed category. With another abuseof notation, we write Poly E also for the double category of polynomials over E ,which has the objects of E as objects, polynomials as horizontal arrows, maps of E as vertical arrows and diagrams of the form X u (cid:15) (cid:15) ¯ F σ o o _(cid:31) θ / / (cid:15) (cid:15) F τ / / φ (cid:15) (cid:15) Y v (cid:15) (cid:15) X ′ ¯ F ′ σ ′ o o θ ′ / / F ′ τ ′ / / Y ′ , where the central square is a pullback, as squares. The bicategory of polynomialsdefined in Example 1.5 is the horizontal bicategory of this double category. The double categories of endomorphisms and monads.
Below, we definethe double category Mnd( C ) of monads in a double category C . After giving thedefinition, we explain how it generalizes the definition of the 2-category Mnd( K )of monads in a 2-category K . In view of our applications, we begin by introducingthe double category End( C ) of endomorphisms in a double category C . Definition 2.3.
Let C be a double category.(i) A horizontal endomorphism is a pair ( X, P ) consisting of an object X anda horizontal arrow P : X → X . Since we consider only horizontal endomor-phisms, we refer to them simply as endomorphisms.(ii) A horizontal endomorphism map ( F, φ ) : (
X, P ) → ( Y, Q ) consists of a hori-zontal arrow F : X → Y and a square X F / / φ Y Q / / YX P / / X F / / Y. (iii) A vertical endomorphism map ( u, ¯ u ) : ( X, P ) → ( X ′ , P ′ ) consists of a verticalarrow u : X → X ′ and a square X P / / u (cid:15) (cid:15) ¯ u X u (cid:15) (cid:15) X ′ P ′ / / X ′ . (iv) An endomorphism square ( X, P ) ( F,φ ) / / ( u, ¯ u ) (cid:15) (cid:15) α ( Y, Q ) ( v, ¯ v ) (cid:15) (cid:15) ( X ′ , P ′ ) ( F ′ ,φ ′ ) / / ( Y ′ , Q ′ )is a square X F / / u (cid:15) (cid:15) α Y v (cid:15) (cid:15) X ′ F ′ / / Y ′ satisfying the condition X / / φ Y / / YX / / (cid:15) (cid:15) ¯ u X / / (cid:15) (cid:15) α Y (cid:15) (cid:15) X ′ / / X ′ / / Y ′ = X / / (cid:15) (cid:15) α Y / / (cid:15) (cid:15) ¯ v Y (cid:15) (cid:15) X ′ / / φ ′ Y ′ / / Y ′ X ′ / / X ′ / / Y ′ . We write End( C ) for the double category of endomorphisms, horizontal endo-morphism maps, vertical endomorphism maps and endomorphism squares. We omitthe straightforward verification that End( C ) is indeed a double category. ONADS IN DOUBLE CATEGORIES 11
Definition 2.4.
Let C be a double category.(i) A monad is an endomorphism ( X, P ) equipped with squares X P / / µ P X P / / XX P / / X X η P XX P / / X satisfying the associativity law X / / µ P X / / X / / XX / / µ P X / / XX / / X = X / / X / / µ P X / / XX / / µ P X / / XX / / X and the unit laws X / / X η P XX / / µ P X / / XX / / X = X / / XX / / X = X η P X / / XX / / µ P X / / XX / / X. As before, we refer to a monad as above by mentioning only its underlyingendomorphism (
X, P ).(ii) A horizontal monad map ( F, φ ) : (
X, P ) → ( Y, Q ) is a horizontal endomor-phism map between the underlying endomorphisms satisfying the followingconditions: X / / Y / / µ Q Y / / YX / / φ Y / / YX / / X / / Y = X / / φ Y / / Y / / YX / / X / / φ Y / / YX / / µ P X / / X / / YX / / X / / YX / / Y η Q YX / / φ Y / / YX / / X / / Y = X η P X / / YX / / X / / Y. (iii) A vertical monad map ( u, ¯ u ) : ( X, P ) → ( X ′ , P ′ ) is a vertical endomorphismmap between the underlying endomorphisms satisfying the following condi-tions: X / / µ P X / / XX / / (cid:15) (cid:15) ¯ u X (cid:15) (cid:15) X ′ / / X ′ = X / / (cid:15) (cid:15) ¯ u X / / (cid:15) (cid:15) ¯ u X (cid:15) (cid:15) X ′ / / µ P ′ X ′ / / X ′ X ′ / / X ′ X η P XX / / (cid:15) (cid:15) ¯ u X (cid:15) (cid:15) X ′ / / X ′ = X (cid:15) (cid:15) X (cid:15) (cid:15) X ′ η P ′ X ′ X ′ / / X ′ . (iv) A monad square is an endomorphism square between the underlying endo-morphism maps.We write Mnd( C ) for the double category of monads, horizontal monad maps,vertical monad maps and monad squares; again, it is straightforward to checkthat Mnd( C ) is a double category. Before giving examples, we clarify the relation-ship between our definitions and those in [23]. Remark 2.5.
Let K be a 2-category and consider the double category H ( K ),that has K as its horizontal 2-category and only identity 1-cells as vertical arrows.Monads in K are the same as monads in H ( K ) and monad maps in K are the sameas horizontal monad maps in H ( K ). Finally, monad 2-cells in K are the same asmonad squares in H ( K ) of the special form( X, P ) ( F,φ ) / / α ( Y, Q )( X, P ) ( F ′ ,φ ′ ) / / ( Y, Q ) . In particular, the horizontal 2-category of Mnd( H ( K )) is the 2-category Mnd( K )of [23]. As we explain in the following examples, the presence of non-trivial verticalarrows in a double category allows us to describe important mathematical structuresas vertical monad maps. Example 2.6.
Let E be a category with finite limits. The category Grph E ofgraphs and graph morphisms internal to E can be identified with the category ofendomorphisms and vertical endomorphism maps in the double category Span E ,while the category Cat E of categories and functors internal to E can be identifiedwith the category of monads and vertical monad maps in Span E . We see here anexample of the benefits of considering monads in a double category rather than in a2-category: while categories can be seen as monads in the bicategory of spans in E ,functors between categories are not the same as monad maps in that bicategory. Example 2.7.
Let E be a locally cartesian closed category with finite disjoint co-products and W-types. We write PolyEnd E for the category of endomorphisms andvertical endomorphism maps in the double category Poly E and write PolyMnd E ONADS IN DOUBLE CATEGORIES 13 for the category of monads and vertical monad maps in
Poly E . If M : E → E isthe free monoid monad in E (which exists by the assumptions on E ), then there isa double category PolyEnd E /M whose objects are endomorphisms with a verticalendomorphism map to M . This is the double category of M -spans in the senseof [3] and [17], while PolyMnd E is the double category of multicategories. Thefree monad on an endofunctor over M is the free multicategory on an M -span.Furthermore, the vertical maps in PolyMnd E are the multifunctors, and hence wesee again the benefits of considering monads in the double categories rather thanjust in 2-categories. Further variations are possible: with a polynomial monad T inthe place of M we get the same result for T -spans and T -multicategories, and in theparticular case where T is the identity monad, we are back to just plain categoriesin E .The function sending a monad ( X, P ) to its underlying object X extends to adouble functor Und : Mnd( C ) → C and the function mapping an object X ∈ C to the identity monad ( X, X ) extends to a double functor Inc : C → Mnd( C ). Itis easy to check that Inc is a horizontal right adjoint to Und, essentially as in the2-categorical formal theory of monads [23, Theorem 1]. The question of when C admits the construction of Eilenberg-Moore objects, that is, of when the doublefunctor Inc has a horizontal right adjoint, will be treated in a sequel to this paper.Here, instead, we focus on the construction of free monads. Free monads in a double category.
We write U : Mnd( C ) → End( C ) for theforgetful double functor mapping a monad to its underlying endomorphism. Definition 2.8.
A double category C is said to admit the construction of freemonads if U : Mnd( C ) → End( C ) has a vertical left adjoint. Remark 2.9.
We now make explicit what it means for a double category C toadmit the construction of free monads. By an analogue of the characterization ofordinary adjunctions in terms of universal arrows [16, Theorem IV.2], to give avertical left adjoint to U amounts to giving the following data in (i)-(iv) satisfyingthe functoriality condition in ( ∗ ).(i) For every endomorphism ( X, P ), a monad ( X ∗ , P ∗ ).(ii) For every endomorphism ( X, P ), a universal vertical endomorphism map( ι X , ι P ) : ( X, P ) → ( X ∗ , P ∗ ) . Universality means that for each vertical endomorphism map ( u, ¯ u ) : ( X, P ) → ( X ′ , P ′ ), where ( X ′ , P ′ ) is a monad, there exists a unique vertical monad map( u ♯ , ¯ u ♯ ) : ( X ∗ , P ∗ ) → ( X ′ , P ′ ) such that X P / / u (cid:15) (cid:15) ¯ u X u (cid:15) (cid:15) X ′ P ′ / / X ′ = X P / / ι X (cid:15) (cid:15) ι P X ι X (cid:15) (cid:15) X ∗ / / u ♯ (cid:15) (cid:15) ¯ u ♯ X ∗ u ♯ (cid:15) (cid:15) X ′ P ′ / / X ′ . (iii) For every horizontal endomorphism map ( F, φ ) : (
X, P ) → ( Y, Q ), a horizontalmonad map ( F ∗ , φ ∗ ) : ( X ∗ , P ∗ ) → ( Y ∗ , Q ∗ ). (iv) For every horizontal endomorphism map ( F, φ ) : (
X, P ) → ( Y, Q ), a universalendomorphism square (
X, P ) ( F,φ ) / / ( ι X ,ι P ) (cid:15) (cid:15) ι ( F,φ ) ( Y, Q ) ( ι Y ,ι Q ) (cid:15) (cid:15) ( X ∗ , P ∗ ) ( F ∗ ,φ ∗ ) / / ( Y ∗ , Q ∗ ) . Universality means that for every endomorphism square(10) (
X, P ) ( F,φ ) / / ( u, ¯ u ) (cid:15) (cid:15) α ( Y, Q ) ( v, ¯ v ) (cid:15) (cid:15) ( X ′ , P ′ ) ( F ′ ,φ ′ ) / / ( Y ′ , Q ′ ) , where ( X ′ , P ′ ), ( Y ′ , Q ′ ) are monads and ( F, φ ′ ) : ( X ′ , P ′ ) → ( Y ′ , Q ′ ) is ahorizontal monad map, there exists a unique monad square( X ∗ , P ∗ ) ( F ∗ ,φ ∗ ) / / ( u ♯ , ¯ u ♯ ) (cid:15) (cid:15) α ♯ ( Y ∗ , Q ∗ ) ( v ♯ , ¯ v ♯ ) (cid:15) (cid:15) ( X ′ , P ′ ) ( F ′ ,φ ′ ) / / ( Y ′ , Q ′ )such that( X, P ) ( F,φ ) / / ( u, ¯ u ) (cid:15) (cid:15) α ( Y, Q ) ( v, ¯ v ) (cid:15) (cid:15) ( X ′ , P ′ ) ( F ′ ,φ ′ ) / / ( Y ′ , Q ′ ) = ( X, P ) ( F,φ ) / / ( ι X ,ι P ) (cid:15) (cid:15) ι ( F,φ ) ( Y, Q ) ( ι Y ,ι Q ) (cid:15) (cid:15) ( X ∗ , P ∗ ) / / ( u ♯ , ¯ u ♯ ) (cid:15) (cid:15) α ♯ ( Y ∗ , Q ∗ ) ( v ♯ , ¯ v ♯ ) (cid:15) (cid:15) ( X ′ , P ′ ) ( F ′ ,φ ′ ) / / ( Y ′ , Q ′ ) . ( ∗ ) The assignments in (i) and (iii) give a functor( − ) ∗ : (cid:0) Obj
End( C ) , Hor
End( C ) (cid:1) → (cid:0) Obj
Mnd( C ) , Hor
Mnd( C ) (cid:1) and the assignments in (ii) and (iv) give a functor ι : (cid:0) Obj
End( C ) , Hor
End( C ) (cid:1) → (cid:0) Ver
End( C ) , Sq End( C ) (cid:1) . Note that the data and the universality in (ii) actually follow from the data andthe universality in (iv) by taking (
F, φ ) to be the horizontal identity on an endo-morphism (
X, P ).A necessary condition for U : Mnd( C ) → End( C ) to have a vertical left adjointis that its vertical part(11) U : Mnd( C ) → End( C ) has a left adjoint. Indeed, this is precisely what items (i) and (ii) of Remark 2.9amount to. Here, End( C ) denotes the category of endomorphisms and verticalendomorphism maps and Mnd( C ) denotes the category of monads and verticalmonad maps. ONADS IN DOUBLE CATEGORIES 15
Example 2.10.
Let E be a category with finite limits. The functor in (11) for thedouble category Span E is the forgetful functor U : Cat E → Grph E mapping acategory in E to its underlying graph. Example 2.11.
Let E be a locally cartesian closed category. The functor in (11) forthe double category Poly E is the forgetful functor U : PolyMnd E → PolyEnd E mapping a polynomial monad to its underlying endofunctor.3. Monads in a framed bicategory
We now proceed to establish some properties of the double category Mnd( C )under the assumption that C is a framed bicategory, leading to our main theorem(Theorem 3.7 below), which provides conditions for C to admit the construction offree monads. We begin by recalling from [21] the definition of a framed bicategoryand some useful facts. Framed bicategories.
For a double category C , the functor( ∂ , ∂ ) : C → C × C , mapping a horizontal arrow F : X → Y to ( X, Y ) and a square as in (9) to ( u, v ) :(
X, Y ) → ( X ′ , Y ′ ), is a Grothendieck fibration if and only if it is a Grothendieckopfibration [21, Theorem 4.1]. When these conditions hold, the double category C is said to be a framed bicategory [21, Definition 4.2]. As explained in [21, Ex-amples 4.4] and [8, Proposition 3.6], the double categories Span E and Poly E areframed bicategories. Lemma 3.1 (Shulman) . If C is a framed bicategory, for every vertical arrow u : X → X ′ there exist horizontal arrows u ! : X → X ′ and u ∗ : X ′ → X to-gether with squares X u ! / / u (cid:15) (cid:15) α u X ′ X ′ X ′ X ′ u ∗ / / β u X u (cid:15) (cid:15) X ′ X ′ X u (cid:15) (cid:15) γ u XX ′ u ∗ / / X X δ u X u (cid:15) (cid:15) X u ! / / X ′ satisfying the equalities X δ u X u (cid:15) (cid:15) X u (cid:15) (cid:15) / / α u X ′ X ′ X ′ = X u (cid:15) (cid:15) X u (cid:15) (cid:15) X ′ X ′ = X u (cid:15) (cid:15) γ u XX ′ / / β u X u (cid:15) (cid:15) X ′ X ′ ,X δ u X / / (cid:15) (cid:15) α u X ′ X / / X ′ X ′ = X u ! / / X ′ X u ! / / X ′ and X ′ / / β u X (cid:15) (cid:15) γ u XX ′ X ′ / / X = X ′ u ∗ / / XX ′ u ∗ / / X. Proof.
See [21, Theorem 4.1]. (cid:3)
Lemma 3.1 can be expressed equivalently by saying that every vertical arrow u in C has an orthogonal companion u ! and an orthogonal adjoint u ∗ in the termi-nology of [11]. Lemma 3.2 (Shulman) . Let C be a framed bicategory. Let u : X → X ′ be a verticalarrow in C . If we define X η u XX u ! / / X ′ u ∗ / / X = def X δ u X (cid:15) (cid:15) γ u XX u ! / / X ′ u ∗ / / X and X ′ u ∗ / / ε u X u ! / / X ′ X ′ X ′ = def X ′ u ∗ / / β u X u ! / / (cid:15) (cid:15) α u X ′ X ′ X ′ X ′ , then the following versions of the triangle identities hold: X ′ u ∗ / / X η u X XX ′ / / ε u X / / X ′ / / XX ′ X ′ u ∗ / / X = X ′ u ∗ / / XX ′ u ∗ / / X,X η u X u ! / / X ′ X / / X ′ / / ε u X / / XX u ! / / X X = X u ! / / X ′ X u ! / / X ′ . Proof.
See [21, Proposition 5.3]. (cid:3)
Monads in framed bicategories.
Let C be a double category. We have thediagram(12) Mnd( C ) U / / ∂ M $ $ IIIIIIIII
End( C ) ∂ E z z vvvvvvvvv C , ONADS IN DOUBLE CATEGORIES 17 where ∂ E and ∂ M send an endomorphism and a monad, respectively, to their un-derlying object and U is the vertical part of the forgetful double functor U ofDefinition 2.8. Proposition 3.3. If C is a framed bicategory, the functors ∂ E : End( C ) → C , ∂ M : Mnd( C ) → C are Grothendieck fibrations and the functor U : Mnd( C ) → End( C ) is a fiberedfunctor relatively to these fibrations.Proof. Writing ∆ : C → C × C for the diagonal functor, the functor ∂ E fits intothe pullback diagram End( C ) / / ∂ E (cid:15) (cid:15) C ∂ ,∂ ) (cid:15) (cid:15) C / / C × C . We then have that ∂ E is a Grothendieck fibration since it is a pullback of ( ∂ , ∂ ),which is a Grothendieck fibration by the hypothesis that C is a framed bicategory.Using Lemmas 3.1 and 3.2, we can define explicitly a base change operation for theGrothendieck fibration ∂ E , as follows. Let u : X → X ′ be a map in C and ( X ′ , P ′ )an endomorphism in C . The base change of ( X ′ , P ′ ) along u is defined to be theendomorphism ( X, P ), where P : X → X is the composite X u ! / / X ′ P ′ / / X ′ u ∗ / / X .
The required cartesian morphism from (
X, P ) to ( X ′ , P ′ ) in End( C ) ( i.e. thecartesian lift of u ) is given by the vertical endomorphism map ( u, ¯ u ) : ( X, P ) → ( X ′ , P ′ ), where ¯ u is the square X u ! / / u (cid:15) (cid:15) α u X ′ P ′ / / X ′ u ∗ / / β u X u (cid:15) (cid:15) X ′ X ′ P ′ / / X ′ X ′ . The verification of the required universal property is straightforward.To show that ∂ M is a Grothendieck fibration, we first observe that if ( X ′ , P ′ ) isa monad, then ( X, P ) inherits a monad structure: its multiplication is the square X u ! / / X ′ P ′ / / X ′ u ∗ / / ε u X u ! / / X ′ P ′ / / X ′ u ∗ / / XX / / X ′ P ′ / / µ X ′ X ′ P ′ / / X ′ / / XX u ! / / X ′ P ′ / / X ′ u ∗ / / X and its unit is the square X η u X X XX u ! / / X ′ η X ′ u ∗ / / XX u ! / / X ′ P ′ / / X ′ u ∗ / / X. The monad axioms are easily verified. Now it only remains to verify that thecartesian lift ( u, ¯ u ) is a vertical monad map and that it is cartesian for ∂ M . Thisverification is straightforward, using Lemmas 3.1 and 3.2. This also shows that U is fibered as claimed. (cid:3) Lemma 3.4.
Let ( X, P ) and ( X ′ , P ′ ) be endomorphisms in a framed bicategory C .There is a bijection between vertical endomorphism maps ( u, ¯ u ) : ( X, P ) → ( X ′ , P ′ ) and horizontal endomorphism maps of the form ( u ∗ , φ ) : ( X ′ , P ′ ) → ( X, P ) , whichrestricts to a bijection between vertical monad maps and horizontal monad mapswhen ( X, P ) and ( X ′ , P ′ ) are monads.Proof. For a vertical endomorphism map ( u, ¯ u ) : ( X, P ) → ( X ′ , P ′ ), define thehorizontal endomorphism map ( u ∗ , φ u ) : ( X ′ , P ′ ) → ( X, P ) by letting φ u be thesquare X ′ u ∗ / / φ u X P / / XX ′ P ′ / / X ′ u ∗ / / X = def X ′ u ∗ / / β u X P / / (cid:15) (cid:15) ¯ u X (cid:15) (cid:15) γ u XX ′ X ′ P ′ / / X ′ u ∗ / / X. In the other direction, given a horizontal endomorphism map ( u ∗ , φ ) : ( X ′ , P ′ ) → ( X, P ),define the vertical endomorphism map ( u, ¯ u φ ) : ( X, P ) → ( X ′ , P ′ ) by letting ¯ u φ bethe square X P / / u (cid:15) (cid:15) ¯ u φ X u (cid:15) (cid:15) X ′ P ′ / / X ′ = def X u (cid:15) (cid:15) γ u X P / / XX ′ / / φ X / / XX ′ / / X ′ / / β u X u (cid:15) (cid:15) X ′ P ′ / / X ′ X ′ . Using Lemma 3.1, it is possible to show that these functions are mutually inverse,that ( u ∗ , φ u ) is a horizontal monad map if ( u, ¯ u ) is a vertical monad map, andthat ( u, ¯ u φ ) is a vertical monad map if ( u ∗ , φ ) is a horizontal monad map. (cid:3) Let us point out that the bijection defined in the proof of Lemma 3.4 is anexample of a cofolding in the sense of [5, Definition 3.16].
Free monads in a framed bicategory.
We now consider the construction of freemonads in a framed bicategory C . Since the functor U in (12) is fibered, a sufficientcondition for it to have a left adjoint is that each of its fibers has a left adjoint. Inthis case, the free monad on an endomorphism ( X, P ) has the form (
X, P ∗ ) and ONADS IN DOUBLE CATEGORIES 19 the component of the unit ( ι X , ι P ) : ( X, P ) → ( X, P ∗ ) is a vertical endomorphismmap of the form (1 X , ι P ) : ( X, P ) → ( X, P ∗ ), where ι P is a square of the form X P / / ι P XX P ∗ / / X. The universal property in the fiber asserts that for every endomorphism square ofthe form X P / / α XX P ′ / / X, where ( X, P ′ ) is a monad, there exists a unique monad square of the form X P ∗ / / α ♯ XX P ′ / / X such that X P / / α XX P ′ / / X = X P / / ι P XX / / α ♯ XX P ′ / / X. The universal property in the fiber implies a more general universal property, withrespect to general monads (and not just monads with X as underlying object)and general vertical endomorphism maps (and not just the special ones consideredabove), as in item (ii) of Remark 2.9. Note, however, that the left adjoint to U constructed from the left adjoints to its fibers need not be a fibered left adjoint,since the so-called Beck-Chevalley conditions are not necessarily satisfied [12, § Example 3.5.
Let us consider the framed bicategory
Span E associated to a cat-egory E with finite limits. The diagram in (12) becomes Cat E U / / ∂ M " " DDDDDDDD
Grph E ∂ E { { xxxxxxxxx E , where ∂ E sends a graph to its object of vertices and ∂ M sends a small category toits object of objects. Since Span E is a framed bicategory, the preceding remarksreduce to the familiar fact that the free category on a graph has the object ofvertices of the graph as its object of objects. Example 3.6.
Let us consider the framed bicategory
Poly E associated to a locallycartesian closed category E with finite disjoint coproducts. For Poly E , the diagram in (12) amounts to: PolyMnd E U / / ∂ M % % KKKKKKKKKK
PolyEnd E ∂ E y y tttttttttt E . In this case, the remarks above amount to the fact, exploited in the proof of [8,Corollary 4.7], that to prove the universal property of the free monad on a poly-nomial endofunctor with respect to maps in
PolyEnd E , it is sufficient to check itwith respect to a special class of them.Theorem 3.7, which is our main result, gives sufficient conditions for a framedbicategory to admit the construction of free monads, facilitating the verification ofthis property in our examples. Recall that, for a double category C , we write C for its category of objects and vertical arrows and C for its category of horizontalarrows and squares. Theorem 3.7.
Let C be a framed bicategory such that the category C has equalizersand the source and target functors ∂ , ∂ : C → C preserve them. If the horizontal2-category of C has local coproducts and admits the construction of free monads,then C admits the construction of free monads. The proof of Theorem 3.7 is given in Section 4. Here, instead, we apply it to ourtwo running examples.
Proposition 3.8. (i) If E is a pretopos with parametrized list objects, the double category Span E admits the construction of free monads.(ii) If E is a locally cartesian closed category with disjoint coproducts and W-types,the double category Poly E admits the construction of free monads.Proof. For both (i) and (ii), we apply Theorem 3.7. For (i), the hypotheses onequalizers are verified because in this case the category C is a category of internalpresheaves, in which equalizers exist and are computed pointwise. For (ii), thehypotheses on equalizers are also satisfied, since pullbacks preserve equalizers. Forboth items, the existence of free monads in the horizontal 2-categories is establishedin Proposition 1.6. (cid:3) Proof of the Main Theorem
Let C be a double category satisfying the hypotheses of Theorem 3.7. We usethe characterization of free monads in a 2-category given in Theorem 1.1 to exhibitthe data listed in Remark 2.9. For items (i) and (ii) of Remark 2.9, let ( X, P ) be anendomorphism. By the existence of free monads in H C , we have a monad ( X, P ∗ )and a square X P / / ι P XX P ∗ / / X satisfying the equivalent conditions in items (i) and (ii) of Theorem 1.1 in H C . Wethen obtain a vertical endomorphism map (1 X , ι P ) : ( X, P ) → ( X, P ∗ ). We need toshow that (1 X , ι P ) enjoys the required universal property. For this, let us consider a ONADS IN DOUBLE CATEGORIES 21 vertical endomorphism map ( u, ¯ u ) : ( X, P ) → ( X ′ , P ′ ), where ( X ′ , P ′ ) is a monad.Here, ¯ u is a square of the form X P / / u (cid:15) (cid:15) ¯ u X u (cid:15) (cid:15) X ′ P ′ / / X ′ . By the cofolding bijection defined in the proof of Lemma 3.4, we have a horizontalendomorphism map ( u ∗ , φ u ) : ( X ′ , P ′ ) → ( X, P ), where φ u is a square of the form X ′ u ∗ / / φ u X P / / XX ′ P ′ / / X ′ u ∗ / / X. By the universal property in item (i) of Theorem 1.1 for (
X, P ∗ ), there exists aunique square X ′ u ∗ / / φ ♯u X P ∗ / / XX ′ P ′ / / X ′ u ∗ / / X such that ( u ∗ , φ ♯u ) : ( X ′ , P ′ ) → ( X, P ∗ ) is a horizontal monad map and X ′ u ∗ / / φ u X P / / XX ′ P ′ / / X ′ u ∗ / / X = X ′ u ∗ / / X P / / ι P XX ′ / / φ ♯u X / / XX ′ P ′ / / X ′ u ∗ / / X. Using again the cofolding bijection of Lemma 3.4, we obtain the vertical monadmorphism ( u, ¯ u ♯ ) : ( X, P ∗ ) → ( X ′ , P ′ ) that factors ( u, ¯ u ) through (1 X , ι P ), asrequired. By the definition of the bijection and Theorem 1.1, the square ¯ u ♯ satisfiesthe equations(13) X η P ∗ XX / / u (cid:15) (cid:15) ¯ u ♯ X u (cid:15) (cid:15) X ′ P ′ / / X ′ = X u (cid:15) (cid:15) X u (cid:15) (cid:15) X ′ η P ′ X ′ X ′ P ′ / / X ′ and(14) X P ∗ / / ν P ∗ X P / / XX / / (cid:15) (cid:15) ¯ u ♯ X (cid:15) (cid:15) X ′ P ′ / / Y = X P ∗ / / (cid:15) (cid:15) ¯ u ♯ X P / / (cid:15) (cid:15) ¯ u X (cid:15) (cid:15) X ′ / / µ P ′ X ′ / / X ′ X ′ P ′ / / X ′ , where the square ν P ∗ is defined by X P ∗ / / ν P ∗ X P / / XX P ∗ / / X = def X P ∗ / / X P / / ι P XX / / µ P ∗ X / / XX P ∗ / / X. For item (iii) of Remark 2.9, let (
F, φ ) : (
X, P ) → ( Y, Q ) be a horizontal endo-morphism map. Exploiting the universal property in item (i) of Theorem 1.1 for(
Y, Q ∗ ), we define X F / / φ ∗ Y Q ∗ / / YX P ∗ / / X F / / X to be the unique square such that ( F, φ ∗ ) : ( X, P ∗ ) → ( Y, Q ∗ ) is a horizontal monadmap and(15) X F / / φ Y Q / / YX / / ι P X / / YX P ∗ / / X F / / Y = X F / / Y Q / / ι Q YX / / φ ∗ Y / / YX P ∗ / / X F / / X. Observe that, by the fact that (
F, φ ∗ ) is a horizontal monad map, we have that(16) X F / / Y η Q ∗ YX / / φ ∗ Y / / YX P ∗ / / X F / / Y = X η P ∗ X F / / YX P ∗ / / X F / / Y ONADS IN DOUBLE CATEGORIES 23 and(17) X F / / Y ν Q ∗ Q ∗ / / Y Q / / YX / / φ ∗ Y / / YX P ∗ / / X F / / Y = X F / / Y φ ∗ Q ∗ / / Y Q / / YX / / X / / Y φ / / YX / / ν P ∗ X / / X / / YX P ∗ / / X F / / Y. In particular, (17) holds by the definitions of ν P ∗ and ν Q ∗ , the first axiom for ahorizontal monad map and (15). For item (iv) of Remark 2.9, the required universalendomorphism square needs to have the form( X, P ) ( F,φ ) / / (1 X ,ι P ) (cid:15) (cid:15) ι ( F,φ ) ( Y, Q ) (1 Y ,ι Q ) (cid:15) (cid:15) ( X, P ∗ ) ( F,φ ∗ ) / / ( Y, Q ∗ ) . Therefore, ι ( F,φ ) has to be a square in C of the form X F / / ι ( F,φ ) YX F / / Y and satisfy the equation(18) X F / / φ Y Q / / YX / / ι P X / / ι ( F,φ ) YX P ∗ / / X F / / Y = X F / / ι ( F,φ ) Y Q / / ι Q YX / / φ ∗ Y / / YX P ∗ / / X F / / Y. We define ι ( F,φ ) to be the identity square on F , so that (18) above is verified by (15).To verify the universal property, we need to show that for an endomorphism square( X, P ) ( F,φ ) / / ( u, ¯ u ) (cid:15) (cid:15) α ( Y, Q ) ( v, ¯ v ) (cid:15) (cid:15) ( X ′ , P ′ ) ( F ′ ,φ ′ ) / / ( Y ′ , Q ′ ) , there exists a unique monad square( X, P ∗ ) ( F,φ ∗ ) / / ( u, ¯ u ♯ ) (cid:15) (cid:15) α ♯ ( Y, Q ∗ ) ( v, ¯ v ♯ ) (cid:15) (cid:15) ( X ′ , P ′ ) ( F ′ ,φ ′ ) / / ( Y ′ , Q ′ )satisfying(19) ( X, P ) ( F,φ ) / / ( u, ¯ u ) (cid:15) (cid:15) α ( Y, Q ) ( v, ¯ v ) (cid:15) (cid:15) ( X ′ , P ′ ) ( F ′ ,φ ′ ) / / ( Y ′ , Q ′ ) = ( X, P ) ( F,φ ) / / (1 X ,ι P ) (cid:15) (cid:15) ι ( F,φ ) ( Y, Q ) (1 Y ,ι Q ) (cid:15) (cid:15) ( X, P ∗ ) / / ( u, ¯ u ♯ ) (cid:15) (cid:15) α ♯ ( Y, Q ∗ ) ( v, ¯ v ♯ ) (cid:15) (cid:15) ( X ′ , P ′ ) ( F ′ ,φ ′ ) / / ( Y ′ , Q ′ ) . First of all, observe that α is a square in C of the form X F / / u (cid:15) (cid:15) α Y v (cid:15) (cid:15) X ′ F ′ / / Y ′ which satisfies the compatibility condition(20) X F / / φ Y Q / / YX / / (cid:15) (cid:15) ¯ u X (cid:15) (cid:15) / / α Y (cid:15) (cid:15) X ′ P ′ / / X ′ F ′ / / Y ′ = X F / / (cid:15) (cid:15) α Y Q / / ¯ v (cid:15) (cid:15) Y (cid:15) (cid:15) X ′ / / φ ′ Y ′ / / Y ′ X ′ P ′ / / X ′ F ′ / / Y ′ . The required monad square α ♯ has to be a square in C of the form X F / / u (cid:15) (cid:15) α ♯ Y v (cid:15) (cid:15) X ′ F ′ / / Y ′ satisfying the compatibility condition(21) X F / / φ ∗ Y Q ∗ / / YX / / (cid:15) (cid:15) ¯ u ♯ X (cid:15) (cid:15) / / α ♯ Y (cid:15) (cid:15) X ′ P ′ / / X ′ F ′ / / Y ′ = X F / / (cid:15) (cid:15) α ♯ Y Q ∗ / / ¯ v ♯ (cid:15) (cid:15) Y (cid:15) (cid:15) X ′ / / φ ′ Y ′ / / Y ′ X ′ P ′ / / X ′ F ′ / / Y ′ . We define α ♯ = def α , so that Equation (19) holds trivially, since ι ( F,φ ) is the identity. ONADS IN DOUBLE CATEGORIES 25
To complete the verification of the universal property of ι ( F,φ ) , it only remainsto show that Equation (21) holds. The idea is to consider the sub-horizontal arrow E of Q ∗ F for which (21) and show that E must be isomorphic to Q ∗ F . Moreprecisely, let us define the horizontal arrow E : X → Y via the following equalizerin the category C of horizontal arrows and squares:(22) E / / θ / / Q ∗ F (¯ u ♯ ,α ) φ ∗ / / φ ′ ( α, ¯ v ♯ ) / / F ′ P ′ . Since the vertical boundaries of the squares in (21) are equal, the assumption thatthe source and target functors ∂ , ∂ : C → C preserve equalizers implies that E is indeed a horizontal arrow from X to Y and that θ has vertical boundaries givenby identity morphisms. The commutativity of the equalizer diagram in (22) can beexpressed as the equation(23) X E / / θ YX / / φ ∗ Y / / YX / / (cid:15) (cid:15) ¯ u ♯ X (cid:15) (cid:15) / / α Y (cid:15) (cid:15) X ′ P ′ / / X ′ F ′ / / Y ′ = X E / / θ YX / / (cid:15) (cid:15) α Y (cid:15) (cid:15) / / ¯ v ♯ Y (cid:15) (cid:15) X ′ / / φ ′ Y ′ / / Y ′ X ′ P ′ / / X ′ F ′ / / Y ′ . To prove Equation (21) we show that θ : E → Q ∗ F is an isomorphism. For this, weexploit the fact (observed in Remark 1.3) that Q ∗ F : X → Y is the initial algebrafor the endofunctor H C ( X, Y ) −→ H C ( X, Y )(24) ( − ) F + Q ( − ) , where H C ( X, Y ) denotes the hom-category of horizontal arrows from X to Y ofthe horizontal 2-category H C of C . Note that here we are using our assumptionthat H C has local coproducts. By the initiality of Q ∗ F , in order to show that θ : E → Q ∗ F is an isomorphism, it is sufficient to show that E admits an algebrastructure for the endofunctor in (24). The required algebra structure is given bythe copair ( λ, ρ ) : F + QE → E , where λ : F → E and ρ : QE → E are determined,via the universal property of the equalizer E , by the commutative diagrams(25) F η Q ∗ F / / Q ∗ F (¯ u ♯ ,α ) φ ∗ / / φ ′ ( α,v ♯ ) / / F ′ P ′ and(26) QE Q θ / / QQ ∗ F ν Q ∗ F / / Q ∗ F (¯ u ♯ ,α ) φ ∗ / / φ ′ ( α,v ♯ ) / / F ′ P ′ , respectively. It remains to show that the diagrams in (25) and (26) commute. Thecommutativity of (25) amounts to the equation(27) X F / / Y η Q ∗ YX / / φ ∗ Y / / YX / / (cid:15) (cid:15) ¯ u ♯ X (cid:15) (cid:15) / / α Y (cid:15) (cid:15) X ′ P ′ / / X ′ F ′ / / Y ′ = X F / / Y η Q ∗ YX / / (cid:15) (cid:15) α Y / / ¯ v ♯ (cid:15) (cid:15) Y (cid:15) (cid:15) X ′ / / φ ′ Y ′ / / Y ′ X ′ P ′ / / X ′ F ′ / / Y ′ . Starting from the left-hand side of Equation (27), we apply Equation (16) in thetop two rows and get X η P ∗ X F / / YX / / (cid:15) (cid:15) ¯ u ♯ X / / (cid:15) (cid:15) α Y (cid:15) (cid:15) X ′ P ′ / / X ′ F ′ / / Y ′ . Then, Equation (13) gives us(28) X (cid:15) (cid:15) X F / / (cid:15) (cid:15) α Y (cid:15) (cid:15) X ′ η P ′ X ′ / / Y ′ X ′ P ′ / / X ′ F ′ / / Y ′ . Considering now the right-hand side of Equation (27), an application of the ana-logue of Equation (13) for ¯ v ♯ gives us X F / / (cid:15) (cid:15) α Y (cid:15) (cid:15) Y (cid:15) (cid:15) X ′ / / Y ′ η Q ′ Y ′ X ′ / / φ ′ Y ′ / / Y ′ X ′ P ′ / / X ′ F ′ / / Y ′ . An application of the second axiom for a horizontal monad map for ( F ′ , φ ′ ) thengives us exactly (28), as required. It remains to show the commutativity of the ONADS IN DOUBLE CATEGORIES 27 diagram in (26), which amounts to the equation(29) P E / / θ Y Q / / YX / / Y / / ν Q ∗ Y / / YX / / φ ∗ Y / / YX / / (cid:15) (cid:15) ¯ u ♯ X (cid:15) (cid:15) / / α Y (cid:15) (cid:15) X ′ P ′ / / X ′ F ′ / / Y ′ = X E / / θ Y Q / / YX / / Y / / ν Q ∗ Y / / YX / / (cid:15) (cid:15) α Y / / ¯ v ♯ (cid:15) (cid:15) Y (cid:15) (cid:15) X ′ / / φ ′ Y ′ / / Y ′ X ′ P ′ / / X ′ F ′ / / Y ′ . Starting from the left-hand side of Equation (29), we use Equation (17) in thesecond and third row to get X E / / θ Y Q / / YX / / Y φ ∗ / / Y / / YX / / X / / Y φ / / YX / / X ν P ∗ / / X / / YX ¯ u ♯ (cid:15) (cid:15) / / X α (cid:15) (cid:15) / / Y (cid:15) (cid:15) X ′ P ′ / / X ′ F ′ / / Y ′ . We then use Equation (14) in the bottom two rows and obtain X E / / θ Y Q / / YX / / Y φ ∗ / / Y / / YX / / X / / X φ / / YX ¯ u ♯ (cid:15) (cid:15) / / X ¯ u (cid:15) (cid:15) / / X α (cid:15) (cid:15) / / Y (cid:15) (cid:15) X ′ µ P ′ / / X ′ / / X ′ / / Y ′ X ′ P ′ / / X ′ F ′ / / Y ′ . We apply Equation (20), which is the assumption that α is an endomorphismsquare, in the third and the fourth row, so as to get X E / / θ Y Q / / YX / / Y φ ∗ / / Y / / YX ¯ u ♯ (cid:15) (cid:15) / / X α (cid:15) (cid:15) / / Y ¯ v (cid:15) (cid:15) / / Y (cid:15) (cid:15) X ′ / / X ′ φ ′ / / Y ′ / / Y ′ X ′ µ P ′ / / X ′ / / X ′ / / Y ′ X ′ P ′ / / X ′ F ′ / / Y ′ . ONADS IN DOUBLE CATEGORIES 29
We now apply Equation (23) in the top three rows and we obtain X E / / θ Y Q / / YX α (cid:15) (cid:15) / / Y (cid:15) (cid:15) ¯ v ♯ / / Y ¯ v (cid:15) (cid:15) / / Y (cid:15) (cid:15) X ′ φ ′ / / Y ′ / / Y ′ / / Y ′ X ′ / / X ′ φ ′ / / Y ′ / / Y ′ X ′ µ P ′ / / X ′ / / X ′ / / Y ′ X ′ P ′ / / Y ′ F ′ / / Y ′ . We use the first axiom for a horizontal monad map (see item (ii) of Definition 2.4)for ( F ′ , φ ′ ) in the bottom three rows so as to get X E / / θ Y Q / / YX α (cid:15) (cid:15) / / Y (cid:15) (cid:15) ¯ v ♯ / / Y ¯ v (cid:15) (cid:15) / / Y (cid:15) (cid:15) X ′ / / Y ′ µ Q ′ / / Y ′ / / Y ′ X ′ φ ′ / / Y ′ / / Y ′ X ′ P ′ / / X ′ F ′ / / Y ′ . We obtain exactly this diagram also by applying the analogue of Equation (14)for ¯ v ♯ to the second and third row of the right-hand side of Equation (29). Thisconcludes the proof of Theorem 3.7. (cid:3) Acknowledgements
We are grateful to Richard Garner for drawing our attention to Sam Staton’scharacterization of free monads. We are also grateful to Mike Shulman for pointingout that an earlier version of this paper contained an inaccuracy regarding fiberedfunctors and that the proof of Theorem 3.7 requires the hypothesis that the sourceand target functors preserve equalizers. We would also like to thank the editor andthe referee for the very rapid handling of the paper.We gratefully acknowledge the support and hospitality of the Centre de Re-cerca Matem`atica during the academic year 2007/08, in occasion of the thematicprogramme on Homotopy Theory and Higher Categories. Thomas M. Fiore was supported at the University of Chicago by NSF Grant DMS-0501208. At the Uni-versitat Aut`onoma de Barcelona he was supported by grant SB2006-0085 of theSpanish Ministerio de Educaci´on y Ciencia under the Programa Nacional de ayu-das para la movilidad de profesores de universidad e investigadores espa˜noles yextranjeros. Thomas M. Fiore was also supported by the Max Planck Institut f¨urMathematik, and he thanks MPIM for its kind hospitality. Joachim Kock waspartially supported by grants MTM2006-11391 and MTM2007-63277 of Spain andSGR2005-00606 of Catalonia.
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Department of Mathematics and Statistics, University of Michigan-Dearborn, 4901Evergreen Road, Dearborn, MI 48128, USA
E-mail address : [email protected] Dipartimento di Matematica e Informatica, via Archirafi 34, 90123 Palermo, Italy,and School of Mathematics, The University of Manchester, Oxford Road, ManchesterM13 9PL, UK
E-mail address : [email protected] Departament de Matem`atiques, Universitat Aut`onoma de Barcelona, 08193 Bellaterra(Barcelona), Spain
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