A note on Frobenius-Eilenberg-Moore objects in dagger 2-categories
aa r X i v : . [ m a t h . C T ] J a n A NOTE ON FROBENIUS-EILENBERG-MOORE OBJECTS INDAGGER 2-CATEGORIES
ROWAN POKLEWSKI-KOZIELL
Abstract.
We define Frobenius-Eilenberg-Moore objects for a dagger Frobenius monadin an arbitrary dagger 2-category, and extend to the dagger context a well-known uni-versal property of the formal theory of monads. We show that the free completion of a2-category under Eilenberg-Moore objects extends to the dagger context, provided oneis willing to work with such dagger Frobenius monads whose endofunctor part suitablycommutes with their unit. Finally, we define dagger lax functors and dagger lax-limitsof such functors, and show that Frobenius-Eilenberg-Moore objects are examples of suchlimits.
1. Preliminaries A dagger category D is a category equipped with a involutive functor † : D op −→ D which is the identity on objects, called the dagger of D . A dagger functor F : D −→ C between dagger categories D , C is a functor which commutes with the daggers on D and C . A 2-category D is a dagger -category when each of the hom-categories D ( A, B ) are notonly (small) categories, but dagger categories. More precisely, given vertically-composable2-cells α and β , and horizontally-composable 2-cells σ and θ in D , the equalities( α · β ) † = β † · α † ( σ ∗ θ ) † = σ † ∗ θ † hold, where, here and elsewhere, · and ∗ denote the vertical and horizontal composition of2-cells, respectively, and where, as we shall do elsewhere, we have dropped all subscriptson daggers † to refer to particular hom-dagger-categories. The dagger 2-category DagCat of small dagger categories, dagger functors and natural transformations is a basic example.Given dagger 2-categories D , C , a 2-functor F : D −→ C is a dagger -functor when foreach pair of objects D , D ′ ∈ D , the functor F D,D ′ : D ( D, D ′ ) −→ C ( F D, F D ′ )is a dagger functor.We shall say that a dagger 2-category D is a full dagger sub- -category of C if there is adagger 2-functor I : D −→ C such that for all objects D , D ′ of D , the component daggerfunctor I D,D ′ : D ( D, D ′ ) −→ C ( ID, ID ′ ) is an isomorphism of dagger categories. Theweaker case of having I D,D ′ only equivalences of categories which are unitarily essentially © Rowan Poklewski-Koziell, 2020. Permission to copy for private use granted. ROWAN POKLEWSKI-KOZIELL surjective has no additional value in our work. The reader is encouraged to consult[Karvonen, 2019, Chapter 3] for a more detailed account of such dagger equivalences.If (
D, t ) is a monad in a dagger 2-category D , it is obviously a comonad too. [HK, 2015,HK, 2016] proposes that in a dagger 2-category, the monads of interest are those thatadditionally satisfy the Frobenius law . [HK, 2016] A monad ( D, t ) (with multiplication -cell µ : t −→ t andunit -cell η : 1 −→ t ) in a dagger -category D is a dagger Frobenius monad when thediagram t t t t tµ † / / µ † t (cid:15) (cid:15) µt (cid:15) (cid:15) tµ / / commutes. Furthermore, DFMnd ( D ) is the dagger -category in which: • -cells are dagger Frobenius monads in D ; • given -cells ( A, s ) and ( D, t ) , a -cell ( f, σ ) : ( A, s ) −→ ( D, t ) consists of a -cell f : A −→ D and a -cell σ : tf −→ f s in D , such that the diagrams: tf s f ssttf tf f s σs / / tσ ; ; ✈✈✈✈✈✈✈✈ µ t f ❍❍❍❍❍❍❍❍❍ fµ s (cid:15) (cid:15) σ / / f ss f stf s ttf tf fµ s / / σs ; ; ✈✈✈✈✈✈✈✈ tσ † ❍❍❍❍❍❍❍❍ σ † (cid:15) (cid:15) µ t f / / (1) tf f sf σ / / η t f b b ❊❊❊❊❊❊❊❊❊❊❊❊ fη s < < ②②②②②②②②②②②② commute, where µ t : t −→ t and µ s : s −→ s are the multiplications of t and s ,respectively, and η t : 1 −→ t and η s : 1 −→ s are the units of t and s , respectively.Composition of -cells is defined as ( g, γ ) · ( f, σ ) = ( gf, gσ · γf ) ; • given -cells ( A, s ) , ( D, t ) and -cells ( f, σ ) , ( g, γ ) : ( A, s ) −→ ( D, t ) in DFMnd ( D ) ,a -cell α : ( f, σ ) −→ ( g, γ ) in DFMnd ( D ) is a -cell α : f −→ g in D , such that NOTE ON FROBENIUS-EILENBERG-MOORE OBJECTS IN DAGGER 2-CATEGORIES the following diagrams tf tgf s gs tα / / σ (cid:15) (cid:15) γ (cid:15) (cid:15) αs / / tg tfgs f s tα † / / γ (cid:15) (cid:15) σ (cid:15) (cid:15) α † s / / commute. Vertical and horizontal composition of -cells is induced by the corre-sponding vertical and horizontal composition of -cells in D , as is the dagger on -cells induced by the dagger on -cells in D .There is an inclusion dagger -functor I : D −→
DFMnd ( D ) , defined on -cells by I ( D ) =( D, , on -cells by I ( f ) = ( f, , and on -cells by I ( α ) = α . A dagger Frobenius monad in the dagger 2-category
DagCat is of course simply amonad (
T, µ, η ) on a dagger category D whose endofunctor part T is a dagger functor,and such that T ( µ D ) µ † T D = µ T D T ( µ † D )for each D in D .One may easily verify that any dagger Frobenius monad is a Frobenius monad in thesense of [Street, 2004] – however, neither that paper nor [Lauda, 2006] explore monads inthe dagger context. In particular, algebras for these monads should satisfy an additionalcondition, so that they may behave quite differently from their non-dagger counterparts.
Let T = ( T, µ, η ) be a dagger Frobenius monad on a dagger category D . A Frobenius-Eilenberg-Moore algebra (or FEM-algebra) for T is an Eilenberg-Moorealgebra ( D, δ ) for T , such that the diagram T ( D ) T ( D ) T ( D ) T ( D ) T ( δ † ) / / µ † D (cid:15) (cid:15) µ D (cid:15) (cid:15) T ( δ ) / / – called the Frobenius law diagram for the algebra ( D, δ ) – commutes. The class ofall Frobenius-Eilenberg-Moore algebras and the class of all homomorphisms of Eilenberg-Moore algebras between FEM-algebras form a dagger category, which is denoted by FEM ( D , T ) . An adjunction in a dagger 2-category D is simply an adjunction in the underlying2-category.For dagger 2-categories A , D , there is 2-category [ A , D ], called the dagger 2-functorcategory , consisting of dagger 2-functors, 2-natural transformations, and modifications. ROWAN POKLEWSKI-KOZIELL
There is no need to specify “dagger 2-natural transformations”: given dagger 2-functors
F, G : A −→ D in [ A , D ], a 2-natural transformation is a family φ = (cid:0) φ A : F A −→ GA (cid:1) A ∈A of 1-cells in D , such that the diagram A ( A, B ) D ( F A, F B ) D ( GA, GB ) D ( F A, GB ) F A,B / / G A,B (cid:15) (cid:15) D ( F A,φ B ) (cid:15) (cid:15) D ( φ A ,GB ) / / commutes for all objects A , B in D , and clearly the representable functors D ( F A, φ B )and D ( φ A , GB ) of this diagram are of course dagger functors themselves. The daggerstructure on D then naturally induces a dagger structure on [ A , D ].A dagger 2-functor F : D −→
DagCat is representable , when there is some D in D and an isomorphism φ : D ( D, − ) −→ F in [ D , DagCat ]. The pair (
D, φ ) is called a representation of F . What is worth remarking is that, for a dagger 2-category C anda dagger 2-functor R : D −→ C , when, for each object C of C , the dagger 2-functor C ( C, R − ) : D −→
DagCat is representable – with representation (
LC, φ C ) – one has thatthe unique (up to 2-natural isomorphism) 2-functor L : C −→ D such that D ( LC, D ) C ( C, RD ) φ C,D / / is 2-natural in both C and D , is also a dagger 2-functor. This is easily seen from thestandard construction of L , as displayed in, say, [Kelly, 2005, Section 1.10]. Furthermore, L is of course the left 2-adjoint of R and such 2-adjunctions correspond bijectively to2-natural isomorphisms φ in the above display.Finally, one also has a Yoneda Lemma for dagger 2-categories: there are dagger 2-functors E , N : [ D op , DagCat ] ×D −→ DagCat , given, respectively, on 0-cells by E ( F, D ) = F ( D ) and N ( F, D ) = [ D op , DagCat ]( D ( − , D ) , F ) and, furthermore, an isomorphism y : N −→ E .Dagger Frobenius monads and categories of Frobenius-Eilenberg-Moore algebras forsuch monads were first considered in [HK, 2015] and [HK, 2016], in which they are shownto include the important example of quantum measurements. In this paper, we continuework initiated in those papers in pursuit of a formal theory of dagger Frobenius monads in the spirit of [Street, 1972] and [LS, 2002].
2. Frobenius-Eilenberg-Moore objects
Let (
D, t ) (with multiplication and unit given, respectively, by µ and η ) be a daggerFrobenius monad in a dagger 2-category D . Then (cid:0) D ( A, D ) , D ( A, t ) (cid:1) is a dagger Frobeniusmonad (with multiplication and unit given, respectively, by D ( A, µ ) and D ( A, η )) in
NOTE ON FROBENIUS-EILENBERG-MOORE OBJECTS IN DAGGER 2-CATEGORIES DagCat , for every object A of D . We may now construct the dagger category of Frobenius-Eilenberg-Moore algebras FEM (cid:0) D ( A, D ) , D ( A, t ) (cid:1) for the dagger Frobenius monad D ( A, t )on the dagger category D ( A, D ). Applying these observations to the case D = DagCat ,we arrive at the following result for a dagger category D and a dagger Frobenius monad( T, µ, η ) on D . Suppose F : A −→ D is a dagger functor, ( T, µ, η ) is a dagger Frobe-nius monad on the dagger category D , and σ : T F −→ F is a natural transformation. ( F A, σ A ) A ∈ A is a family of Frobenius-Eilenberg-Moore algebras for T if and only if ( F, σ ) is a Frobenius-Eilenberg-Moore algebra for the dagger Frobenius monad DagCat ( A , T ) onthe dagger category DagCat ( A , D ) . Furthermore, given another such Frobenius-Eilenberg-Moore algebra ( G, γ ) for DagCat ( A , T ) , and a natural transformation α : F −→ G , thefamily (cid:0) α A : ( F A, σ A ) −→ ( GA, γ A ) (cid:1) A ∈ A is a family of Eilenberg-Moore algebra homo-morphisms if and only if α : ( F, σ ) −→ ( G, γ ) is a homomorphism of Eilenberg-Moorealgebras for the monad DagCat ( A , T ) . Proof.
A routine calculation shows that, for every object A in A , the diagram T ( F A ) T ( F A ) T ( F A ) F A F A µ F A / / T ( σ A ) (cid:15) (cid:15) σ A (cid:15) (cid:15) σ A / / η F A o o ✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉ commutes if and only if the diagram DagCat ( A , T ) ( F ) DagCat ( A , T )( F ) DagCat ( A , T )( F ) F F
DagCat ( A ,µ )( F ) / / DagCat ( A ,T )( σ ) (cid:15) (cid:15) σ (cid:15) (cid:15) σ / / DagCat ( A ,η )( F ) o o ♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥ commutes. That is, the family ( F A, σ A ) A ∈ A is a family of Eilenberg-Moore algebras for T if and only if ( F, σ ) is an Eilenberg-Moore algebra for the monad
DagCat ( A , T ) on the(dagger) category DagCat ( A , D ). Likewise, for every object A in A , the diagram T ( F A ) T ( F A ) T ( F A ) T ( F A ) T ( σ † A ) / / µ † F A (cid:15) (cid:15) µ F A (cid:15) (cid:15) T ( σ A ) / / ROWAN POKLEWSKI-KOZIELL commutes if and only if the diagram
DagCat ( A , T )( F ) DagCat ( A , T ) ( F ) DagCat ( A , T ) ( F ) DagCat ( A , T )( F ) DagCat ( A ,T )( σ † ) / / DagCat ( A ,µ ) † ( F ) (cid:15) (cid:15) DagCat ( A ,µ )( F ) (cid:15) (cid:15) DagCat ( A ,T )( σ ) / / commutes. The second part of the proposition is similarly proved. Suppose ( T, µ, η ) is a dagger Frobenius monad on the dagger category D . For every dagger category A , there is an isomorphism of dagger categories DagCat ( A , FEM ( D , T )) ∼ = FEM ( DagCat ( A , D ) , DagCat ( A , T )) which is -natural in each of the arguments. Proof.
Each dagger functor F : A −→ FEM ( D , T ) determines a dagger functor F = U T F : A −→ D and a family (cid:0) F A, σ A (cid:1) A ∈ A of FEM-algebras, where U T : FEM ( D , T ) −→ D is the forgetful (dagger) functor. Since F is a functor, the family σ = (cid:0) σ A : T F A −→ F A (cid:1) is a natural transformation σ : T F −→ F . Therefore, by Proposition 2.1, ( F, σ )is a FEM-algebra for the dagger Frobenius monad
DagCat ( A , T ) on the dagger category DagCat ( A , D ).Conversely, given a dagger functor F : A −→ D and a natural transformation σ : T F −→ F such that ( F, σ ) is a FEM-algebra for the dagger Frobenius monad
DagCat ( A , T ), for each object A of A , (cid:0) F A, σ A (cid:1) is a FEM-algebra for T , again by Proposi-tion 2.1. Since σ : T F −→ F is a natural transformation, for each morphism f : A −→ B of A , F f : F A −→ F ( B ) is a morphism (cid:0) F A, σ A (cid:1) −→ (cid:0) F B, σ B (cid:1) of Eilenberg-Moorealgebras. This now defines a functor F : A −→ FEM ( D , T ).Next, the second part of Proposition 2.1 similarly establishes correspondences betweennatural transformations F −→ G and homomorphisms ( F, σ ) −→ ( G, γ ) of Eilenberg-Moore algebras for the monad
DagCat ( A , T ), which preserve daggers.Clearly, these correspondences are inverses of each other. It is routine to show thateach is 2-natural in each of the arguments.The previous theorem suggests our main definition. For a dagger -category D , a dagger Frobenius monad ( D, t ) in D issaid to have a Frobenius-Eilenberg-Moore object (or FEM-object) if the dagger -functor FEM (cid:0) D ( − , D ) , D ( − , t ) (cid:1) : D op −→ DagCat whose object-part is defined by A FEM (cid:0) D ( A, D ) , D ( A, t ) (cid:1) , is representable. A choiceof a representing object in D , denoted FEM ( D, t ) , is called the Frobenius-Eilenberg-Mooreobject for ( D, t ) . D is further said to have Frobenius-Eilenberg-Moore objects if everydagger Frobenius monad ( D, t ) in D has a Frobenius-Eilenberg-Moore object. NOTE ON FROBENIUS-EILENBERG-MOORE OBJECTS IN DAGGER 2-CATEGORIES Suppose ( D, t ) is a dagger Frobenius monad in the dagger -category D . For every object A of D , there is an isomorphism of dagger categories DFMnd ( D )(( A, , ( D, t )) ∼ = FEM ( D ( A, D ) , D ( A, t )) (2)2 -natural in each of the arguments.
Proof.
One easily shows that, to give a pair ( f, σ ) in which f : A −→ D is a 1-cell and σ : tf −→ f a 2-cell in D satisfying the top-left and bottom diagrams (1) for the monads( A,
1) and (
D, t ) is exactly to give an Eilenberg-Moore algebra for the monad D ( A, t ) on D ( A, D ). (
F, σ ) is, moreover, a morphism of dagger Frobenius monads ( A, −→ ( D, t ),exactly when, by the top-right diagram (1), σ † · σ = µf · tσ † , which is the statementthat σ † : ( f, σ ) −→ ( tf, µf ) = ( D ( A, t )( f ) , D ( A, µ )( f )) is a homomorphism of Eilenberg-Moore algebras for the monad D ( A, t ). By [HK, 2016, Lemma 6.8], this is exactly to saythat ( f, σ ) is a FEM-algebra for the dagger Frobenius monad D ( A, t ).Finally, for a second morphism ( g, γ ) : ( A, −→ ( D, t ) of dagger Frobenius monads,to give a 2-cell α : ( f, σ ) −→ ( g, γ ) in DFMnd ( D ) is exactly to give a homomorphism( f, σ ) −→ ( g, γ ) of Eilenberg-Moore algebras for the monad D ( A, t ), by [HK, 2016, Lemma6.7]. [HK, 2016] A dagger -category D admits the construction of Frobenius-Eilenberg-Moore algebras when the inclusion dagger -functor I : D −→
DFMnd ( D ) has aright -adjoint, which is denoted FEM : DFMnd ( D ) −→ D . From Proposition 2.4, the following result is immediate.
A dagger -category D admits the construction of Frobenius-Eilenberg-Moore algebras if and only if D has Frobenius-Eilenberg-Moore objects. In particular, togive a right adjoint to I : D −→
DFMnd ( D ) is precisely to give a choice, for each daggerFrobenius monad in D of a Frobenius-Eilenberg-Moore-object. Theorems 2.2 and 2.6 now give the following known result. [HK, 2016, Theorem 7.5]
DagCat admits the construction of Frobenius-Eilenberg-Moore algebras.
When a dagger Frobenius monad (
D, t ) in D has a FEM-object, the dagger iso-morphism (2) uniquely determines a morphism of dagger Frobenius monads ( u t , ξ ) : (cid:0) FEM ( D, t ) , (cid:1) −→ (cid:0) D, t (cid:1) , in which we think of the 1-cell u t as the “forgetful” 1-cell.Moreover, if D further admits the construction of Frobenius-Eilenberg-Moore algebras,then the component of the counit of the 2-adjunction evaluated at the dagger Frobeniusmonad ( D, t ) is ( u t , ξ ). In particular, in the case that D = DagCat , the forgetful 1-cell U T is of course the usual forgetful dagger functor FEM ( D , T ) −→ D .[Street, 1972] shows that much of the 1-dimensional theory of monads can be describedby several important universal properties in a 2-dimensional context. We next show thatin passing to the dagger context, there are corresponding universal properties. ROWAN POKLEWSKI-KOZIELL
For an adjunction f ⊣ u in a dagger -category D , the monad generated bythe adjunction f ⊣ u is a dagger Frobenius monad. Proof. If f ⊣ u is an adjunction in a dagger 2-category D , with counit ǫ : f u −→ η : 1 −→ uf , then we also have u ⊣ f , with counit η † : uf −→ ǫ † : 1 −→ f u . [Lauda, 2006, Corollary 2.22] now says that the monad ( D, uf ) generatedby the adjunction f ⊣ u is a dagger Frobenius monad.Following this proposition we call ( D, uf ) the dagger Frobenius monad generated bythe adjunction f ⊣ u . [HK, 2016, Theorem 7.4] Every dagger Frobenius monad in a dagger -category D having a Frobenius-Eilenberg-Moore object is generated by an adjunction. When a dagger Frobenius monad (
D, t ) in a dagger 2-category D has a FEM-object,the isomorphism of dagger categories D (cid:0) A, FEM ( D, t ) (cid:1) −→ DFMnd ( D ) (cid:0) ( A, , ( D, t ) (cid:1) (3)is defined by f ( u t f, ξf ) on 1-cells and σ u t σ on 2-cells, for the unique morphism( u t , ξ ) : (cid:0) FEM ( D, t ) , (cid:1) −→ ( D, t ) of dagger Frobenius monads. The proof of Theorem 2.9shows that, for a dagger Frobenius monad (
D, t ) in a dagger 2-category D , if ( D, t ) hasa FEM-object, there exists a unique 1-cell f t : D −→ FEM ( D, t ) such that t = u t f t and µ = ξf t , and a unique 2-cell ǫ t : f t u t −→ u t ǫ t = ξ . Furthermore, f t is a leftadjoint of u t and generates the dagger Frobenius monad ( D, t ). In the notation above, suppose the dagger Frobenius monad ( D, t ) gen-erated by the adjunction f ⊣ u has a Frobenius-Eilenberg-Moore object. Then, there existsa unique -cell n : A −→ FEM ( D, t ) such that u t n = u and uǫ = ξn , where ǫ is the counitof the adjunction f ⊣ u . Moreover, this n satisfies nf = f t and nǫ = ǫ t n . Proof.
One easily verifies that ( u, uǫ ) : ( A, −→ ( D, t ) is a morphism of monads. Itremains to verify that it is a morphism of dagger Frobenius monads. From the top-rightdiagram of (1), ( u, uǫ ) is a morphism of dagger Frobenius monads if and only if u ( ǫ † · ǫ ) = uǫ † · uǫ = uǫf u · uf uǫ † = u ( ǫf u · f uǫ † )But, a straightforward application of the interchange law gives the equalities f uǫ · ǫ † f u = ǫ † · ǫ = ǫf u · f uǫ † And so, ( u, uǫ ) is indeed a morphism of dagger Frobenius monads. The rest of theproof proceeds identically to the similar proof in [Street, 1972]. Since ( u, uǫ ) : ( A, −→ ( D, t ) is a morphism of dagger Frobenius monads, there exists a unique 1-cell n : A −→ NOTE ON FROBENIUS-EILENBERG-MOORE OBJECTS IN DAGGER 2-CATEGORIES FEM ( D, t ) such that the diagram (
D, t )( A, (cid:0) FEM ( D, t ) , (cid:1) ( n, / / ( u,uǫ ) (cid:31) (cid:31) ❄❄❄❄❄❄❄❄❄❄❄❄ ( u t ,ξ ) (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ commutes. Therefore, u t n = u and uǫ = ξn , so that u t ( nǫ ) = uǫ = ξn = u t ( ǫ t n ).Therefore, by the dagger isomorphism (3), we have nǫ = ǫ t n . Finally, u t ( nf ) = uf = tξ ( nf ) = uǫf = µ By the property which uniquely determines f t , we have f t = nf .Since DagCat admits the construction of Frobenius-Eilenberg-Moore algebras, the fol-lowing result is immediate. [HK, 2016, Theorem 6.9] Suppose F and U are dagger adjoints be-tween dagger categories A and D , with T the dagger Frobenius monad generated by F ⊣ U .Then, there exists a unique dagger functor N : A −→ FEM (cid:0) D , T (cid:1) such that U T N = U and N F = F T . The unique -cell n : A −→ FEM ( D, t ) of Theorem 2.10 is called theright comparison -cell of the adjunction f ⊣ u . If this -cell is a dagger equivalence (thatis, there is a -cell m : FEM ( D, t ) −→ A , and -cell unitaries nm ∼ = 1 and ∼ = mn ), thenthe adjunction f ⊣ u is said to be monadic. Note that 2-functors between 2-categories send adjunctions to adjunctions. The for-mulation of Frobenius-Eilenberg-Moore objects as representing objects for a representabledagger 2-functor in the previous section now gives our final important result, whose proofis identical to that of [Street, 1972, Corollary 8.1].
Suppose the dagger Frobenius monad generated by an adjunction f ⊣ u in a dagger -category D has a Frobenius-Eilenberg-Moore object. The adjunction f ⊣ u is monadic if and only if, for each object X of D , the adjunction D ( X, f ) ⊣ D ( X, u ) in DagCat is monadic.
3. Free completions under FEM-objects
Dually, we define
Frobenius-Kleisli objects for dagger Frobenius monads in a dagger 2-category.0
ROWAN POKLEWSKI-KOZIELL
A Frobenius-Kleisli object for a dagger Frobenius monad ( D, t ) in adagger -category D is a Frobenius-Eilenberg-Moore object for ( D, t ) considered as a daggerFrobenius monad in D op . A Frobenius-Kleisli object for ( D, t ) , when it exists, is denotedby FK ( D, t ) , and in particular satisfies, for each object X in D , the following isomorphismof dagger categories D (cid:0) FK ( D, t ) , X (cid:1) ∼ = FEM (cid:0) D ( D, X ) , D ( t, X ) (cid:1) -natural in each of the arguments. D is said to have Frobenius-Kleisli objects if everydagger Frobenius monad in D has a Frobenius-Kleisli object. From [HK, 2016, Lemma 6.1] we know that the Kleisli category D T for a daggerFrobenius monad ( T, µ, η ) on a dagger category D carries a canonical dagger structure,given by (cid:0) f : C −→ T D (cid:1) (cid:0) T ( f † ) µ † D η D : D −→ T C (cid:1) (4)which commutes with the canonical dagger functors D −→ D T and D T −→ D . In fact,this makes D T a Frobenius-Kleisli object for ( D , T ). Each dagger Frobenius monad T = ( T, µ, η ) on a dagger category D hasa Frobenius-Kleisli object, which is the Kleisli category D T of the monad T . Proof.
Let F T : D −→ FEM ( D , T ) and F T : D −→ D T denote the canonical free (dag-ger) functors. For a dagger category X and a dagger functor S ′ : F T ( D ) −→ X , thepair ( S ′ F T , S ′ µ ) is a Frobenius-Eilenberg-Moore algebra for the dagger Frobenius monad DagCat ( T, X ) on the dagger category DagCat ( D , X ). Indeed, since for each object D in D , F T ( D ) = (cid:0) T ( D ) , µ D (cid:1) is an Eilenberg-Moore algebra for the monad T , ( S ′ F T , S ′ µ )is surely an Eilenberg-Moore algebra for DagCat ( T, X ). Furthermore, since T is a dag-ger Frobenius monad, ( S ′ F T , S ′ µ ) is additionally a Frobenius-Eilenberg-Moore algebra.Sending a natural transformation α : S ′ −→ S ′′ : F T ( D ) −→ X to the homomorphism αF T : S ′ F T −→ S ′′ F T of Eilenberg-Moore algebras then determines a dagger functor DagCat (cid:0) F T ( D ) , X (cid:1) −→ FEM (cid:0)
DagCat ( D , X ) , DagCat ( T, X ) (cid:1) (5)On the other hand, if ( S, φ ) is a Frobenius-Eilenberg-Moore algebra for the dagger Frobe-nius monad
DagCat ( T, X ), the mappings D SD, (cid:0) f : C −→ T D (cid:1) (cid:0) φ D Sf : SC −→ SD (cid:1) yield a dagger functor S : D T −→ X . For, given morphisms g : B −→ T C and f : C −→ T D in D , the composite morphism f · g in D T is given by the morphism µ D T ( f ) g : B −→ T D in D , and so S ( f · g ) = φ D S ( µ D ) ST ( f ) S ( g )= φ D φ T ( C ) ST ( f ) S ( g )= φ D S ( f ) φ C S ( g ) = S ( f ) S ( g ) NOTE ON FROBENIUS-EILENBERG-MOORE OBJECTS IN DAGGER 2-CATEGORIES
S, φ ) being an Eilenberg-Moore alge-bra, and the third equality by the fact that φ : ST −→ S is a natural transformation.Furthermore, since η D : D −→ T D in D is the identity morphism D −→ D in D T , S (1 D ) = φ D S ( η D ) = 1 SD again by definition of ( S, φ ) being an Eilenberg-Moore algebra, and so S is indeed a func-tor. Finally, note that since ( S, φ ) is Frobenius-Eilenberg-Moore algebra for
DagCat ( T, X ),one has that for each D in D , φ T D S ( µ † D ) = S ( µ D ) φ † T D (6)Therefore, for f : C −→ T D in D , S ( f † ) = φ C ST ( f † ) S ( µ † D ) S ( η D )= S ( f † ) φ T D S ( µ † D ) S ( η D ) ( ) = S ( f † ) S ( µ D ) φ † T D S ( η D )= S ( f † ) S ( µ D ) ST ( η D ) φ † D = S ( f † ) φ † D = (cid:0) φ D S ( f ) (cid:1) † = S ( f ) † Sending a homomorphism of Eilenberg-Moore algebras ψ : ( P, ρ ) −→ ( S, φ ) to its under-lying natural transformation ψ : P −→ S now determines a dagger functor FEM (cid:0)
DagCat ( D , X ) , DagCat ( T, X ) (cid:1) −→ DagCat (cid:0) D T , X (cid:1) (7)These two dagger functors (5) and (7) determine an isomorphism of dagger categories DagCat (cid:0) D T , X (cid:1) ∼ = FEM (cid:0)
DagCat ( D , X ) , DagCat ( T, X ) (cid:1) free completion under Frobenius-Eilenberg–Moore objects of a dagger 2-category D . What we mean by this free completionwill be clear from Theorem 3.5 below. Informally, however, it will manifest a ‘dagger-enriched’ version of the closure K of a 2-category K in [ K op , Cat ] under Eilenberg-Mooreobjects, as detailed in [Street, 1976, Section 4]. Rather than attempting to extend tothe dagger context the sophisticated machinery of [Street, 1976], we more directly ap-proach the current situation, while equally following very closely the similar argument in[LS, 2002].Given a dagger 2-category D , each dagger Frobenius monad ( F, φ ) in [ D op , DagCat ]has a Frobenius-Kleisli object, which we denote FK ( F, φ ). Indeed, Theorem 3.2 showsthat there exists a dagger 2-functor FK : DFMnd ( DagCat ) −→ DagCat – which is in2
ROWAN POKLEWSKI-KOZIELL fact a left 2-adjoint of the inclusion dagger 2-functor I : DagCat −→ DFMnd ( DagCat )– and so one constructs a dagger 2-functor FK ( F, φ ) : D op −→ DagCat in the obviousfashion of specifying 0-cells, 1-cells and 2-cells in
DFMnd ( DagCat ) determined by thepointwise values of (
F, φ ), and then taking their images under the dagger 2-functor FK : DFMnd ( DagCat ) −→ DagCat . Finally, since we now have, for each D in D , a 2-naturalisomorphism of dagger categories DagCat (cid:0) FK ( F D, φ D ) , SD (cid:1) ∼ = FEM (cid:0)
DagCat ( F D, SD ) , DagCat ( φ D , SD ) (cid:1) we surely have a 2-natural isomorphism of dagger categories[ D op , DagCat ]( FK ( F, φ ) , S ) ∼ = FEM ([ D op , DagCat ]( F, S ) , [ D op , DagCat ]( φ, S ))for each dagger 2-functor S : D op −→ DagCat .Now, we proceed by a familiar transfinite process of, starting with the collection of allrepresentable dagger 2-functors D ( − , D ) in [ D op , DagCat ] and adding to this collection ateach step thereafter, all Frobenius-Kleisli objects of dagger Frobenius monads involvingobjects of the collection at the previous step. Since the argument presented in [LS, 2002]boils down to the fact that the free functor D ( X, D ) −→ D ( X, D ) T to the Kleisli categoryfor a monad T on D ( X, D ) is bijective on objects, the same argument applies mutismutandis in our dagger case, so that this transfinite process in fact also terminates afterthe first step.In conclusion, taking the replete full dagger sub-2-category of [ D op , DagCat ] of ob-jects resulting from the single step of this process produces a dagger 2-category hav-ing Frobenius-Kleisli objects. Furthermore, since each representable D ( − , D ) is itself aFrobenius-Kleisli object for a dagger Frobenius monad on a representable (for example,the identity monad on D ( − , D )), every object of this dagger 2-category is a Frobenius-Kleisli object for a dagger Frobenius monad on a representable. We shall denote thisdagger 2-category by FK ( D ).A simplification is possible which allows us to give an explicit description of FK ( D ). Each -cell FK (cid:0) D ( − , D ) , D ( − , t ) (cid:1) −→ FK (cid:0) D ( − , C ) , D ( − , s ) (cid:1) in FK ( D ) is a pair ( f, σ ) in which f : D −→ C is a -cell in D and σ : f t −→ sf a -cell in D NOTE ON FROBENIUS-EILENBERG-MOORE OBJECTS IN DAGGER 2-CATEGORIES which make the following diagrams sf t ssff tt f t sf sσ / / σt ; ; ✈✈✈✈✈✈✈✈ fµ t ❍❍❍❍❍❍❍❍❍ µ s f (cid:15) (cid:15) σ / / ssf sfsf t f tt f t µ s f / / sσ ; ; ✈✈✈✈✈✈✈✈ σ † t ❍❍❍❍❍❍❍❍ σ † (cid:15) (cid:15) fµ t / / f t sff σ / / fη t b b ❊❊❊❊❊❊❊❊❊❊❊❊ η s f < < ②②②②②②②②②②②② (8) commute. Furthermore, each -cell in FK ( D ) between such -cells ( f, σ ) , ( g, γ ) is a -cell α : f −→ sg in D such that the diagram sf ssgf tsgt ssg sg sα / / σ / / αt (cid:15) (cid:15) sγ / / µ s g (cid:15) (cid:15) µ s g / / (9) commutes. Proof.
We proceed by similar arguments presented in [LS, 2002]. For the objects FK (cid:0) D ( − , D ) , D ( − , t ) (cid:1) and FK (cid:0) D ( − , C ) , D ( − , s ) (cid:1) in FK ( D ), determined, respectively, by the dagger Frobeniusmonads ( D, t ) and (
C, s ) in D , a 1-cell FK (cid:0) D ( − , D ) , D ( − , t ) (cid:1) −→ FK (cid:0) D ( − , C ) , D ( − , s ) (cid:1) (10)is a FEM-algebra for the dagger Frobenius monad FK (cid:0) D (cid:1)(cid:16) D ( − , t ) , FK (cid:0) D ( − , C ) , D ( − , s ) (cid:1)(cid:17) (11)on the dagger category FK (cid:0) D (cid:1)(cid:16) D ( − , D ) , FK (cid:0) D ( − , C ) , D ( − , s ) (cid:1)(cid:17) (12)by the definition of Frobenius-Kleisli objects. By the Yoneda lemma for dagger 2-categories, (12) is 2-naturally isomorphic to FK (cid:0) D ( D, C ) , D ( D, s ) (cid:1) , while the daggerFrobenius monad corresponding to (11) is denoted FK (cid:0) D ( t, C ) , D ( t, s ) (cid:1) . By a similar4 ROWAN POKLEWSKI-KOZIELL argument, a 2-cell between 1-cells (10) is simply a morphism of Eilenberg-Moore algebrasbetween the corresponding FEM-algebras. That is, there is an isomorphism of dagger cate-gories between the dagger category FK ( D ) (cid:16) FK (cid:0) D ( − , D ) , D ( − , t ) (cid:1) , FK (cid:0) D ( − , C ) , D ( − , s ) (cid:1)(cid:17) and the dagger category FEM (cid:16) FK (cid:0) D ( D, C ) , D ( D, s ) (cid:1) , FK (cid:0) D ( t, C ) , D ( t, s ) (cid:1)(cid:17) which is 2-natural in the arguments.Now, the dagger category FK (cid:0) D ( D, C ) , D ( D, s ) (cid:1) has as objects 1-cells f : D −→ C in D , and as morphisms 2-cells σ : f −→ sg in D . Composition is given by the usual Kleislicomposition. Turning to the dagger Frobenius monad FK (cid:0) D ( t, C ) , D ( t, s ) (cid:1) , its (dagger)endofunctor part acts on objects by f f t and on morphisms ( σ : f −→ sg ) ( σt : f t −→ sgt ). The component at some g : D −→ C of the multiplication part of this daggerFrobenius monad is given by η s gt · gµ t . Likewise, the component at g : D −→ C of theunit part is given by η s gt · gη t .Therefore, a 1-cell (10) is a pair ( f, σ ) in which f : D −→ C is a 1-cell in D , and σ : f t −→ sf a 2-cell in D satisfying the associative, unit and Frobenius laws for a FEM-algebra for the dagger Frobenius monad FK (cid:0) D ( t, C ) , D ( t, s ) (cid:1) – the first two laws of whichgive the top-left and bottom diagrams of (8).It remains only to calculate the Frobenius law diagram for ( f, σ ). By [HK, 2016,Lemma 6.8], this is exactly the commutativity of the diagram f tsf sf ttsf t τ / / ρ / / σ (cid:15) (cid:15) µ s ft · sσt (cid:15) (cid:15) in which τ = sf µ t † · sη s † f t · µ s † f t · η s f t and ρ = µ s f t · ssσ † · sµ s † f · sη s f . The top path is µ s f t · sσt · τ = µ s f t · sσt · sf µ t † · sη s † f t · µ s † f t · η s f t = µ s f t · sσt · sf µ t † · η s f t = µ s f t · s ( σt · f µ t † ) · η s f t = µ s f t · η s sf t · σt · f µ t † = σt · f µ t † while, using the Frobenius law for the dagger Frobenius monad ( C, s ), for the bottom
NOTE ON FROBENIUS-EILENBERG-MOORE OBJECTS IN DAGGER 2-CATEGORIES ρ · σ = µ s f t · ssσ † · sµ s † f · sη s f · σ = sσ † · µ s sf · sµ s † f · sη s f · σ = sσ † · sµ s f · µ s † sf · sη s f · σ = sσ † · sµ s f · ssη s f · µ s † f · σ = sσ † · µ s † f · σ That is, the commutativity of the Frobenius law diagram for ( f, σ ) is the equality σ † · µ s f · sσ = f µ t · σ † t which is exactly the top-right diagram (8).Finally, as in the (possibly) non-dagger case presented in [LS, 2002], to give a 2-cell α :( f, σ ) −→ ( g, γ ) seen as FEM-algebras for the dagger Frobenius monad FK (cid:0) D ( t, C ) , D ( t, s ) (cid:1) is to give a 2-cell α : f −→ sg in D satisfying (9). The dagger α † , considered as a 2-cell in FK ( D ), is calculated from the canonical dagger (4) as the 2-cell sα † · µ s † g · η s g : g −→ sf in D .We may now take the 0-cells of FK ( D ) to be dagger Frobenius monads in D , and 1- and2-cells in FK ( D ) to be as described in the above proposition. Furthermore, the Yonedaembedding dagger 2-functor induces a (2-)fully faithful dagger 2-functor I : D −→ FK ( D )whose action on 0-cells is given by D ( D, D . Furthermore, we now define FEM ( D ) = KL ( D op ) op . A 0-cell in FEM ( D ) is once again adagger Frobenius monad in D , while 1-cells are the same as 1-cells in DFMnd ( D ). A 2-cell( f, σ ) −→ ( g, γ ) : ( D, t ) −→ ( C, s ) is a 2-cell α : f −→ gt in D such that the diagram f t gttsfsgt gtt gt αt / / σ / / sα (cid:15) (cid:15) γt / / gµ t (cid:15) (cid:15) gµ t / / (13)commutes. Again, the restricted Yoneda embedding dagger 2-functor induces a dagger2-functor I : D −→
FEM ( D ) whose action on 0-cells is given by D ( D, Consider
FEM ( D ) for the case that the dagger 2-category D has Frobenius-Eilenberg-Moore objects. As usual, it has 0-cells as dagger Frobenius monads in D . Givendagger Frobenius monads ( D, t ) and (
C, s ) in D , there is a bijection between the set of1-cells ( f, σ ) : ( D, t ) −→ ( C, s ) of
DFMnd ( D ) (and hence FEM ( D )) and the set of pairs6 ROWAN POKLEWSKI-KOZIELL ( f, f ) of 1-cells in D such that the diagram FEM ( D, t ) FEM ( C, s ) D C f / / u t (cid:15) (cid:15) u s (cid:15) (cid:15) f / / commutes, where u t and u s are the forgetful 1-cells. To see this, first fix the 1-cell f : D −→ C . To give a 1-cell f : FEM ( D, t ) −→ FEM ( C, s ) such that the above di-agram commutes is, by the definition of the FEM-object
FEM ( C, s ), to give a FEM-algebra ( f u t , ξ ) for the dagger Frobenius monad D ( FEM ( D, t ) , s ) on the dagger category D ( FEM ( D, t ) , C ). But the adjunction f t ⊣ u t in D of course induces an adjunction D ( u t , C ) ⊣ D ( f t , C ) in DagCat , so that there is a bijection D ( FEM ( D, t ) , C )( sf u t , f u t ) ∼ = D ( D, C )( sf, f t )So, by [HK, 2016, Lemma 6.8], to give such a FEM-algebra ( f u t , ξ ) for the dagger Frobe-nius monad D ( FEM ( D, t ) , S ) is exactly to give a morphism ( f, σ ) : ( D, t ) −→ ( C, s ) ofdagger Frobenius monads.In other words, 1-cells (
D, t ) −→ ( C, s ) in
FEM ( D ) are pairs ( f, f ) of 1-cells in D satisfying f u t = u s f .As is true in the (possibly) non-dagger case in [LS, 2002], a 2-cell ( f, f ) −→ ( g, g ) in FEM ( D ) from ( D, t ) to (
C, s ) is simply a 2-cell f −→ g in D .Next, suppose that tη t = η t t . We show that under this condition, this correspondenceof 2-cells preserves daggers. Indeed, for 1-cells ( f, f ), ( g, g ) in FEM ( D ), to give a 2-cell α : f −→ g in D is exactly to give a 2-cell u s αf t · f η t = α : f −→ gt in FEM ( D ). Therefore,to give the 2-cell α † : g −→ f in D is exactly to give the 2-cell u s α † f t · gη t : g −→ f t . But α † is calculated as the 2-cell( u s αf t · f η t ) † t · gµ t † · gη t = f η t † t · u s α † f t t · gµ t † · gη t in D . Therefore, it remains to show that u s α † f t · gη t = f η t † t · u s α † f t t · gµ t † · gη t , which isthe case when tη t † = η t † t . Let D be a dagger -category, and let C be a dagger -category such that,for every dagger Frobenius monad ( C, s ) in C , the equality sη s = η s s holds. Then, if C has Frobenius-Eilenberg-Moore objects, composition with the daggerinclusion -functor I induces an equivalence of categories [ FEM ( D ) , C ] FEM ≈ [ D , C ] between the dagger -functor category [ D , C ] and the full subcategory of the dagger -functor category [ FEM ( D ) , C ] FEM of dagger -functors which preserve Frobenius-Eilenberg-Moore objects. NOTE ON FROBENIUS-EILENBERG-MOORE OBJECTS IN DAGGER 2-CATEGORIES Proof.
Since C has FEM-objects, a FEM-object-preserving dagger 2-functor F : FEM ( D ) −→ C extending F : D −→ C must be defined (up to 2-natural isomorphism) on 0-cellsby F (cid:0) D, t (cid:1) = FEM (cid:0)
F D, F t (cid:1) while its action on 1-cells and 2-cells must be defined by the action of the composite ofthe dagger 2-functor
FEM ( C ) −→ C of Example 3.4 with the dagger 2-functor FEM ( F ) : FEM ( D ) −→ FEM ( C ) induced by F .On the other hand, these requirements can be used as a definition of such a dagger2-functor F : FEM ( D ) −→ C . Therefore, the desired extension does exist and is uniqueup to a 2-natural isomorphism. If the inclusion dagger -functor functor I : D −→
FEM ( D ) has aright -adjoint, then D has Frobenius-Eilenberg-Moore objects. Proof.
We prove the dual result for Frobenius-Kleisli objects. Suppose I has a left2-adjoint L : FK ( D ) −→ D . Then, for any dagger Frobenius monad ( D, t ) in D , D (cid:0) L (cid:0) FK (cid:0) D ( − , D ) , D ( − , t ) (cid:1)(cid:1) , X (cid:1) ∼ = FK ( D ) (cid:0) FK (cid:0) D ( − , D ) , D ( − , t ) (cid:1) , I ( X ) (cid:1) = FK ( D ) (cid:0) FK (cid:0) D ( − , D ) , D ( − , t ) (cid:1) , D ( − , X ) (cid:1) ∼ = FEM (cid:0) FK ( D ) (cid:0) D ( − , D ) , D ( − , X ) (cid:1) , FK ( D ) (cid:0) D ( − , t ) , D ( − , X ) (cid:1)(cid:1) ∼ = FEM (cid:0) D (cid:0) D, X (cid:1) , D (cid:0) t, X (cid:1)(cid:1) Therefore, the object L (cid:0) FK (cid:0) D ( − , D ) , D ( − , t ) (cid:1)(cid:1) is a Frobenius-Kleisli object for ( D, t ).
4. Dagger lax functors and dagger lax-limits
In this section, we extend the notion of a lax functor between 2-categories to the daggercontext. This will allow us to describe the universal properties of FEM-objects in Section2 as dagger analogues of lax-limits of lax functors.
Given dagger -categories D , C , a lax functor F : D −→ C – havingfamilies γ A,B,C : c C · ( F A,B × F B,C ) −→ F A,C · c D and δ A : u C −→ F A,A · u D of ‘comparison’ natural transformations – is a dagger lax functor when, for each A , B in D , the functors F A,B : D ( A, B ) −→ C ( F A, F B ) are dagger functors, and the families γ and δ additionally satisfy the Frobenius axiom : For every triple of arrows
A B C D f / / g / / h / / ROWAN POKLEWSKI-KOZIELL in D , the following diagram in C F ( h ) · F ( g · f ) F ( h ) · F ( g ) · F ( f ) F ( h · g · f ) F ( h · g ) · F ( f ) F h ∗ γ † f,g / / γ g,h ∗ F f (cid:15) (cid:15) γ g · f,h (cid:15) (cid:15) γ † f,h · g / / (14) commutes, where ∗ indicates the horizontal composition of -cells in C and, for simplicity,we have written γ f,g instead of (cid:0) γ A,B,C (cid:1) ( f,g ) . Let us clarify that the composite
D C B F / / G / / dagger lax functor is indeed well-defined. For, given two such dagger lax functors, thecomposite family γ GF is determined via the pasting operation D ( A, B ) × D ( B, C ) D ( A, C ) C (cid:0) F A, F B (cid:1) × C (cid:0)
F B, F C (cid:1) C (cid:0) F A, F C (cid:1) B (cid:0) GF A, GF B (cid:1) × B (cid:0)
GF B, GF C (cid:1) B (cid:0) GF A, GF C (cid:1) c D / / F A,B × F B,C (cid:15) (cid:15) F A,C (cid:15) (cid:15) c C / / G F A,F B × G F B,F C (cid:15) (cid:15) c B / / G F A,F C (cid:15) (cid:15) γ F K S γ G K S That is, for f : A −→ B and g : B −→ C in D , the composite γ GF comparison family isgiven by γ GFf,g = G ( γ Ff,g ) · γ GF f,F g
Then with f , g as above, and h : C −→ D in D , the following diagram in C GF ( h ) · GF ( g · f ) GF ( h ) · G (cid:0) F ( g ) · F ( f ) (cid:1) G (cid:0) F ( h ) · F ( g · f ) (cid:1) G (cid:0) F ( h ) · F ( g ) · F ( f ) (cid:1) GF h ∗ G (cid:0) γ F † f,g (cid:1) / / γ GF g · F f,F h (cid:15) (cid:15) γ GF ( g · f ) ,F h (cid:15) (cid:15) G (cid:0) F h ∗ γ F † f,g (cid:1) / / GF ( h ) · GF ( g ) · GF ( f ) G (cid:0) F ( h ) · F ( g ) (cid:1) · GF ( f ) GF h ∗ γ G † F f,F g / / γ GF g,F h ∗ GF f (cid:15) (cid:15) γ G † F f,F h · F g / / GF ( h · g · f ) G (cid:0) F ( h · g ) · F ( f ) (cid:1) G (cid:0) γ Fg,h ∗ F f (cid:1) (cid:15) (cid:15) G (cid:0) γ Fg · f,h (cid:1) (cid:15) (cid:15) G (cid:0) γ F † f,h · g (cid:1) / / GF ( h · g ) · GF ( f ) G (cid:0) γ Fg,h (cid:1) ∗ GF f (cid:15) (cid:15) γ G † F f,F ( h · g ) / / commutes. Therefore, one easily sees that γ GF does indeed satisfy the Frobenius axiom(14) of Definition 4.1. NOTE ON FROBENIUS-EILENBERG-MOORE OBJECTS IN DAGGER 2-CATEGORIES For a dagger 2-category D , [HV, 2019, Lemma 5.4] shows that a daggerlax functor −→ D from the terminal dagger 2-category to D is exactly a daggerFrobenius monad ( D, t ) in D . Moreover, each dagger 2-functor F : D −→ C is of coursea dagger lax functor, in which the comparison families γ F and δ F are simply the identityfamilies of natural transformations. Since dagger lax functors compose, this providesan immediate proof of the fact that dagger 2-functors send dagger Frobenius monads todagger Frobenius monads. Consider two dagger lax functors
F, G : D −→ C between dagger -categories D , C . A lax-natural transformation α : F −→ G – having a family τ A,B : C ( α A , · G A,B −→ C (1 , α B ) · F A,B of natural transformations – is a dagger lax-natural transformation when τ satisfies thefollowing additional coherence axiom: For every pair of arrows A B C f / / g / / in D , the following diagram in C G ( g ) · G ( f ) · α A G ( g ) · α B · F ( f ) α C · F ( g ) · F ( f ) G ( g · f ) · α A α C · F ( g · f ) G ( g ) ∗ τ † f o o τ g ∗ F ( f ) / / γ Gf,g ∗ αA (cid:15) (cid:15) αC ∗ γ Ff,g (cid:15) (cid:15) τ † g · f o o commutes, where, for simplicity, we have written τ f instead of (cid:0) τ A,B (cid:1) f . Vertical composi-tion of dagger lax-natural transformations is defined as for usual lax-natural transforma-tions: given a dagger lax functor H : D −→ C and a dagger lax-natural transformation β : G −→ H , the composite dagger lax-natural transformation δ = β · α : F −→ H isdefined by the family of -cells (cid:0) δ A = β A · α A : F ( A ) −→ H ( A ) (cid:1) A ∈D in C , and the family of -cells (cid:16) τ δf = (cid:0) β B ∗ τ αf (cid:1) · (cid:0) τ βf ∗ α A (cid:1) : H ( f ) · δ A −→ δ B · F ( f ) (cid:17) f ∈D ( A,B ) in C . Consider two dagger lax functors
F, G : D −→ C between dagger -categories D , C , and two dagger lax-natural transformations α, β : F −→ G . A modifi-cation Ξ : α β of the underlying lax-natural transformations is a dagger modification ROWAN POKLEWSKI-KOZIELL when the following additional property is satisfied: for every parallel pair f, g : A −→ B of -cells in D and every -cell φ : f −→ g in D , the following diagram in C G ( f ) · α A G ( g ) · β A α B · F ( f ) β B · F ( g ) G ( φ ) † ∗ Ξ † A o o τ βg (cid:15) (cid:15) τ αf (cid:15) (cid:15) Ξ † B ∗ F ( φ ) † o o commutes. The vertical and horizontal composition of modifications is defined as forusual modifications. Furthermore, the dagger on -cells in C induces a dagger on daggermodifications. We have already seen that a dagger lax functor T : −→ D is a daggerFrobenius monad ( D, t ) in D . Given another dagger lax functor S : −→ D , a dagger lax-natural transformation F : T −→ S is exactly a morphism of dagger Frobenius monads( D, t ) −→ ( C, s ) in D . Given another such dagger lax-natural transformation G : T −→ S ,a dagger modification F G is exactly a morphism in DFMnd ( D ) (cid:0) ( D, t ) , ( C, s ) (cid:1) from themorphism of dagger Frobenius monads corresponding to F , to the morphism of daggerFrobenius monads corresponding to G . For dagger -categories D , C , let DagLax D , C denote the dagger -category of dagger lax functors D −→ C , dagger lax-natural transformations between them,and dagger modifications between dagger lax-natural transformations. Let ∆ C : D −→ C denote the constant dagger -functor on an object C in C . The dagger lax-limit of a dag-ger lax functor F : D −→ C , if it exists, is a pair ( L, π ) where L is an object of C and π : ∆ L −→ F is a dagger lax-natural transformation such that, for each object C in C ,the dagger functor C ( C, L ) −→ DagLax D , C [∆ C , F ] of composition with π is an isomorphism of dagger categories, -natural in C . Suppose a dagger Frobenius monad (
D, t ) in a dagger 2-category D has aFrobenius-Eilenberg-Moore object. The dagger lax-limit of ( D, t ), considered as a daggerlax functor −→ D , is the pair (cid:0) FEM ( D, t ) , π (cid:1) , where π = ( u t , ξ ) is the pair as definedbelow Theorem 2.7. For, to say that ( L, π ) is a dagger lax-limit of (
D, t ) – when it exists– is to give a pair π = ( h, σ ) with h : L −→ D a 1-cell and σ : th −→ h a 2-cell in D suchthat ( h, σ ) : ( L, −→ ( D, t ) is a morphism of dagger Frobenius monads, and such thatthe following universal property is satisfied: for any C in D , 1-cell g : C −→ D and 2-cell γ : tg −→ g in D such that ( g, γ ) : ( C, −→ ( D, t ) is a morphism of dagger Frobeniusmonads, there exists a unique 1-cell n : C −→ L such that hn = g and σn = γ . But,by Proposition 2.4 this is exactly to say that L is a Frobenius-Eilenberg-Moore object for( D, t ). NOTE ON FROBENIUS-EILENBERG-MOORE OBJECTS IN DAGGER 2-CATEGORIES References
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Department of Mathematics and Applied Mathematics, University of Cape TownRondebosch 7701