Enrichment and internalization in tricategories, the case of tensor categories and alternative notion to intercategories
aa r X i v : . [ m a t h . C T ] J a n Categories internal in tricategories:B ¨ohm’s weak pseudomonoidand tensor categories
Bojana Femi´c
Mathematical Institute ofSerbian Academy of Sciences and ArtsKnez Mihailova 35,11 000 Belgrade, Serbia
Abstract
This is the second paper that completes the series started by [8]. In the for-mer, we constructed a tricategory DblPs of double categories and announced thenotion of categories internal in tricategories of the type of DblPs, as alternativeto the notion of intercategories of Grandis and Par´e, so that monoids from [1] fitas their particular case. We complete this project in the current paper and showthat weak pseudomonoids in a tricategory which has B ¨ohm’s monoidal categoryas the underlying 1-category are internal categories in DblPs. Moreover, we provethat categories enriched over certain type of tricategories may be made categoriesinternal in them, and show how the tricategory of tensor categories is an exam-ple of this over the ambient tricategory 2- Cat wk of 2-categories, pseudofunctors,pseudonatural transformations and modifications. In [1, Section 4] a Gray type monoidal structure on the category of strict double cate-gories was constructed, leading to a monoidal category (
Dbl , ⊗ ). In our previous paper[8] we presented a motive to introduce an alternative notion of intercategories from[11], which is to have a setting to which monoids in ( Dbl , ⊗ ) fit, as opposed to the othernotions of monoidal double categories which fitted the setting of intercategories. Weannounced that this alternative notion would be a category internal in the tricategory DblPs, of strict double categories, double pseudofunctors (from [21]), double pseudo-natural transformations and double modifications, which we introduced in [8, Section4]. In the present paper we formalize this, introducing at the first place a notion of acategory internal in a tricategory V , which is of a similar type as DblPs. Most impor-tantly, V has an underlying 1-category, a property to which we refer to as . Wedescribe the structure of a category internal in DblPs, and, similarly to intercategories,we give a geometric interpretation of it in the form of cubes. Moreover, we upgrade1he monoidal category ( Dbl , ⊗ ) to a (1-strict) tricategory ( Dbl , ⊗ ) and show how “weakpseudomonoids” in Dbl are categories internal in DblPs.As bicategories (which are categories enriched over the 2-category Cat of cate-gories) embed into double categories (which are categories internal in Cat ), we studywhat happens in one dimension higher. For the tricategory Tens of tensor categories,bimodule categories over the latter, bimodule functors and bimodule natural trans-formations, we show that it is a category enriched over the tricategory 2- Cat wk , of2-categories, pseudofunctors, weak natural transformations and modifications. More-over, we show that Tens is a part of the structure of a category internal in 2- Cat wk .Motivated by this example, we prove in Proposition 5.2 that under mild conditionscategories enriched over 1-strict tricategories can be made into categories internal inthem. This generalizes to tricategories analogous results from [6] and [3]. Since 2- Cat wk embeds into DblPs, this (1 × Dbl , ⊗ )and weak pseudomonoids in ( Dbl , ⊗ ) fit this setting, and we present the announcedgeometric interpretation on cubes. In the last section we prove that categories enrichedover certain type of 1-strict tricategories V are special cases of categories internal in V ,and we illustrate this result on the tricategory of tensor categories. We first recall the data that comprise a general tricategory, introduced in [10]. Havingin mind coherence Theorems 1.5 and 1.6 of [10] we consider that a tricategory consistsof the following (the direction of r in 8. and accordingly µ and ρ in 10. are slightlychanged with respect to [10], the details for 11.-13. see in loc.cit. ):1. 0-cells p , q .. , 1-cells A , B .. , 2-cells F , G ... and 3-cells α, β... ;2. a strictly associative and unital transversal composition · of 3-cells with the identity3-cell Id F for 2-cells F ;3. a strictly associative and unital vertical composition ⊙ of 2- and 3-cells with theidentity 2-cell id A for each 1-cell A , such that on 3-cells it holds:( a ) Id G ⊙ Id F = Id G ⊙ F and ( b ) ( β ′ · β ) ⊙ ( α ′ · α ) = ( β ′ ⊙ α ′ ) · ( β ⊙ α );4. horizontal composition ⊗ of 1-, 2- and 3-cells such that on 3-cells it holds:( a ) Id G ⊗ Id F = Id G ⊗ F and ( b ) ( β ′ ⊙ β ) ⊗ ( α ′ ⊙ α ) = ( β ′ ⊗ α ′ ) ⊙ ( β ⊗ α ) ;5. for pairwise horizontally and vertically composable 2-cells: p ⇓ FA (cid:28) (cid:28) / / ⇓ F ′ A ′′ B B q ⇓ GB (cid:29) (cid:29) / / ⇓ GB ′′ A A r there are natural isomorphism 3-cells (controlling the interchange law) ξ : ( G ′ ⊗ F ′ ) ⊙ ( G ⊗ F ) ⇛ ( G ′ ⊙ G ) ⊗ ( F ′ ⊙ F )2 : id B ⊗ A ⇛ id B ⊗ id A ;6. for each 0-cell p there is a 1-cell I p and a 2-cell ι P : I p −→ I p so that there is anisomorphism 3-cell Id I p (cid:27) ι p ;7. we have associativity of ⊗ on the levels of 1-, 2- and 3-cells:2-cells a C , B , A : ( C ⊗ B ) ⊗ A ⇒ C ⊗ ( B ⊗ A ) with quasi-inverse a ′ C ′ , B ′ , A ′ and invertible3-cells a C , B , A ⊙ a ′ C , B , A ⇛ Id ( C ⊗ B ) ⊗ A and Id C ⊗ ( B ⊗ A ) ⇛ a ′ C , B , A ⊙ a C , B , A ;invertible 3-cells (natural in H , G , F ): a C ′ , B ′ , A ′ ⊙ (cid:16) ( H ⊗ G ) ⊗ F (cid:17) a H , G , F ⇛ (cid:16) H ⊗ ( G ⊗ F ) (cid:17) ⊙ a C , B , A a ′ C ′ , B ′ , A ′ ⊙ (cid:16) H ⊗ ( G ⊗ F ) (cid:17) a ′ H , G , F ⇛ (cid:16) ( H ⊗ G ) ⊗ F (cid:17) ⊙ a ′ C , B , A ;the following equation of the transversal compositions of 3-cells holds: a H ′ , G ′ , F ′ · (cid:16)(cid:16) γ ⊗ ( β ⊗ α ) (cid:17) ⊙ Id (cid:17) = (cid:16) Id ⊙ (cid:16) ( γ ⊗ β ) ⊗ α (cid:17)(cid:17) · a H , G , F ;8. unity laws for ⊗ ;2-cells l A : I q ⊗ A ⇒ A and r A : A ⊗ I p ⇒ A and their quasi-inverses l ′ A and r ′ A , withinvertible 3-cells l A ⊙ l ′ A ⇛ id A , id I q ⊗ A ⇛ l ′ A ⊙ l A , r A ⊙ r ′ A ⇛ id A , id A ⊗ I p ⇛ l ′ A ⊙ l A ;invertible 3-cells (natural in F ): l B ⊙ (Id I q ⊗ F ) l F ⇛ F ⊙ l A , l ′ B ⊙ F l ′ F ⇛ (Id I q ⊗ F ) ⊙ l ′ A , r B ⊙ ( F ⊗ Id I p ) r F ⇛ F ⊙ r A , r ′ B ⊙ F r ′ F ⇛ ( F ⊗ Id I p ) ⊙ r ′ A ;for a 3-cell α : F ⇛ G one has the identities: l G · (cid:16) Id l B ⊙ (Id Id Iq ⊗ α ) (cid:17) = ( α ⊙ Id l A ) · l F r G · (cid:16) Id r B ⊙ ( α ⊗ Id Id Ip ) (cid:17) = ( α ⊙ Id r A ) · r F ;9. an invertible 3-cell a D , C , B ⊗ id A a D , CB , A id D ⊗ a C , B , A π ⇛ a DC , B , A a D , C , BA so that the following two transversal compositions of 3-cells coincide: a ⊗ id a • , •• , • id ⊗ aJ ⊗ ( H ⊗ ( G ⊗ F )) π Id ⇛ a •• , • , • a • , • , •• J ⊗ ( H ⊗ ( G ⊗ F )) Id a J , H , GF ⇛ a •• , • , • ( J ⊗ H ) ⊗ ( G ⊗ F ) a • , • , •• a JH , G , F Id ⇛ (( J ⊗ H ) ⊗ G ) ⊗ Fa •• , • , • a • , • , •• ⇓ IdId id ⊗ a ⊗ id a • , •• , • J ⊗ (( H ⊗ G ) ⊗ F )id ⊗ a Id a • , •• , • Id ⇛ a ⊗ id( J ⊗ ( H ⊗ G )) ⊗ Fa • , •• , • id ⊗ a IdId π ∗ ⇛ a ⊗ id( J ⊗ ( H ⊗ G )) ⊗ Fa − ⊗ id a •• , • , • a • , • , •• a J , H , G ⊗ idId ≡ ⇛ (( J ⊗ H ) ⊗ G ) ⊗ Fa ⊗ id a − ⊗ id a •• , • , • a • , • , •• where the fractions denote vertical compositions of both 2- and 3-cells and the2-cells a • , • , • are evaluated at 1-cells A , B , C , D ;10. there exist 3-cells: r B ⊗ id A µ B , A ⇛ a B , Id , A id B ⊗ l A l B ⊗ id A λ B , A ⇛ a Id , B , A id ⊗ l BA a B , A , Id id B ⊗ r A ρ B , A ⇛ r BA so that the following three pairs of transversal compositions of 3-cells coincide,the first one involving µ : ( ( G ⊗ Id) ⊗ Fr B ′ ⊗ id A ′ ) Id µ B ′ , A ′ ⇛ ( G ⊗ Id) ⊗ Fa • , id , • id B ′ ⊗ l A ′ a − G , Id , F Id ⇛ a • , id , • G ⊗ (Id ⊗ F )id B ′ ⊗ l A ′ Id ξ ⇛ a • , id , • [ G id B ′ ] ⊗ [ Id ⊗ Fl A ′ ] ⇓ ξ ⇓ IdId ⊗ l F (cid:26) [ G ⊗ Id r B ′ ] ⊗ [ F id A ′ ] (cid:27) r G ⊗ Id ⇛ (cid:26) [ r B G ] ⊗ [ id A F ] (cid:27) ξ − ⇛ (cid:26) r B ⊗ id A G ⊗ F (cid:27) µ B , A Id ⇛ a • , id , • id B ⊗ l A G ⊗ F Id ξ ⇛ a • , id , • [ id B G ] ⊗ [ l A F ] the second one involving λ : a id , • , • Id ⊗ ( G ⊗ F ) a − , • , • l B ′ ⊗ id A ′ a Id , G , F Id = ⇛ (Id ⊗ G ) ⊗ Fa id , • , • a − , • , • l B ′ ⊗ id A ′ ≡ ⇛ ( (Id ⊗ G ) ⊗ Fl B ′ ⊗ id A ′ ) ξ ⇛ (cid:26) [ Id ⊗ Gl B ′ ] ⊗ [ F id A ′ ] (cid:27) ⇓ Id = λ ∗ B ′ A ′ ⇓ l G ⊗ Id a id , • , • Id ⊗ ( G ⊗ F ) l B ′ A ′ Id l GF ⇛ a id , • , • a − , • , • l B ⊗ id A G ⊗ F ≡ ( l B ⊗ id A G ⊗ F ) ξ ⇛ ( [ l B G ] ⊗ [ id A F ] ) ρ : ( G ⊗ F ) ⊗ Id a • , • , id id B ′ ⊗ r A ′ a − G , F , Id Id ⇛ a • , • , id G ⊗ ( F ⊗ Id)id B ′ ⊗ r A ′ Id ξ ⇛ a • , • , id [ G id B ′ ] ⊗ [ F ⊗ Id r A ′ ] ⇓ Id ρ B ′ A ′ ⇓ IdId ⊗ r F ( ( G ⊗ F ) ⊗ Id r B ′ A ′ ) r GF ⇛ (cid:26) r BA G ⊗ F (cid:27) ρ − BA Id ⇛ a • , • , id id B ⊗ r A G ⊗ F Id ξ ⇛ a • , • , id [ id B G ] ⊗ [ r A F ] ;11. (TA1) non abelian 4-cocycle condition for π ;12. (TA2) left normalization for the 4-cocycle π , and13. (TA3) right normalization for the 4-cocycle π .A 3-category consists of the data 1. - 8., where (the equivalences) a , l and r from 7.and 8. are identities, the 3-cells ξ and ξ from 5. are identities, and 1-cells I p and 2-cells ι P : I p −→ I p from 6. are identities, too. A Gray-category di ff ers from a 3-category in thatthe interchange 3-cells ξ and ξ are not identities.As we are interested in considering “categories internal in the tricategory DblPs from[8]”, let us consider a generic tricategory V of the same type, meaning the following.Let V be weaker from a 3-category in that the horizontal associativity and unitality of 2-cells and the interchange law work up to an isomorphism, and so that the associativityand unitality of the composition of 1-cells are strict (that is, the 2-cells a C , B , A from 7. and l A and r A from 8. are identities). Then V comprises of the data 1.-5., identity 1-cells on0-cells, the conditions 11.-13. are trivially fulfilled, and the rest of the data from 7.-10.come down to the following. There are invertible 3-cells (natural in H , G , F ):( H ⊗ G ) ⊗ F a H , G , F ⇛ H ⊗ ( G ⊗ F )satisfying: a H ′ , G ′ , F ′ · (cid:16) γ ⊗ ( β ⊗ α ) (cid:17) = (cid:16) ( γ ◦ β ) ◦ α (cid:17) · a H , G , F ; invertible 3-cells (natural in F ):Id id q ⊗ F l F ⇛ F , F ⊗ Id id p r F ⇛ F , so that for a 3-cell α : F ⇛ G the following identities hold: l G · (Id Id idq ⊗ α ) = α · l F , r G · ( α ⊗ Id Id idp ) = α · r F . In 9. the 3-cell π is identity, and the equation of 3-cells comes down to a pentagonalequation. In 10. the 3-cells λ, ρ, µ are identities, and the three equations of 3-cells comedown to a septagonal and two pentagonal equations, respectively.The 3-cells l F in DblPs are actually identities, see [8, Section 4.8]. One could considera bit stricter version of the tricategory V , where both horizontal unity constraints areidentities, see Remark 3.2, e). 5 Categories internal to tricategory V When considering a category internal to an ambient weak n-category W , where n isa natural number, one should in particular have 0-cells B and B , 1-cell c : B × B B −→ B and a 2-cell α : c ( c × B id B ) −→ c ( id B × B c ) in W , where B × B B is some kind of apullback in W . We are not sure if e.g. the 1-cell c × B id B makes sense if c is not a “realmorphism” between 0-cells in W , that is, if 1-cells of W are not strictly associative andthe identity 1-cell strictly unital. (Take for W for example the bicategory of rings, ringbimodules and bimodule maps.) For this reason, for internalization we will consideronly those ambient weak n-categories W which have an underlying 1-category. Thenthe pullbacks in W we will consider to be the pullbacks in the underlying 1-categoryof W . This is in accordance with [5, Remark 2.]. Such a weak n-category we will call , as we already did in [8]. A folklore example of internal categories are pseudodouble categories for which this condition is fulfilled: they are internal categories in the2-category of categories. Also the tricategory Bicat of bicategories, pseudofunctors,pseudonatural transformations and modifications is 1-strict.Let now V be a tricategory from the end of the previous section. It is 1-strict andit has pullbacks if so does its underlying 1-category. We want to define a categoryinternal in V . In [5, Definition 2.11] an internal category in a Gray-category wasdefined. Therein, the definition of a Gray-category is based on whisker, so that insteadof a full interchange law there appears an isomorphism 3-cell sw (with an additionalrule for whiskering). In contrast to [5, Definition 2.11], working in V we will have to useassociativity constraint a , left and right unity constraints l and r on 2-cells of V and thefull interchange law ξ at the appropriate places. Let (V4-1) - (V4-5) denote the axiomscorresponding to (C4-1) - (C4-5) from [5, Definition 2.11] obtained by modifying thelatter as indicated above, adding a , l and r on 2-cells of V and ξ where it corresponds.Writing out the diagrams in our style for these axioms would take several pages, so weomit to type them out. From the point of view of V , the 3-cell sw can be defined as thefollowing transversal composition of 3-cells: (cid:18) [ α | Id][Id | β ] (cid:19) ξ −→ (cid:20)(cid:16) α Id (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:16) Id β (cid:17)(cid:21) (cid:27) (cid:20)(cid:16) Id α (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:16) β Id (cid:17)(cid:21) ξ − −→ (cid:18) [Id | β ][ α | Id] (cid:19) , for 2-cells α and β , where the middle isomorphism stands for the composition of one“vertical” unity constraint with the inverse of the other in the appropriate order, inboth coordinates. Here notation [ α | β ], closer to the diagramatical “bubble” notation forbicategories, stands for the horizontal composition β ⊗ α . As we said at the beginningof Section 2, we consider by the coherence Theorem [10, Theorem 1.5] that these unityconstraints are identities, so sw will be identity. In the next definition, to simplify thenotation, the unsubscribed symbol × will stand for × B at certain places. Definition 3.1
Let V be a 1-strict tricategory of the type evoked above with pullbacks. Acategory internal in V consists of:1. 0-cells B and B of V;2. four 1-cells B −→←−−→ B , B × B B c −→ B in V, where we call the two 1-cells s , t : B −→ B source and target morphisms, u : B −→ B the unit (or identity) morphism andc : B × B B −→ B composition; . equivalence 2-cells a ∗ : c ⊗ ( id B × B c ) ⇒ c ⊗ ( c × B id B ) , l ∗ : c ⊗ ( u × B id B ) ⇒ id B andr ∗ : c ⊗ ( id B × B u ) ⇒ id B in V;4. 3-cells π ∗ : Id c ⊗ (Id id B × a ∗ ) a ∗ ⊗ Id × c × Id c ⊗ ( a ∗ × Id id B ) ⇛ a ∗ ⊗ Id × × c Id c ⊗ Nat ( c × × × c ) a ∗ ⊗ Id c × × µ ∗ : Id c ⊗ (Id id B × r ∗ ) ⇛ a ∗ ⊗ Id id B × u × id B Id c ⊗ ( l ∗ × id B ) λ ∗ : Id c ⊗ (Id id B × l ∗ ) ⇛ a ∗ ⊗ Id id B × id B × u Id c ⊗ Nat ( c × id B ) ⊗ ( id B × id B × u ) l ∗ ⊗ Id c ρ ∗ : a ∗ ⊗ Id u × id B × id B Id c ⊗ ( r ∗ × id B ) ⇛ Id c ⊗ Nat ( c × id B ) ⊗ ( u × id B × id B ) r ∗ ⊗ Id c Nat id B ⊗ c ǫ ∗ : l ∗ ⊗ Id u ⇛ Id c ⊗ Nat ( id B × u ) ⊗ u r ∗ ⊗ Id u where the 2-cells Nat are all identities, and their subscripts are their 1-cell domains (seethe Remark below), which satisfy axioms (V4-1) - (V4-5) and symmetric versions of(V4-1), (V4-3) and (V4-4);the above data should moreover satisfy the following compatibility conditions:sp = tp , su = id B = tu , sc = sp , tc = tp , s ◦ l ∗ = id s = s ◦ r ∗ , t ◦ l ∗ = id t = t ◦ r ∗ , s ◦ a ∗ = id sp , t ◦ a ∗ = id tp , s • π ∗ = Id id sp , t • π ∗ = Id id tp , s • µ ∗ = s • λ ∗ = s • ρ ∗ = Id id sp , t • µ ∗ = t • λ ∗ = t • ρ ∗ = Id id tp , where p i , i = , , , , are 1-cells projections from the corresponding pullbacks in V. Remark 3.2
When writing out the 3-cells (and the axioms) in our definition, we omitedwriting the (co)domain 1-cells on which the 2-cells act, but we want to draw reader’sattention to the following few things.a) By the pullback properties in the underlying 1-category of V , for all 1-cells F , G , H , K in V it holds: ( G ⊗ F ) × ( H ⊗ K ) = ( G × H ) ⊗ ( F × K ) . (1)b) By the pullback properties it holds: ( id B × u ) ⊗ c = c × u , and c ⊗ id B = c and similarlyfor u and their symmetric counterparts.7) To understand the (co)domains of the 2-cells Id • × a ∗ , Id • × r ∗ , Id • × l ∗ and their sym-metric counterparts in item 3 in the definition above, have in mind the property(1).d) In order to be precise, in some vertical compositions of horizontal compositions of2-cells one should take into account certain naturality 2-cells Nat • , whose subscripts(usually) are their 1-cell domains. They will actually all be identities by (1) andalso by the properties like in the item b) above. For example: Nat ( c × id B ) ⊗ ( id B × id B × u ) :( c × id B ) ⊗ ( id B × id B × u ) ⇒ ( id B × u ) ⊗ c , by (1) it is easily seen that both (co)domain1-cells equal c × u .e) In order to simplify the diagrams and the definition, we could want the followingtwo vertical compositions of horizontal compositions of 2-cells to be equal: (cid:25) (cid:25) ⇓ Id C C (cid:25) (cid:25) ⇓ Id C C (cid:25) (cid:25) ⇓ Id C C (cid:25) (cid:25) ⇓ α C C (cid:25) (cid:25) ⇓ Id C C (cid:25) (cid:25) ⇓ α C C (cid:25) (cid:25) ⇓ Id C C (cid:25) (cid:25) ⇓ Id C C When V = DblPs, applying [8, Propositions 4.13 and 4.14] one can see that the twocompositions above di ff er by a modification given by the globular 2-cells δ α i , id for i = ,
2. Thus one could restrict to a full sub tricategory of V whose 1-cells are doublepseudofunctors F which applied to the identity 1h- and 1v-cells give identities. Thenone could also consider that their distinguished 2-cells F A and F A (see Subsection 4.2below) are identities (for all 0-cells A of the domain strict double category of F ), thusthe unity constraints for the horizontal composition would both be identities (see [8,Section 4.8]), and one could also consider that 2-cells of the sub tricategory are thosedouble pseudonatural tranformations α of V whose associated globular 2-cells δ α i , id for i = , Remark 3.3
Let us comment the axioms (V4-1) - (V4-5). We do it for the case of thefull sub tricategory of V from point e) in the above Remark, let us denote it by V ∗ .Although the 3-cell sw is identity in our context, we will mention it, as it helps to betterunderstand technically how the compositions of 3-cells are made in the axioms.(V4-1) comprises of λ ∗ u , ε ∗ , µ ∗ u , sw and the first 1-cell in its domain 2-cell is id B ⊗ u ⊗ u (in the symmetric version it is u ⊗ u ⊗ id B ).(V4-2) comprises of λ ∗ , ρ ∗ , sw , a , ξ and the first 1-cell in its domain 2-cell is u ⊗ id B ⊗ u .(V4-3) comprises of λ ∗ , π ∗ , sw , a , ξ and the first 1-cell in its domain 2-cell is id B ⊗ id B ⊗ id B ⊗ u (in the symmetric version it is u ⊗ id B ⊗ id B ⊗ id B ). It corresponds to the normalization in the first and the fourth coordinate .(V4-4) comprises of µ ∗ , λ ∗ , π ∗ , sw , a , ξ and the first 1-cell in its domain 2-cell is id B ⊗ id B ⊗ u ⊗ id B (in the symmetric version it is id B ⊗ u ⊗ id B ⊗ id B ). It corresponds to the normalization in the second and the third coordinate .(V4-5) comprises of π ∗ , sw , ξ . It corresponds to the ∗ .Observe in these axioms that the 3-cells sw , a , ξ are the distinguished 3-cells from theambient tricategory V . 8 Intercategories versus categories internal in
DblPs
We first recall the structure of intercategories and then turn to categories internal inDblPs.
Intercategories are defined in [11] as pseudocategories in the 2-category
LxDbl of pseudodouble categories, lax double functors and horizontal transformations. In loc.cit. ina pseudo double category the vertical direction is considered bicategorical, that is, itscategory of objects consists of objects and horizontal morphisms, whereas the categoryof morphisms consists of vertical morphisms and squares, so that objects, verticalmorphisms and squares form a bicategory.Let A be an intercategory. As an internal category in LxDbl , it consists of pseudodou-ble categories D and D , lax double functors s , t : D −→ D , u : D −→ D , m : D × D D −→ D and horizontal transformations a ∗ : m ( id × B m ) −→ m ( m × B id ) , l ∗ : m ( u × B − ) −→ − and r ∗ : m ( − × B u ) −→ − satisfying the corresponding properties. As done byGrandis and Par´e we denote this structure formally by D × D D −→−→−→ D −→←−−→ D (2)where the additional two arrows D × D D −→ D stand for the two projections. Thetwo projections as well as s and t are taken to be strict double functors. As these are threepseudodouble categories, that is categories internal in the 2-category of categories, eachof them themselves can be represented by an analogous diagram but now of categoriesand functors. To make this representation simpler, it is more convenient to write (2) as: C −→−→−→ B −→←−−→ A (3)and then indicate the pseudo double category structure of A by A × A A −→ A −→←−−→ A with its respective functors s A , t A , u A and m A , and analogously for B and C . Then wemay obtain the following grid of categories and functors: (4) C B ✲✲✲ M p p ✲ A ✛ ✲ s u t C B ✲✲✲ M p p ✲ A ✛ ✲ s u t ✻❄✻ s C u C t C ✻❄✻ s B u B t B ✻❄✻ s A u A t A ✲ C × C C B × B B ✲✲ p × p M × M p × p ✲ A × A A ✲✲ s × s u × u t × t p C m C p C ✻✻✻ ✻✻✻ p B m B p B ✻✻✻ p A m A p A where C (cid:27) B × A B , C (cid:27) B × A B C × C C (cid:27) ( B × B B ) × A × A A ( B × B B ) (cid:27) ( B × A B ) × B × A B ( B × A B ) . That is, each of the three rows above is a pseudo double category, except from the firstone which is strict, (related with condition (1) in [11, Section 3]), and the same holdsfor the three columns (related with conditions (2)-(4) in loc.cit. ). In this grid note thatthe functors s i , t i , u i , M i , i = , s , t , u , m from (2). The functors u and M are equipped with naturaltransformations expressing that they both are lax multiplicative and lax unital. The fourquadrants carry natural transformations determining the latter four functor structures:on the left down the lax multiplicative structure of M ((20) from [11, Section 3]) andon the left up the lax unitality structure of M (19), right down the lax multiplicativestructure of u (18) and right up its lax unitality structure (17), right down the strictmultiplicative structure for s , t (8), and right up their strict unitality structures (7).The rest of the natural (identity) transformations in the four quadrants represent thecompatibility of the source, target and projections strict functors of A , B and C , on onehand, with projections, unity and composition of the pseudo double category (3) above(rows of the grid): (9)-(10), (13)-(14) and (15)-(16) from [11, Section 3] respectively, andwith its source and target functors (5)-(6), on the other hand. Also the compatibility ofunity and composition functors of B and C with the projection functors of (3) above arerepresented, which corresponds to (11)-(12) of [11, Section 3].For the lax functors u and m from (2) above the corresponding natural transformationsatisfies a coherence hexagonal and two triangular conditions, leading to additional sixconditions (21)-(26) in [11, Section 3]. For each of the horizontal transformations a ∗ , l ∗ , r ∗ two axioms should be satisfied, which adds another 6 conditions: (27)-(32) in [11,Section 3].Two notions of monoidal double categories, prior to the construction of (monoids in)the Gray type monoidal category ( Dbl , ⊗ ) in [1], are examples of intercategories. Theseare monoids in the Cartesian monoidal category of strict double categories and strictdouble functors from [2], and pseudomonoids in the Cartesian monoidal 2-category PsDbl of pseudo double categories, pseudo double functors and vertical transforma-tions from [20]. As explained in [13, Section 3.1], a pseudomonoid in
PsDbl consists ofa pseudo double category D and pseudo double functors m : D × D −→ D and u : 1 −→ D which make D an intercategory D × D −→−→−→ D −→←−−→ D = D , for D take the trivial double category 1, the pullback D × D D comesdown to the above Cartesian product).In a monoid A in ( Dbl , ⊗ ), equipped with a strict double functor m : A ⊗ A −→ A ,horizontal and vertical 1-cells of the type ( f ⊗ ⊗ g ) and (1 ⊗ g )( f ⊗ f ⊛ g by m (of some 1-cell in A ⊗ A ): as m (cid:16) ( f ⊗ ⊗ g ) (cid:17) , or m (cid:16) (1 ⊗ g )( f ⊗ (cid:17) . Both choices yield a double pseudo functor from the Cartesian product double category ⊛ : A × A −→ A . As double pseudofunctors do not appear in the 2-category LxDbl , ⊛ can not be taken for a composition10n the pullback as in (5), and monoids in ( Dbl , ⊗ ) can not be seen as particular cases ofintercategories. For this reason we substituted LxDbl by another category which hasdouble pseudo functors for 1-cells (which are psuedo in both directions), rather thanlax double functors (which are strict in one direction and lax in the other), as in
LxDbl .This led us to obtain the tricategory DblPs in [8], and propose to consider internalcategories in DblPs, so that monoids in (
Dbl , ⊗ ) can fit this new gadget. DblPs
An internal category in DblPs consists of strict double categories D and D , strictdouble functors S , T : D −→ D , double pseudo functors U : D −→ D , M : D × D D −→ D , double pseudonatural transformations a ∗ , l ∗ , r ∗ and double modifications π ∗ , µ ∗ , λ ∗ , ρ ∗ , ε ∗ , satisfying the corresponding axioms from [8, Section 4]. Both doublepseudo functors U and M are equipped with distinguished globular 2-cells (set F forany of U and M ): F ( A ) F ( C ) ✲ F ( g f ) F ( A ) F ( B ) ✲ F ( f ) F ( C ) ✲ F ( g ) ❄ = ❄ = F g f F ( A ) F ( A ) ✲ F ( id A ) ❄ = F ( A ) F ( A ) ✲ = ❄ = F A F ( A ) F ( A ) ✲ = F ( A ′ ) ❄ F ( u ) F ( A ′′ ) F ( A ′′ ) ✲ = ❄ F ( v ) ❄ F ( vu ) F vu F ( A ) F ( A ) ✲ = ❄ F ( id A ) F ( A ) F ( A ) ✲ = ❄ = F A satisfying the following axioms, where f , g , h are composable 1h-cells, u , v , w are com-posable 1v-cells, a and b are 2-cells composable horizontally and a and a ′ are 2-cellscomposable vertically (note that here, for simplicity of the notation, we are denotingboth 1h- and 1v-composition by juxtaposition, the di ff erence is clear from the lettersdenoting 1-cells). Coherence in the 1h-direction: F ( A ) F ( C ) ✲ F ( g f ) F ( D ) ✲ F ( h ) F ( A ) F ( B ) ✲ F ( f ) F ( C ) ✲ F ( g ) F ( D ) ✲ F ( h ) ❄ = ❄ = ❄ = F g f Id F ( A ) F ( D ) ✲ F ( h ( g f )) ❄ = ❄ = F h ( g f ) = F ( A ) F ( B ) ✲ F ( f ) F ( D ) ✲ F ( hg ) F ( A ) F ( B ) ✲ F ( f ) F ( C ) ✲ F ( g ) F ( D ) ✲ F ( h ) ❄ = ❄ = ❄ = Id F hg F ( A ) F ( D ) ✲ F (( hg ) f ) ❄ = ❄ = F ( hg ) f F ( A ) F ( A ) ✲ F ( id A ) F ( B ) ✲ F ( f ) ❄ = F ( A ) F ( A ) ✲ = F ( B ) ✲ F ( f ) ❄ = ❄ = F A Id = F ( A ) F ( A ) ✲ F ( id A ) F ( B ) ✲ F ( f ) F ( A ) F ( B ) ✲ F ( f id A ) ❄ = ❄ = F fid A F ( A ) F ( B ) ✲ F ( f ) F ( B ) ✲ F ( id B ) ❄ = F ( A ) F ( B ) ✲ F ( f ) F ( B ) ✲ = ❄ = ❄ = F B Id = F ( A ) F ( B ) ✲ F ( f ) F ( B ) ✲ F ( id B ) F ( A ) F ( B ) , ✲ F ( id B f ) ❄ = ❄ = F id B f F ( A ) F ( A ) ✲ = F ( A ) ✲ = F ( A ′ ) ❄ F ( u ) F w ( vu ) F ( A ′′ ) ✲ = ❄ F ( v ) F ( A ′′ ) ❄ F ( vu ) F vu ❄ F ( w )Id ❄ F ( w ) F ( A ′′′ ) ✲ = F ( A ′′′ ) ✲ = G ( B ′ ) ❄ F ( w ( vu )) = F ( A ) F ( A ) ✲ = F ( A ) ✲ = F ( A ′ ) ❄ F ( u ) F ( wv ) u F ( A ′′ ) ❄ F ( v ) ❄ F ( wv ) F ( A ′ ) ❄ F ( u )Id ❄ F ( w ) F ( A ′′′ ) ✲ = F ( A ′′′ ) ✲ = ✲ = F ( A ′′′ ) ❄ F (( wv ) u ) F wv F ( A ) F ( A ) ✲ = F ( A ) F ( A ) ✲ = F ( A ′ ) F ( A ′ ) ✲ = ❄ = ❄ F ( u ) ❄ F ( id A ) ❄ F ( u ) F A Id = F ( A ) F ( A ) ✲ = F ( A ′ ) F ( A ′ ) ✲ = ❄ F ( uid A ) F ( A ) ❄ F ( id A ) ❄ F ( u ) F uid A and F ( A ) F ( A ) ✲ = F ( A ′ ) F ( A ′ ) ✲ = F ( A ′ ) F ( A ′ ) ✲ = ❄ F ( u ) ❄ = ❄ F ( u ) ❄ F ( id A ′ )Id F A ′ = F ( A ) F ( A ) ✲ = F ( A ′ ) F ( A ′ ) , ✲ = ❄ F ( id A ′ u ) F ( A ′ ) ❄ F ( u ) ❄ F ( id A ′ ) F id A ′ u coherence for the composition of and unity 2-cells, horizontally: F ( A ) F ( C ) ✲ F ( g f ) F ( A ) F ( B ) ✲ F ( f ) F ( C ) ✲ F ( g ) F g f ❄ F ( v ) ❄ = ❄ = F ( a ) F ( b ) F ( A ′ ) F ( B ′ ) ✲ F ( f ′ ) F ( C ′ ) ✲ F ( g ′ ) ❄ F ( u ) ❄ F ( w ) = F ( A ) F ( C ) ✲ F ( g f ) F ( A ′ ) F ( C ′ ) ✲ F ( g ′ f ′ ) ❄ F ( u ) ❄ F ( w ) F ( a | b ) F g ′ f ′ F ( A ′ ) F ( B ′ ) ✲ F ( f ′ ) F ( C ′ ) ✲ F ( g ′ ) ❄ = ❄ = F ( A ) F ( A ) ✲ = F ( A ) F ( A ) ✲ F ( id A ) F ( A ′ ) F ( A ′ ) ✲ F ( id A ′ ) ❄ = ❄ F ( u ) ❄ = ❄ F ( u ) F − A F (Id u ) = F ( A ) F ( A ) ✲ = F ( A ′ ) F ( A ′ ) ✲ = F ( A ′ ) F ( A ′ ) ✲ F ( id A ′ ) ❄ F ( u ) ❄ = ❄ F ( u ) ❄ = F (Id u ) F − A ′ and vertically: F ( A ) F ( A ) ✲ = ✲ F ( f ) F ( A ′ ) ❄ F ( u ) F vu F ( A ′′ ) F ( A ′′ ) ✲ = ❄ F ( v ) ❄ F ( vu ) F ( aa ′ ) F ( B ) F ( B ′′ ) ❄ F ( v ′ u ′ ) ✲ F ( h ) = F ( A ) F ( B ) ✲ F ( f ) ✲ = ❄ F ( u ) F v ′ u ′ F ( A ′′ ) ✲ F ( h ) ❄ F ( v ) F ( B ′ ) ❄ F ( u ′ ) F ( B ′′ ) ❄ F ( v ′ ) F ( a ) F ( A ′ ) ✲ F ( g ) F ( B ) F ( B ′′ ) ❄ F ( v ′ u ′ ) ✲ = F ( a ′ ) ( A ) F ( A ) ✲ = F ( B ) ✲ F ( f ) ❄ F ( id A ) F ( A ) F ( A ) ✲ = F ( B ) ✲ F ( f ) ❄ = ❄ F ( id B ) F A F (Id f ) = F ( A ) F ( B ) ✲ F ( f ) F ( B ) ✲ = F ( A ) F ( B ) ✲ F ( f ) F ( B ) . ✲ = ❄ = ❄ = ❄ F ( id B )Id F ( f ) F B The above three coherences in the 1v-direction for U and M correspond to axioms(21)-(26) of [11, Section 3], respectively. The analogous six coherences in the 1h-directiondo not appear there. The two (horizontally) globular 2-cells F vu and F A for U and M correspond to natural transformations (17)-(20): U A = τ, U vu = µ, M A = δ, M vu = χ ,and the above two coherences for the composition of and unity 2-cells in the verticaldirection for U and M correspond to naturalities of (17)-(20). One can analogouslyformulate natural transformations in the horizontal direction, introducing additionaltwo (vertically) globular 2-cells F g f and F A for U and M and the above two coherencesfor the composition of and unity 2-cells in the horizontal direction, which correspondto their naturalities. (To formulate these natural transformations in the horizontaldirection change the roles of vertical and horizontal cells in the definition of two cate-gories determining a strict double category.) For the sake of comparing this structureto intercategories, for mnemotechnical reasons we could denote these distinguished(vertically) globular 2-cells as follows: U A = τ ′ , U g f = µ ′ , M A = δ ′ , M g f = χ ′ .Summing up, for the double pseudo functors U and M we have eight globular2-cells: U g f , U A , U vu , U A , M g f , M A , M vu , M A , which satisfy in total 20 axioms named above. We will denote their actions as follows.Let us denote the image under M : D × D D −→ D of ( y , x ) by ( x | y ) for any of thefour types of cells ( y , x ) ∈ D × D D . Moreover, let us denote by Id hx the image under U : D −→ D of any of the four types of cells x in D . Now for 1h-cells g , g ′ , f , f ′ and1v-cells u , u ′ , v , v ′ of D (for the action of M ), respectively of D (for the action of U ) wewill write: χ : ( u | u ′ )( v | v ′ ) ⇒ ( uv | u ′ v ′ ) , δ : id v ( A | A ′ ) ⇒ ( id vA | id vA ′ ) , µ : Id hu Id hv ⇒ Id h uv , τ : id v Id hA ⇒ Id hid vA (6) ( g f | g ′ f ′ ) ⇓ χ ′ ( g | g ′ )( f | f ′ ) ( id hA | id hA ′ ) ⇓ δ ′ id h ( A | A ′ ) Id hg f ⇓ µ ′ Id hg Id hf Id hid hA ⇓ τ ′ id h Id hA (7)here id vA denotes the identity 1v-cell on A (observe that the composition in the juxtapo-sitions is read from right to left, while in ( −|− ) it is done the other way around!).A double pseudonatural transformation α : F ⇒ G between double pseudo functors F and G consists of a vertical pseudonatural transformation α : F ⇒ G and a horizontalpseudonatural transformation α : F ⇒ G , both of which by [8, Definition 4.1] are givenby two distinguished globular 2-cells δ α , u and δ α , f and satisfy 5 axioms (two of themare trivial and one is simplified in the context of intercategories), two distinguished2-cells t α f and r α u for every 1v-cell u and 1h-cell f , which have to satisfy 6 axioms intotal, by [8, Definition 4.10]. Comparing such a structure of a double pseudonatural13ransformation with the context of intercategories, that is, comparing 2-cells of thetricategory DblPs and the 2-category LxDbl , one finds that in the latter context only α appears (with δ α , f trivial), being the resting data α , four 2-cells and 6 axioms new inour context.Thus each of double pseudonatural transformations a ∗ : M (Id × D M ) −→ M ( M × D Id) , l ∗ : M ( U × D − ) −→ − and r ∗ : M ( − × D U ) −→ − is equipped with 6 distinguished2-cells for every 1v-cell u and 1h-cell f and satisfies 16 axioms. This makes 18 dis-tinguished 2-cells and 48 axioms. As commented in [8, Subsection 4.3], if doublepseudonatural transformations come from Θ -double pseudonatural transformations(the 2-cells t α f and r α u come from a 2-cell Θ α A ), as indicated in [8, Proposition 4.16], thentwo axioms become trivially fulfilled for each double pseudonatural transformation,reducing the amount of axioms to 42. The 6 conditions (27)-(32) from [11, Section 3]for horizontal transformations, corresponding to our a ∗ , l ∗ , r ∗ , together with the corre-sponding three naturality conditions, so 9 in total, are substituted by 42 or 48 axiomsin our context.We present next the half of axioms for the double pseudonatural transformation a ∗ : M (Id × D M ) −→ M ( M × D Id), the other half is analogous and symmetric. Theaxioms for l ∗ and r ∗ are much simpler, it would occupy too much space to write out allthe axioms for all of the transformations here, so we omit them.For the double pseudonatural transformation a ∗ : (( −|− ) |− ) ⇒ ( −| ( −|− )) let usshorten: L = ( −|− ) |− = (( −|− ) |− ) and R = −| ( −|− ) = ( −| ( −|− )). The 1v- and 1h-composition in D we will denote by fractions and juxtapositions: uv and g f , respec-tively. The distinguished globular 2-cells for the double pseudonatural transformations L and R are given by: L vu = (cid:16) ( u | u ′ ) | u ′′ ( v | v ′ ) | v ′′ χ •• , • ⇒ ( u | u ′ )( v | v ′ ) | u ′′ v ′′ χ | ⇒ (cid:16) uv | u ′ v ′ (cid:17) | u ′′ v ′′ (cid:17) , R vu = (cid:16) u | ( u ′ | u ′′ ) v | ( v ′ | v ′′ ) χ • , •• ⇒ uv | ( u ′ | u ′′ )( v ′ | v ′′ ) | χ ⇒ uv | (cid:16) u ′ v ′ | u ′′ v ′′ (cid:17)(cid:17) L A = (cid:16) Id v ( A | A ′ ) | A ′′ δ •• , • ⇒ [Id v ( A | A ′ ) | Id vA ′′ ] [ δ | ⇒ [Id vA | Id vA ′ ] Id vA ′′ (cid:17) R A = (cid:16) Id vA | ( A ′ | A ′′ ) δ • , •• ⇒ [Id vA | Id v ( A ′ | A ′′ ) ] [1 | δ ] ⇒ Id vA [Id vA ′ | Id vA ′′ ] (cid:17) L g f = (cid:16) ( f | g ) | ( f ′ | g ′ ) (cid:17) | ( f ′′ | g ′′ ) ⇓ χ ′ | Id( f ′ f | g ′ g ) | ( f ′′ | g ′′ ) ⇓ χ ′ f ′′ ( f ′ f ) | g ′′ ( g ′ g ) R g f = ( f | g ) | (cid:16) ( f ′ | g ′ ) | ( f ′′ | g ′′ ) (cid:17) ⇓ Id | χ ′ ( f | g ) | ( f ′′ f ′ | g ′′ g ′ ) ⇓ χ ′ f ′′ ( f ′ f ) | g ′′ ( g ′ g ) L A = ( id hA | id hA ′ ) | id hA ′′ ⇓ δ ′ | Id id hA | A ′ | id hA ′′ ⇓ δ ′ id h ( A | A ′ ) | A ′′ R A = id hA | ( id hA ′ | id hA ′′ ) ⇓ Id | δ ′ id hA | id hA ′ | A ′′ ⇓ δ ′ id hA | ( A ′ | A ′′ ) a ∗ are the following(the order of subindices in the 1-cells ( a ∗ ) • , • , • accompanies the notation of 1h-cells usedin the diagrams, ˜ f stands shortly for a triple f , f ′ , f ′′ and so on): ✲ ( f | f ′ ) | f ′′ ✲ ( a ∗ ) B , B ′ , B ′′ ✲ ( g | g ′ ) | g ′′ ✲ ( a ∗ ) B , B ′ , B ′′ ❄ ( u | u ′ ) | u ′′ ❄ ( v | v ′ ) | v ′′ ❄ v | ( v ′ | v ′′ )( a | a ′ ) | a ′′ ( a ∗ ) ˜ v ✲ ( a ∗ ) A , A ′ , A ′′ ✲ g | ( g ′ | g ′′ ) ❄ = ❄ = δ a ∗ , ˜ g = ❄ = ❄ ( u | u ′ ) | u ′′ ✲ ( f | f ′ ) | f ′′ ✲ ( a ∗ ) B , B ′ , B ′′ ✲ ( a ∗ ) A , A ′ , A ′′ ✲ f | ( f ′ | f ′′ ) ❄ = δ a ∗ , ˜ f u | ( u ′ | u ′′ ) ❄ ( a ∗ ) ˜ u ✲ ( a ∗ ) A , A ′ , A ′′ ✲ g | ( g ′ | g ′′ ) ❄ v | ( v ′ | v ′′ ) a | ( a ′ | a ′′ ) ✲ = ✲ a ∗ A , A ′ , A ′′ ❄ ( u | u ′ ) | u ′′ L ˜ v ˜ u ✲ = ❄ ( v | v ′ ) | v ′′ ❄ ( uv | u ′ v ′ ) | u ′′ v ′′ ( a ∗ ) ˜ v ˜ u ❄ uv | ( u ′ v ′ | u ′′ v ′′ ) ✲ a ∗ A , A ′ , A ′′ = ✲ a ∗ A , A ′ , A ′′ ✲ = ❄ ( u | u ′ ) | u ′′ R ˜ v ˜ u ✲ a ∗ A , A ′ , A ′′ ❄ ( v | v ′ ) | v ′′ ❄ v | ( v ′ | v ′′ ) ❄ u | ( u ′ | u ′′ ) a ∗ ˜ u ✲ a ∗ A , A ′ , A ′′ ❄ uv | ( u ′ v ′ | u ′′ v ′′ ) ✲ = a ∗ ˜ v ✲ = ✲ a ∗ A , A ′ , A ′′ ❄ L ( id ˜ A ) ✲ = ✲ a ∗ A , A ′ , A ′′ ❄ = ❄ R ( id ˜ A ) L ˜ A a ∗ id ˜ A = ✲ a ∗ A , A ′ , A ′′ ✲ = ❄ = ✲ a ∗ A , A ′ , A ′′ ✲ = ❄ = ❄ R ( id ˜ A )Id a ∗ A , A ′ , A ′′ R ˜ A ✲ ( g f | g ′ f ′ ) | g ′′ f ′′ ✲ a ∗ C , C ′ , C ′′ ✲ a ∗ A , A ′ , A ′′ ✲ g f | ( g ′ f ′ | g ′′ f ′′ ) ❄ = ❄ = δ a ∗ , ˜ g ˜ f = ✲ ( g f | g ′ f ′ ) | g ′′ f ′′ ✲ ( f | f ′ ) | f ′′ ✲ ( g | g ′ ) | g ′′ ✲ a ∗ C , C ′ , C ′′ ❄ = ❄ = L ˜ g ˜ f δ a ∗ , ˜ g ✲ ( f | f ′ ) | f ′′ ✲ a ∗ B , B ′ , B ′′ ✲ g | ( g ′ | g ′′ ) ❄ = ❄ = ✲ a ∗ A , A ′ , A ′′ ✲ f | ( f ′ | f ′′ ) ✲ g | ( g ′ | g ′′ ) ❄ = ❄ = δ a ∗ , ˜ f R − g ˜ f ✲ g f | ( g ′ f ′ | g ′′ f ′′ ) ❄ = ❄ = = ❄ = ❄ = ✲ L ( id ˜ A )( L ˜ A ) − ✲ a ∗ A , A ′ , A ′′ ✲ a ∗ A , A ′ , A ′′ ✲ R ( id ˜ A ) ❄ = ❄ = δ a ∗ , id ˜ A R ˜ A ❄ = ❄ = ✲ = = ✲ a ∗ A , A ′ , A ′′ ✲ a ∗ A , A ′ , A ′′ ❄ = ❄ = Id ˜ A and there are five analogous axioms for vertical pseudonatural transformation in a ∗ ,then there are axioms (T3-1), (T3-2) and (T3-3) of which one half is:(T3-1) ✲ ( f | f ′ ) | f ′′ ✲ ( a ∗ ) B , B ′ , B ′′ ✲ ( g | g ′ ) | g ′′ ✲ ( a ∗ ) B , B ′ , B ′′ ❄ ( u | u ′ ) | u ′′ ❄ ( v | v ′ ) | v ′′ ❄ v | ( v ′ | v ′′ )( a | a ′ ) | a ′′ ( a ∗ ) ˜ v ✲ g | ( g ′ | g ′′ ) ❄ a ∗ ( ˜ A ) ❄ = t a ∗ ˜ g = ✲ = ❄ ( u | u ′ ) | u ′′ δ a ∗ , ˜ u ✲ = ❄ a ∗ ( ˜ A ) ❄ a ∗ ( ˜ A ) ❄ u | ( u ′ | u ′′ ) ✲ ( f | f ′ ) | f ′′ ✲ ( a ∗ ) B , B ′ , B ′′ ✲ f | ( f ′ | f ′′ ) ❄ = t a ∗ ˜ f ✲ g | ( g ′ | g ′′ ) ❄ v | ( v ′ | v ′′ ) a | ( a ′ | a ′′ )(T3-2) ✲ ( g | g ′ ) | g ′′ ✲ ( a ∗ ) C , C ′ C ′′ ✲ ( f | f ′ ) | f ′′ ✲ ( a ∗ ) B , B ′ , B ′′ ✲ g | ( g ′ | g ′′ ) ❄ = ❄ = δ a ∗ , ˜ g t a ∗ ˜ f ✲ f | ( f ′ | f ′′ ) ❄ a ∗ ( ˜ A ) ❄ = = ✲ ( f | f ′ ) | f ′′ ✲ ( g | g ′ ) | g ′′ ✲ ( a ∗ ) C , C ′ C ′′ ( a ∗ ) ˜ f ❄ a ∗ ( ˜ A ) t a ∗ ˜ g ❄ a ∗ ( ˜ B ) ❄ = ✲ f | ( f ′ | f ′′ ) ✲ g | ( g ′ | g ′′ )(T3-3) t a ∗ ˜ g ˜ f = ✲ ( g f | g ′ f ′ ) | g ′′ f ′′ ✲ ( f | f ′ ) | f ′′ ✲ ( g | g ′ ) | g ′′ ✲ ( a ∗ ) C , C ′ C ′′ ❄ = ❄ = L ˜ g ˜ f ( a ∗ ) f , f ′ , f ′′ ❄ ( a ∗ ) A , A ′ A ′′ t a ∗ ˜ g ❄ ( a ∗ ) B , B ′ B ′′ ❄ = ✲ f | ( f ′ | f ′′ ) ✲ g | ( g ′ | g ′′ ) ❄ = ❄ = R − g ˜ f ✲ g f | ( g ′ f ′ | g ′′ f ′′ )and the other half is symmetric to these (exchange f ↔ u , a ∗ ↔ a ∗ etc.)16inally, we have double modifications π ∗ , µ ∗ , λ ∗ , ρ ∗ , ε ∗ , satisfying the correspondingaxioms. Each modification is given via pairs of a horizontal and a vertical globular2-cell for every 0-cell of the domain double category (copies of D ) which need to satisfy6 axioms in total, [8, Definition 4.18]. So, for example π ∗ : Id c ⊗ (Id id B × a ∗ ) a ∗ ⊗ Id × c × Id c ⊗ ( a ∗ × Id id B ) = : Λ ⇛ P : = a ∗ ⊗ Id × × c Id c ⊗ Nat ( c × × × c ) a ∗ ⊗ Id c × × is given via a horizontally globular 2-cell π ∗ ( A ) : Λ ( A ) ⇒ P ( A ) and a verticallyglobular 2-cell π ∗ ( A ) : Λ ( A ) ⇒ P ( A ), so that π ∗ ( A ) satisfies: ✲ = ✲ F ( f ) ✲ = ✲ G ( f ) ❄ Λ ( A ) ❄ P ( A ) ❄ P ( B ) π ∗ ( A ) ( P ) f = ✲ F ( f ) ✲ = ✲ G ( f ) ✲ = ❄ Λ ( A ) ❄ Λ ( B ) ❄ P ( B )( Λ ) f π ∗ ( B ) ✲ Λ ( A ) ❄ = ❄ F ( u ) ❄ G ( u ) ❄ = a ( A ) ✲ P ( A ) ✲ P ( A ′ )( P ) u = ✲ Λ ( A ) ❄ F ( u ) ❄ G ( u ) ❄ = ❄ = ( Λ ) u ✲ Λ ( A ′ ) ✲ P ( A ′ ) a ( A ′ ) ✲ Λ ( B ) ❄ = ❄ = a ( B ) ✲ = ✲ F ( f ) ✲ P ( B ) ❄ = ✲ = ✲ G ( f ) ❄ Λ ( A ) ❄ P ( A ) π ∗ ( A ) t Pf = ✲ F ( f ) ✲ Λ ( B ) ✲ G ( f ) ❄ Λ ( A ) ❄ = t Λ f and π ∗ ( A ) satisfies three symmetric axioms.Now, π ∗ , µ ∗ , λ ∗ , ρ ∗ and ε ∗ should satisfy axioms (V4-1) - (V4-5) and symmetric ver-sions of (V4-1), (V4-3) and (V4-4). ( Dbl , ⊗ ) as a category internal in DblPs
From our discussion from the end of Subsection 4.1 we see that in order to view a monoid A in ( Dbl , ⊗ ) as a category internal in DblPs, the double pseudo functor ⊛ : A × A −→ A is a good candidate for a desired composition on the pullback ( M : D × D D −→ D ,with D = A and D = m : A ⊗ A −→ A is a strict double functor on the Gray type monoidalproduct on ( Dbl , ⊗ ), while ⊛ : A × A −→ A is a double pseudo functor on the Cartesianproduct of double categories. The image m ( x ⊗ y ) we denoted by x ⊛ y , for any of thefour types of cells x , y . Let us set f ⊛ g = m (cid:16) (1 ⊗ g )( f ⊗ (cid:17) (recall the discussion fromthe end of Subsection 4.1). Since m is strictly multiplicative in both directions, we find m (cid:16) (1 ⊗ g )( f ⊗ (cid:17) = m (1 ⊗ g ) m ( f ⊗ ⊛ g )( f ⊛ = f ⊛ g . (We showed that ⊛ is a double pseudo functor in [8, Section 3.3] by evaluating it on the composition of1-cells ( g ′ , f ′ ) , ( g , f ) when we obtained: ( g ′ g ) ⊛ ( f ′ f ) = m (1 ⊗ g ′ ) m (1 ⊗ g ) m ( f ′ ⊗ m ( f ⊗ f ′ ⊛ g ′ )( f ⊛ g ) = m (1 ⊗ g ′ ) m ( f ′ ⊗ m (1 ⊗ g ) m ( f ⊗ f ′ = g =
1, we recover the strict identity that we obtainedabove.)Now direct computation shows: h ⊛ ( g ⊛ f ) = ( h ⊛ g ) ⊛ f in both vertical andhorizontal direction of 1-cells: use the distributive law of the tensor with respect tothe composition of 1-cells in the Gray type tensor product A ⊗ A (see the descriptionof this tensor product after [8, Definition 3.2]), the fact that associativity of the lattercompositions is strict and that m is strictly associative [1, (iii) of Section 4.3]). Thisyields an analogous result on 0- and double cells, then for the double pseudonaturaltransformation a ∗ : ⊛ (Id × ⊛ ) −→ ⊛ ( ⊛ × Id) we may set to be identity: ( a ∗ ) C , B , A = id v ( A | B ) | C and ( a ∗ ) C , B , A = id h ( A | B ) | C , ( a ∗ ) f ′′ , f ′ , f = Id ( f | f ′ ) | f ′′ = t a ∗ f = t a ∗ f ′′ , f ′ , f and ( a ∗ ) u ′′ , u ′ , u = Id ( u | u ′ ) | u ′′ = r a ∗ u = r a ∗ u ′′ , u ′ , u , and L A = L A ′′ , A ′ , A = ( A | A ′ ) | A ′′ , the same for R A , with the notation fromSubsection 4.2, here C , B , A are 0-cells, u ′′ , u ′ , u f ′′ , f ′ , f A .Observe here that after notation in Subsection 4.2 it is M = ⊛ , M ( y , x ) = y ⊛ x = ( x | y ).Let I denote the image 0-cell of the strict double functor u : ∗ −→ A . Observe that: m ( A , I ) = ⊛ ( A , I ) = A ⊛ I and similarly the other way around, for any 0-cell A ∈ A . Nowby [1, (iii) of Section 4.3] we deduce that left and right unity constraints l ∗ and r ∗ for ⊛ : A × A −→ A are identities. As a matter of fact, as a monoid in a 1-category it cannot have 2- and 3-cells for the constraints, so we have that a monoid A in ( Dbl , ⊗ ) isnot only a category internal in DblPs, but even a category internal in the underlying1-category of DblPs, which is the category from [21, Section 6].Let us consider a monoidal 2-category made out of the monoidal category ( Dbl , ⊗ )from [1] by adding as 2-cells vertical transformations, whose 1v-cell components have companions . Recall from [12, Section 1.2], [20, Section 3] that a companion for a 1v-cell u : A −→ A ′ is a 1h-cell u ∗ : A −→ A ′ together with certain 2-cells ε and η satisfying[ η | ε ] = Id u ∗ and ηε = Id u . We denote this 2-category by ( Dbl , ⊗ ). Let us now considerpeudomonoids in this 2-category. We repeat the analogous arguments as in the abovecomputations. The di ff erence appears when computing associativity on the 1-cells:now m is not strictly associative, rather there is an isomorphism a m : ⊗ ( id × ⊗ ) −→⊗ ( ⊗ × id ). We have to take into account the form of (horizontal and vertical) 1-cellsin A ⊗ A ⊗ A , we find: h ⊛ ( g ⊛ f ) = (cid:16) h ⊛ (1 ⊛ (cid:17) [ (cid:16) ⊛ ( g ⊛ (cid:17) · (1 ⊛ (1 ⊛ f ) (cid:17) ] and( h ⊛ g ) ⊛ f = [ (cid:16) ( h ⊛ ⊛ (cid:17) · (cid:16) (1 ⊛ g ) ⊛ (cid:17) ] (cid:16) (1 ⊛ ⊛ f (cid:17) , where the square brackets may beomitted, and the dot denotes the composition of 1-cells (in the corresponding direction).18hen we define the 2-cell ( a ∗ ) h , g , f as the following 2-cell:( a ∗ ) h , g , f = ✲ ( f | | ✲ (1 | g ) | ✲ (1 | | h ( a m ) f , , ❄ ( a m ) A , B , C ( a m ) , g , ( a m ) , , h ❄ ( a m ) A ′ , B , C ❄ ( a m ) A ′ , B ′ , C ❄ ( a m ) A ′ , B ′ , C ′ ✲ f | (1 | ✲ | ( g | ✲ | (1 | h ) (8)so that on 0-cells we have: ( a ∗ ) A , B , C = ( a m ) A , B , C . (On the right hand-side of the identity (8)the indices are read from the left to the right, to accompany the notation of the 1h-cellsused here.) In [8, Section 5.2] we proved for what here are 2-cells of ( Dbl , ⊗ ) thatthey can be turned into double pseudonatural transformations, that is, 2-cells in thetricategory DblPs. Let a m denote the obtained (strong) horizontal transformation, and t mh , g , f and r mw , v , u the obtained distinguished 2-cells making a m = ( a m , a m , t m , r m ) a doublepseudonatural transformation. We define the 2-cell ( a ∗ ) w , v , u , for 1v-cells u , v , w , in theanalogous way as we did for ( a ∗ ) h , g , f above. The 2-cells t m , r m are constructed due to [8,Proposition 4.16] as follows: t m ˜ f = F ( ˜ A ) F ( ˜ B ) ✲ F ( ˜ f ) G ( ˜ B ) ✲ a m ( ˜ B ) G ( ˜ A ) G ( ˜ B ) ✲ G ( ˜ f ) G ( ˜ B ) ✲ = ❄ a m ( ˜ A ) ❄ a m ( ˜ B ) ❄ = ( a m ) ˜ f ε m ˜ B and r m ˜ u = F ( ˜ A ) G ( ˜ A ) ✲ a m ( ˜ A ) F ( ˜ A ′ ) ❄ F ( ˜ u ) G ( ˜ A ′ ) ✲ = ❄ a m ( ˜ A ′ ) G ( ˜ A ′ ) ❄ = G ( ˜ A ′ ) ❄ G ( ˜ u )( a m ) ˜ u ✲ a m ( ˜ A ′ ) ε m ˜ A ′ (9)where F = ⊗ ( id × ⊗ ) and G = ⊗ ( ⊗ × id ), ˜ f and ˜ u are 1h- and 1v-cell in A × A × A ,respectively, and ε mA is the 2-cell from the data that a m ( A ) is a companion of a m ( A ). Weconstruct t ∗ and r ∗ by the same recipe: substitute ( a m ) ˜ f from (9) by ( a ∗ ) h , g , f from (8),and set ε ∗ C ′ , B ′ , A ′ = ε mC ′ , B ′ , A ′ to define t ∗ h , g , f , analogously for r ∗ w , v , u . Then a ∗ = ( a ∗ , a ∗ , t ∗ , r ∗ )constitutes a 2-cell in DblPs.For the unity constraints l ∗ , r ∗ the argument is simpler. Since A ⊛ I is an image bothby m : A ⊗ A −→ A and by ⊛ : A × A −→ A , as we argued above, we just set l ∗ = l m and r ∗ = r m , being the right hand-sides unity constraints for m . Analogously as above,these vertical transformations can be made into double pseudonatural transformations,hence l ∗ and r ∗ are indeed 2-cells in DblPs.For the 3-cells in Definition 3.1 we take to be identities and get that a pseudomonoidin ( Dbl , ⊗ ) is indeed a category internal in DblPs.In order to have an example with non-trivial 3-cells from Definition 3.1, one cantake a “weak pseudomonoid” in the tricategory ( Dbl , ⊗ ), which is obtained from the2-category ( Dbl , ⊗ ) by adding invertible vertical modifications as 3-cells, i.e. invertiblemodifications of vertical transformations.Let us now prove that invertible vertical modifications give rise to invertible hori-zontal modifications, so that together they make (invertible) 3-cells in the tricategoryDblPs. Then the 3-cells constraints for m , which are π m , µ m , λ m , ρ m , can be upgraded to3-cells π ∗ , µ ∗ , λ ∗ , ρ ∗ corresponding to the desired 3-cells in Definition 3.1, and we wouldhave this desired example. 19ecall that vertical modifications are given by 2-cells b ( A ) as on the left hand-sidebelow, then let the inverses of horizontal modifications be given via the 2-cells b − ( A )on the right hand-side below: F ( A ) F ( A ) ✲ = G ( A ) G ( A ) ✲ = ❄ α ( A ) ❄ β ( A ) b ( A ) b − ( A ) = F ( A ) F ( A ) ✲ = η α ( A ) ❄ α ( A ) ❄ β ( A ) F ( A ) F ( A ) ✲ = G ( A ) ✲ β ( A ) G ( A ) ✲ = G ( A ) ✲ = ❄ = b ( A ) ε β A F ( A ) G ( A ) ✲ α ( A ) ❄ = (in the obvious way b ( A ) is given via b − ( A ); recall that η and ε come from the data ofcompanions). It is straightforward to prove that this defines horizontal modifications(one uses ε - η -properties and the construction of a horizontal transformation out of avertical one from [8, Proposition 5.1]; recall that for vertical transformations α thedistinguished 2-cells δ α , u are identities, for 1v-cells u ). Finally, the two compatibilityconditions between a horizontal and a vertical modification from [8, Definition 4.18] aredirectly proved. In the second condition one uses the third identity in [8, Corollary 5.4]which is fulfilled in this context. This finishes the proof that a “weak pseudomonoid”in the tricategory ( Dbl , ⊗ ) is a category internal in the tricategory DblPs.The examples of intercategories from [13] which do not rely on laxness of the doublefunctors in LxDbl , as duoidal categories do, are all examples of categories internal inthe tricategory DblPs (so that 3-cells for the internal structure are trivial). These are e.g. monoidal double categories of [20], cubical bicategories of [9], Verity doublebicategories from [22], Gray categories [14].
DblPs
Let us denote this structure as in Subsection 4.1 formally by D × D D −→−→−→ D −→←−−→ D or: C −→−→−→ B −→←−−→ A (10)and we have a similar grid of categories as in (4). The di ff erence is that now the threecolumns are strict double categories and the rows di ff er in that now also U and M are equipped with natural transformations expressing their lax multiplicativity and laxunitality.Let us see a geometrical representation of this alternative notion to intercategorieson a cube. Considering source and target functors, as well as arrows from morphismsto objects in the categories A , A , B , B , one sees that the objects of A are the lowestand morphisms of B are the highest in this hierarchy, so we may present the formerby vertices of a cube and the latter by the whole cube. For the rest of gadgets there is achoice, we will fix the one as in [11, Section 4], so that we have:20ertices - objects of D horizontal arrows - objects of D ,vertical arrows - 1v-cells of D ,transversal arrows - 1h-cells of D ,horizontal cells - 1h-cells of D ,lateral cells - 2-cells of D ,basic cells - 1v-cells of D andcube - 2-cells of D . ✟✟✟✟✟✟✟✟✯ • • ✲✟✟✟✟✟✟✟✟✯ h o f D vertices : ob jects o f D • • ✲ ob jects o f D • ❄ v o f D • • ✲❄ v o f D cube : 2- cells o f D • ❄✟✟✟✟✟✟✟✟✯ h o f D cells o f D From here we see that vertical and transversal arrows compose in their respectivedirections, horizontal cells compose in the transversal direction, basic cells composein the vertical direction, and lateral cells both in vertical and transversal directons.All of them compose strictly associatively and unitary. The pullback D × D D canbe represented by horizontal cennecting of cubes, and accordingly the functor M : D × D D −→ D corresponds to the horizontal composition of cubes.The globular 2-cells (6) of D are thus cubes whose only non-identity cells are thebasic ones, and we will consider that they map from the back towards the front. Theycompose in the transversal direction. On the other hand, the globular 2-cells (7) of D are cubes whose only non-identity cells are the horizontal ones, they map from top tobottom, and compose in the vertical direction.The double pseudofunctor U applied to a 2-cell a of D gives a cube Id ha which ishorizontal identity cube on the lateral cell a , and the rest of the cells are identities onthe corresponding 1h- and 1v-cells at the borders of a .A 2-cell in D is a cube whose lateral cells are identities, top and bottom correspondto its source and target 1h-cells, while front and back basic cells correspond to its sourceand target 1v-cells.For all the laws described in Subsection 4.2 observe that horizontal compositionof 2-cells in D corresponds to the transverasl composition of cubes, and that verticalcomposition of 2-cells in D corresponds to the vertical composition of cubes. When writing [8], we had another important motivating example for an internal cat-egory in a tricategory. Namely, analogously to the double category of rings, in onedimension higher we have a (1 × e.g. [20]). It is an internal category in a suitable tricategory V , so that the category of objectsconsists of tensor categories, tensor functors and tensor natural transformations (thusthe vertical direction is strictly associative and unital), while the category of objectsconsists of bimodule categories, bimodule functors, and bimodule natural transforma-21ions. Since the associativity for the relative tensor product of bimodule categories is anequivalence (and not an isomorphism!), the horizontal direction of this (1 × × Cat wk , of 2-categories, pseudofunctors, weak natural transformations and modifica-tions. Note that 2- Cat wk is 1-strict. Remark 5.1
We comment that in order to avoid saying “a tricategory enriched over a(1-strict) tricategory V ” one could say “a tricategory weakly enriched over the under-lying category of V ”. In the latter style it is said that an arbitrary tricategory, [10], isweakly enriched over Bicat, seen as the category of bicategories and pseudofunctors.Nevertheless, one indeed has a structure enriched over the (1-strict) tricategory Bicat of bicategories, pseudofunctors, pseudo natural transformations and modifications,but of course this can not be used as a definition of the notion of a tricategory.We will also show how the tricategory Tens forms a part of a category internal inthe same tricategory 2 Cat wk , in the sense of Definition 3.1. Inspired by this example wewould like to have a general result stating that “enriched” is a special case of “internal”in the setting of tricategories. Strict versions of such a statement were proved in [6] and[3] for 1-categories. Let W be a 1-strict monoidal tricategory with the Cartesian monoidal product. Wedenote by W the underlying 1-category of W , by · the composition of 1-cells in W andsuppose that W has a terminal object ∗ . Moreover, let V be a 1-strict tricategory withpullbacks. In the following table we list the defining data for two structures: in the leftcolumn for a category T enriched over W , and in the right one for a category internalin V . For T suppose that there is a set O b T whose elements are called objects of T :22 is a category enriched over W a category internal to V ∀ A , B ∈ O b T : T ( A , B ) 0-cell in W T , T V ◦ : T ( A , B ) × T ( B , C ) −→ T ( A , C ) s , t : T −→←−−→ T : u , I A : ∗ −→ T ( A , A ) T × T T c −→ T W V α : − ◦ ( − ◦ − ) −→ ( − ◦ − ) ◦ − α ∗ : c ( id T × T c ) −→ c ( c × T id T ) ∀ A , B ∈ O b T λ : I B · T ( A , B ) −→ T ( A , B ) , λ ∗ : c ( u × T id T ) −→ id T ,ρ : 1 T ( A , B ) · I A −→ T ( A , B ) ρ ∗ : c ( id T × T u ) −→ id T equivalence 2-cells in W equivalence 2-cells in V π, µ, λ, ρ, ε invertible 3-cells in W π ∗ , µ ∗ , λ ∗ , ρ ∗ , ε ∗ invertible 3-cells in V π, µ, λ, ρ, ε and π ∗ , µ ∗ , λ ∗ , ρ ∗ , ε ∗ are analogous to the 3-cells in point 4. of Definition 3.1The 2-cells α ∗ , λ ∗ , ρ ∗ and the 3-cells π ∗ , µ ∗ , λ ∗ , ρ ∗ , ε ∗ in the right column above a prioriare not related with the respective cells in the left column. The source and target 1-cellsin V should obey the expected rules, and the whole right column basically depictsDefinition 3.1. Proposition 5.2
Let V be a 1-strict Cartesian monoidal tricategory so that its underlying1-category V has finite limits and small coproducts preserved by pullbacks. Assumemoreover that the functors − × X , X × − : V −→ V preserve small coproducts for everyobject X of V . A category T enriched over V is a particular case of a category internalto V . Proof.
The existence of finite limits assures in particular the existence of a terminalobject I and of pullbacks in V . Set T = ∐ A ∈O b T I A - the coproduct of copies of theterminal object indexed by the objects of T . Moreover, set T = ∐ A , B ∈O b T T ( A , B ).By the first preservation assumption we have that the pullback T × T T is given by ∐ A , B , C ∈O b T T ( A , B ) × T ( B , C ), and by iteration: T × T T × T T (cid:27) ∐ A , B , C , D ∈O b T T ( A , B ) ×T ( B , C ) × T ( C , D ) (in both versions regarding associativity). Take the composition c : T × T T −→ T to be induced by ◦ : T ( A , B ) × T ( B , C ) −→ T ( A , C ), as indicated in the23iagram: ✘✘✘✘✘✘✘✘✘✘✘✘✘✘✘✿ ˜ ι ✟✟✟✟✟✟✟✟✯ (cid:27) T × T T T ( B , C ) × T ( C , D ) ∐ B , C , D ∈O b T T ( B , C ) × T ( C , D ) ✲ ι ❄ ◦∐ B , D ∈O b T T ( B , D ) ✲ j T ( B , D ) ❄ ◦ ❄ c (start with the left lower rectangular, then ι induces ˜ ι and ◦ induces c via the before-mentioned isomorphism). Observe this diagram of 0- and 1-cells in V as a diagramin the 1-category V . Apply to it the functor T ( A , B ) × − , on one hand, and on theother, the functor − × T ( D , E ). The diagram chasing together with the properties athand yields that the equivalence 2-cell α : − ◦ ( − ◦ − ) −→ ( − ◦ − ) ◦ − in V induces anequivalence 2-cell α ∗ : c ( id T × T c ) −→ c ( c × T id T ). It is even easier to show that theunity equivalence 2-cells λ and ρ induce corresponding equivalences λ ∗ and ρ ∗ (for thenotation see the left and right column in the table above). In the similar fashion oneshows that the 3-cells π, µ, λ, ρ, ε , controlling the behavior of the equivalences α, λ and ρ , induce desired 3-cells π ∗ , µ ∗ , λ ∗ , ρ ∗ , ε ∗ . The 1-cells source and target are clear, and wehave obtained that T and T with the described data form a category internal to V .Observe that by the construction of T in the above proof, if V is a 1-strict tricategorywhose 0-cells are bicategories, then 0-cells of T are the same as 0-cells of T . Moreover,1-cells of T are the identities on its 0-cells and 2-cells are identities on the latter, i.e. theobject of objects T is discrete.The construction in the above proposition can be carried out in 2-categories: con-sider 3-cells in the table above to be identities, and consider equivalence 2-cells to bebijective. Then we obtain: Corollary 5.3
Let V be a Cartesian monoidal 2-category so that its underlying 1-categoryV has finite limits and small coproducts preserved by pullbacks. Assume moreover that thefunctors − × X , X × − : V −→ V preserve small coproducts for every object X of V . A category T enriched over V is a particular case of a category internal to V. Example 5.4
A category enriched over the 2-category V = Cat of categories, functorsand natural transformations is a bicategory, and it is well-known that a bicategoryembeds into a pseudodouble category, which is a category internal in Cat (this is sortof a weak version of the result from [6]). Example 5.5
Let V = PsDbl be the 2-category of pseudodouble categories, pseudodou-ble functors and vertical transformations, [20, 9]. A category enriched over
PsDbl isa locally cubical bicategory from [9, Definition 11]. A category internal to
PsDbl is aversion of an intercategory. Corollary 5.3 applied to V = PsDbl uses the argumentationsimilar to [13, Section 3.5], where a locally cubical bicategory is shown to be a particularcase of an intercategory.Truncating the result of Corollary 5.3 to 1-categories one recovers a version ofthe above-mentioned results in [6] and [3]. As a particular case of this we have the24ollowing. A Gray-category is a category enriched over the monoidal category
Gray with the Gray monoidal product. This product is defined as an image of a cubicalfunctor defined on the Cartesian product of two 2-categories. In [13, Section 5.2] itis shown how a Gray-category can be seen as an intercategory, a category internal in
LxDbl . As an intermediate step one can see how a Gray-category is made a categoryinternal in
Gray .In the above two examples we can embed the 1-category
Gray and the 2-categories
Cat and PsDbl to our tricategory DblPs from Subsection 4.2 and we get three examplesof categories internal in DblPs. (1 × -category of tensor categories Let us recall and discuss the structure of a tricategory Tens of tensor categories.1. For every two tensor categories C an D we have a 2-category Bimod( C , D );2. given two C - D -bimodule categories M , N there is a category Bimod( C , D ) (cid:16) M , N (cid:17) = C Fun D ( M , N ) whose composition of morphisms is given by the vertical compo-sition of C - D -bimodule natural transformations, which we denote by · (this is thetransversal composition of 3-cells in Tens);3. given a third C - D -bimodule category L there is a functor ◦ : C Fun D ( N , L ) × C Fun D ( M , N ) −→ C Fun D ( M , L ) given by the composition of C - D -bimodule func-tors and C - D -bimodule natural transformations; thus the horizontal compositionof 2-cells in Bimod( C , D ) is given by the usual horizontal composition of naturaltransformations (this is the vertical composition of 2- and 3-cells in Tens); by thefunctor properties of ◦ , on this 2-category level we have the strict interchangelaw: ( ζ ′ · ω ′ ) ◦ ( ζ · ω ) = ( ζ ′ ◦ ζ ) · ( ω ′ ◦ ω ) for accordingly composable naturaltransformations ω, ω ′ , ζ, ζ ′ ;4. given a third tensor category E there is a pseudofunctor ⊠ D : Bimod( D , E ) × Bimod( C , D ) −→ Bimod( C , E ), so the composition of 1-cells and the horizontalcomposition of 2- and 3-cells in Tens is given by the relative tensor product ofbimodule categories. Let ( M , N ) , ( M ′ , N ′ ) ∈ Bimod( D , E ) × Bimod( C , D ), thenset for the hom-setBimod( D , E ) × Bimod( C , D ) (cid:16) ( M , N ) , ( M ′ , N ′ ) (cid:17) = C Fun D - bal E (cid:16) ( M , N ) , ( M ′ , N ′ ) (cid:17) , which is the category of D -balanced C - E -bimodule functors and natural transfor-mations. Then there is a functor e : C Fun D - bal E (cid:16) ( M , N ) , ( M ′ , N ′ ) (cid:17) −→ C Fun E ( N ⊠ D M , N ′ ⊠ D M ′ ) and there are natural isomorphisms e G ◦ e F (cid:27) ] G ◦ F and Id N ⊠ D M (cid:27) ] Id ( M , N ) for all F ∈ C Fun D - bal E (cid:16) ( M , N ) , ( M ′ , N ′ ) (cid:17) and G ∈ C Fun D - bal E (cid:16) ( M ′ , N ′ ) , ( M ′′ , N ′′ ) (cid:17) (this corresponds to the bimodule case of [19, Proposition 3.3.2]). In particular,the latter natural isomorphisms imply that we have the interchange law at thislevel holding up to an isomorphism: ( F ′ ⊠ D G ′ ) ◦ ( F ⊠ D G ) (cid:27) ( F ′ ◦ F ) ⊠ D ( G ′ ◦ G )for according bimodule functors, and also: Id N ⊠ D M (cid:27) Id N ⊠ D Id M . The abovefunctor property implies in particular: ( ζ ′ ◦ ω ′ ) ⊠ D ( ζ ◦ ω ) = ( ζ ′ ⊠ D ζ ) ◦ ( ω ′ ⊠ D ω )for according bimodule natural transformations, and Id G ⊠ D F = Id G ⊠ D Id F ;25. for 0-, 1- and 2-cells C , M and F respectively there are identity 1-, 2- and 3-cells C , id M and Id F , respectively;6. there are pseudonatural equivalences a , l , r so that concretely for the correspond-ing bimodule categories one has equivalence functors: a M , N , L : ( M ⊠ C N ) ⊠ D L (cid:27) −→M ⊠ C ( N ⊠ D L ) , l N : C ⊠ C N (cid:27) −→ N and r N : N ⊠ D D (cid:27) −→ N (observe that therespective naturalities hold up to natural isomorphisms);7. there are modifications π, µ, λ and ρ which evaluated at bimodule categories givenatural isomorphisms π : (id K ⊠ C a N , M , L ) ◦ a K , N ⊠ D M , L ◦ ( a K , N , M ⊠ E id L ) ⇛ a K , N , M ⊠ E L ◦ a K ⊠ C N , M , L ,µ M , D , L : r M ⊠ D id N ⇛ (id M ⊠ D l N ) ◦ a M , DN ,λ C , M , N : l M ⊠ D id N ⇛ l M ⊠ D N ◦ a C , M , N ,ρ C , M , E : (id M ⊠ D r N ) ◦ a M , N , E ⇛ r M ⊠ D N , similar to those in (vi)-(ix) of [19, Theorem 3.6.1] and they satisfy three axiomsanalogous to those in (x) of loc.cit. . Remark 5.6
To see that the functors on the two sides in the isomorphism ( F ′ ⊠ D G ′ ) ◦ ( F ⊠ D G ) (cid:27) ( F ′ ◦ F ) ⊠ D ( G ′ ◦ G ) are a priori not equal, observe the following. As functorsacting on the relative tensor product, they both are given up to an isomorphism by thedefining functors ( F ′ ×G ′ ) ◦ ( F ×G ) and ( F ′ ◦F ) × ( G ′ ◦G ), respectively, which are clearlyequal between themselves. Since both functors are determined up to an isomorphismby the same functor, they only can be isomorphic between themselves, and one cannot claim that they are equal. This applies to the point 4. above. By the same reasonnaturalities in the point 6. above hold only up to an isomorphism. Remark 5.7
For a fixed tensor category C it was proved in [15] that Bimod( C , C ) forms amonoidal 2-category in the sense of [16], which is a non-semistrict monoidal bicategory,namely, it is weaker than a Gray monoid. Though, [15] follows the approach of [7]where the relative tensor product of bimodule categories is defined in such a way thata functor from such tensor product is defined uniquely by a balanced functor, whereasin [4] it is defined up to a unique isomorphism . This has for a consequence that manyof the structure isomorphisms in Bimod( C , C ) in [15] result to be identities (coherence3-cells: π for the associativity constraint, [15, Proposition 4.9], and λ and ρ for theleft and right unity constraints [15, Proposition 4.11]), and moreover the associativityconstraint a itself is an isomorphism instead of being an equivalence (see the proof of[15, Proposition 4.4]). Substituting tensor categories by fusion categories (semisimpletensor categories), in [19] it was proved that these form a tricategory (in a weaker sensethan in [15], as we just pointed out). Semisimplicity does not influence the argumentsof the proof, so we may take it as a proof that Tens is a tricategory. Note that the authoruses the term “2-functor” for a pseudofunctor, [19, Definition A 3.6].From the items 1, 4, 5, 6 and 7 above it is clear that Tens is a category enrichedover the tricategory of 2-categories, pseudofunctors, weak natural transformations andmodifications, which we denoted earlier by 2- Cat wk .26ow let us explain the (1 × i.e. of a cate-gory internal in 2- Cat wk . To do so we will give 2-categories C and C , pseudofunctors s , t , u and c , weak natural equivalences α ∗ , λ ∗ and ρ ∗ and modifications π ∗ , µ ∗ , λ ∗ , ρ ∗ , ε ∗ .As we announced at the beginning of this section, let C be the 2-category of tensor cate-gories, tensor functors and tensor natural transformations, and let C be the 2-categoryof bimodule categories, bimodule functors and bimodule natural transformations. Fixtensor categories C and D . To give a source and target 2-functors s , t : C −→ C , let M be a C - D -bimodule category, F a C - D -bimodule functor, and ω a C - D -bimodule nat-ural transformation. Set s ( M ) = C , t ( M ) = D , s ( F ) = id C , t ( F ) = id D and s ( ω ) = Id id C and t ( ω ) = Id id D - the identity functors on C and D are obviously tensor functors, andthe identity natural transformations on these two identity functors are obviously tensorones. It is also clear that thus defined source and target functors are strict 2-functors. Todefine the identity 2-functor u : C −→ C , take tensor categories C , D , tensor functors F , G : C −→ D and a tensor natural transformation ζ : F −→ G , and for C , C ′ , C ′′ ∈ C let C ⊲ C ′ denote the left action of C on C ′ and C ′ ⊳ C ′′ the right action of C ′′ on C ′ . Set u ( C ) = C as a C -bimodule category, u ( F ) = F as a C -bimodule functor where D is a C -bimodule category through F , that is: C ⊲ D ⊳ C ′ = F ( C ) ⊗ D ⊗ F ( C ′ ) for an object D ∈ D and where ⊗ denotes the tensor product in D (a well-known fact), then F is clearly C -bilinear. Finally, set u ( ζ ) = ζ , then similarly as for functors, ζ is a C -bilinear naturaltransformation. To see that u is indeed a 2-functor, take a further tensor category E anda tensor functor G : D −→ E , then it is clear that GF as a C -bimodule functor is equal tothe composition of G as a D -bimodule functor and F as a C -bimodule functor.The rest of the structure (a pseudofunctor c , pseudonatural equivalences α ∗ , λ ∗ , ρ ∗ and modifications π ∗ , µ ∗ , λ ∗ , ρ ∗ , ε ∗ ) are given as in Proposition 5.2. That c is a pseudo-functor and not a 2-functor follows from Remark 5.6. For this reason the tricategoryTens is an internal category in the tricategory 2- Cat wk , rather than in the Gray 3-category2 CAT nwk , as conjectured in [5, Example 2.14] (1-cells in 2
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