Formal aspects of Grays tensor product of 2-categories

F ORMAL ASPECTS OF GRA Y’S TENSOR PR ODUCT OF 2-CA TEGORIES ALEXANDRU E. ST ANCULESCU Abstract. The category of s m all 2-categories has tw o monoidal structures due to John Gray: one bi closed and one closed. W e pr opose a formalisation of the construction of the right in ternal and int ernal homs of these monoidal structures. 1. Introduction Let Cat be the category of small catego ries. In [7], Gray int ro duced t wo mono idal structures on the category 2- Cat o f sma ll categorie s e nr iched over Cat : one biclose d and o ne clo sed. The coherence a x ioms for these mo no idal structures were prov ed in [8 ]; see also [3]. W e recall that the rig ht in ter nal hom of the biclosed monoidal structure on 2- Cat consists of 2-functors, quasi-natura l (also called lax natural) transformatio ns a nd mo difications. This pap er is an a ttempt to understand these monoidal structur e s and so me facts surrounding them. In this resp ect w e prop os e a forma lisation of the construction of the (r ight) internal hom of 2- Cat , a s outlined below. Let V be a closed category and let V - Cat b e the categor y of small V -ca tegories. Given a comonoid C • in the category of cosimplicia l o b jects in V equipp ed with the Day co nv o lutio n pro duct, and tw o small V -ca tegories A and B , w e c onstruct a small V - category C oh C • ( A , B ) whose ob jects ar e the V -functor s from A to B and whose homs are the ob jects of V -coheren t transformatio ns with r esp ect to C • . The nota tion C oh is b or row ed from the work o f Cordier and Porter [4], who made a simila r construction when V is the ca tegory of simplicial sets, exc ept that in their case C oh ( A , B ) is not a categor y enriched ov er s implicial sets. When C • is the cons tant cos implicial ob ject with v alue the unit ob ject of V , we r ecov er the standard internal hom of V - Cat consisting of the V -na tural tra nsformations. When V = Cat and C • is wha t w e call the standard coc ategory (cogroup oid) in ter v al in Cat , our construction recov ers the right internal (internal) ho m of 2- Cat . W e a c tually c o nstruct a V - c ategory C oh C ( A , B ) for every como no id C in the categ ory o f coa ugmented cosimplicial o b jects in V . The constructio n is natura l in all three v ar iables, and we present it decomp osed in as many steps as we could. F or instance, the endofunctor C oh C ( A , − ) is a co mpo site of four natural functors, each one having le ft adjoints. As a n outcome of this for malisation w e obtain a formula for Gray’s tensor pro duct(s) of 2-categ o ries. O ther outcomes will be detailed in [15]. The pa p er is or ganised as follo ws. Sections 2 and 3 are prepa ratory , and consist of r ecollections of facts regarding the Da y conv olution pro ducts on certa in functor categor ies, a ctions of monoidal categor ies and T ensor-Hom situations. In section 4 we exhibit a chain of monoidal functors which will b e part of the construction o f C oh C ( A , − ). The most impo rtant one is a familiar cosimplicia l cobar co nstruction. In section 5 we construct C oh C ( A , B ) and show that, as a functor of three v ariables, it is pa rt of a T ensor-Hom situation. W e end by showing that if we take V = Cat , equipp e d with the car tesian closed structure, then the standard c o category in terv a l in Cat , to b e denoted by I • , is a comonoid in the categor y of cosimplicial ob jects in Cat , and that C oh I • ( A , B ) is precisely the right internal hom of 2- Cat . 2. Bac kgr ound, p ar t one W e denote by ∆ the categ o ry of finite non- empt y or dinals and order preserving maps. The o rdinal n + 1 = { 0 , ..., n } will b e denoted by [ n ]. W e let ∆ + be the categ o ry of a ll finite ordinals and or der preserv ing maps. The ordinal n will be denoted by n . ∆ + has a monoidal pro duct given by the ordinal addition, with unit the ordinal 0. W e deno te by i the inclusion ∆ ⊂ ∆ + . W e let ∆( n ) b e the n -th trunca tion of ∆ and ∆( n ) mon be ∆( n ) without the co degener acies s i : [ n ] → [ n − 1 ], n ≥ 1. If V is a monoidal category , we deno te b y C omon ( V ) the categ o ry of co monoids in V . Throughout this section ( V , ⊗ , I ) is a co complete closed category . Researc h s upported by the Mi nistry of Education of the Czech Republic under grant LC505. 1 2 A. E. ST ANCULESCU 2.1. Coreflexiv e graphs. The ca tegory V ∆(1) admits the non-symmetric Day conv olution pro duct ⋆ . W e recall its construction in deta il. F or X • , Y • ∈ V ∆(1) , one has ( X • ⋆ Y • ) 0 = X 0 ⊗ Y 0 and ( X • ⋆ Y • ) 1 is the pushout of the diagram X 0 ⊗ Y 1 i X 0 ,Y 1 / / ( X • ⋆ Y • ) 1 X 0 ⊗ Y 0 d 1 ⊗ 1 Y 0 / / 1 X 0 ⊗ d 0 O O X 1 ⊗ Y 0 i X 1 ,Y 0 O O The unit o f ⋆ is c stI , the constant 1-tr uncated cosimplicia l ob ject with v a lue I . The co faces are D 0 = i X 1 ,Y 0 ( d 0 ⊗ 1 Y 0 ) and D 1 = i X 0 ,Y 1 (1 X 0 ⊗ d 1 ). The codegener acy is obtained using the universal pro pe r ty of the pusho ut. The asso ciativity iso morphism can be seen from the diagra m X 0 ⊗ Y 0 ⊗ Z 0 d 1 ⊗ 1 Z 0 / / 1 X 0 ⊗ d 0 ) ) ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ 1 X 0 ⊗ d 0 ⊗ 1 Z 0   1 X 0 ⊗ d 1 ⊗ 1 Z 0   X 1 ⊗ Y 0 ⊗ Z 0   X 0 ⊗ Y 0 ⊗ Z 1   X 0 ⊗ Y 1 ⊗ Z 0 i X 0 ,Y 1 ⊗ 1 Z 0 / / 1 X 0 ⊗ i Y 1 ,Z 0 ( ( ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ( X • ⋆ Y • ) 1 ⊗ Z 0 ) ) ❘ ❘ ❘ ❘ ❘ ❘ ❘ X 0 ⊗ ( Y • ⋆ Z • ) 1 / / ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ( X • ⋆ Y • ⋆ Z • ) 1 where all the faces a re pusho uts. The ob ject ( X • ⋆ ( Y • ⋆ Z • )) 1 is obtained from the back face and the ma p 1 X 0 ⊗ i Y 1 ,Z 0 ; the ob ject (( X • ⋆ Y • ) ⋆ Z • ) 1 is obtained from the left face and the map i X 0 ,Y 1 ⊗ 1 Z 0 . The monoidal pro duct ⋆ restricts to a monoidal product ⋆ on V ∆(1) mon . 2.2. (Coaugmented) cosim plicial ob jects. The catego ry V ∆ has the non-symmetric Day co nv olutio n pro duct ⋆ , see [1], [4 ], [1 4]. W e detail tw o of its presentations. If X • , Y • ∈ V ∆ , one has ( X • ⋆ Y • ) 0 = X 0 ⊗ Y 0 . F or n ≥ 1, ( X • ⋆ Y • ) n is the co equaliser ` p + q = n − 1 X p ⊗ Y q v / / u / / ` r + s = n X r ⊗ Y s X p ⊗ Y q 1 X p ⊗ d 0 / / inj n − 1 p,q O O d p +1 ⊗ 1 Y q ( ( ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ X p ⊗ Y q +1 inj n p,q +1 O O X p +1 ⊗ Y q inj n p +1 ,q > > ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ FORMAL ASPECTS OF GRA Y’S TENSOR P R ODUCTS OF 2-CA TEGORIES 3 where u inj n − 1 p,q = inj n p +1 ,q ( d p +1 ⊗ 1 Y q ) and v inj n − 1 p,q = inj n p,q +1 (1 X p ⊗ d 0 ). F or 0 ≤ k ≤ n + 1, the c o face map D k is determined by the diagr am X r ⊗ Y s − 1 1 X r ⊗ d 0 / / inj n − 1 r,s − 1 ' ' P P P P P P P P P P P P P P 1 X r ⊗ d k − r − 1   X r ⊗ Y s 1 X r ⊗ d k − r   inj n r,s ' ' P P P P P P P P P P P P P P ` p + q = n − 1 X p ⊗ Y q v / / ` r + s = n X r ⊗ Y s X r ⊗ Y s 1 X r ⊗ d 0 / / inj n r,s ' ' P P P P P P P P P P P P P P X r ⊗ Y s +1 inj n +1 r,s +1 ' ' P P P P P P P P P P P P P P ` p + q = n X p ⊗ Y q v / / ` r + s = n +1 X r ⊗ Y s if r < k , and b y the diagram X r − 1 ⊗ Y s d r ⊗ 1 Y s / / inj n − 1 r − 1 ,s ' ' P P P P P P P P P P P P P P d k ⊗ 1 Y s   X r ⊗ Y s d k ⊗ 1 Y s   inj n r,s ' ' P P P P P P P P P P P P P P ` p + q = n − 1 X p ⊗ Y q u / / ` r + s = n X r ⊗ Y s X r ⊗ Y s d r +1 ⊗ 1 Y s / / inj n r,s ' ' P P P P P P P P P P P P P P X r +1 ⊗ Y s inj n +1 r +1 ,s ' ' P P P P P P P P P P P P P P ` p + q = n X p ⊗ Y q u / / ` r + s = n +1 X r ⊗ Y s 4 A. E. ST ANCULESCU if r ≥ k . The co deg eneracies ar e defined simila rly , see [1 ],[1 4]. The unit o f ⋆ is cstI , the co nstant cosimplicial ob ject with v alue I . ( X • ⋆ Y • ) n ( n ≥ 1) can also b e calculated [1] as the colimit of the diagram X 0 ⊗ Y n X 0 ⊗ Y n − 1 1 X 0 ⊗ d 0 O O d 1 ⊗ 1 Y n − 1 / / X 1 ⊗ Y n − 1 ... O O / / X p ⊗ Y q +1 X p ⊗ Y q 1 X p ⊗ d 0 O O d p +1 ⊗ 1 Y q / / X p +1 ⊗ Y q O O ... X n − 1 ⊗ Y 0 1 X n − 1 ⊗ d 0 O O d n ⊗ 1 Y 0 / / X n ⊗ Y 0 where p + q = n − 1. It is this present ation that w e shall use the most. An y monoida l (re sp. opmo noidal and co contin uous) functor V 1 → V 2 betw een co co mplete closed catego ries induces a monoidal (res p. opmonoidal) functor V ∆ 1 → V ∆ 2 . There are v ar ious adjoint pairs b etw e en V and V ∆ , which we summarise as sk ⊣ ev 0 ⊣ cst ⊣ H 0 Here cst deno tes the constant cosimplicial ob ject functor, ev 0 is the ev aluation a t [0] and sk ( A ) n = ⊔ ∆([0] , [ n ]) A . The functors cst and ev 0 are strong monoidal, and sk is opmonoidal for formal r easons. W e hav e induced adjoint pa irs sk : C omon ( V ) ⇄ C omon ( V ∆ ) : ev 0 and ev 0 : C omon ( V ∆ ) ⇄ C omon ( V ) : cst The functor e v 0 : C omon ( V ∆ ) → C omon ( V ) is a (Grothendieck) bifibration, provided that V is sufficiently complete. The s a me adjunctions and the same fa c ts co ncerning the tw o categ ories of como no ids hold if one r e pla ces ∆ by ∆(1). The functor category V ∆ + has the Day c o nv o lution pr o duct ⋆ . Its unit is F ∆ + (0 , − ), where F : S e t → V is F ( S ) = ⊔ S I . The functor i ∗ : V ∆ + → V ∆ is strong monoidal. (T o see this it s uffices [10, Theorem 5 .1] to show that ∆([0] , − ) is a monoid in ( S e t ∆ , ⋆ ) op , that is, a comonoid in ( S et ∆ , ⋆ ). But ∆([0] , − ) = sk (1 ).) Therefor e i ! is opmonoidal for formal reasons. W e hav e an induced adjoint pair i ! : C omon ( V ∆ ) ⇄ C omon ( V ∆ + ) : i ∗ 3. Bac kgr ound, p ar t two Let ( V , ⊗ , I ) b e a monoidal category . W e recall that an action of V on a categor y E is the data co nsisting o f a functor ∗ : V × E → E and na tur al isomo rphisms α and λ with comp onents α A,B ,X : ( A ⊗ B ) ∗ X → A ∗ ( B ∗ X ) and λ X : I ∗ X → X , sub ject to certa in coher ence co nditions (see, for example, [11]). Let E i (1 ≤ i ≤ 3) b e a category . W e r ecall from [9] that a TH-s ituation consists of t w o functors T : E 1 × E 2 → E 3 H : E op 2 × E 3 → E 1 FORMAL ASPECTS OF GRA Y’S TENSOR P R ODUCTS OF 2-CA TEGORIES 5 and natural isomorphisms E 3 ( T ( X 1 , X 2 ) , X 3 ) ∼ = E 1 ( X 1 , H ( X 2 , X 3 )) If ∗ : V × W → W H : W op × W → V is a TH-situation with ∗ a n action of V on a categor y W , then W b ecomes a tensor ed V -categor y . If, in addition, W is a monoidal ca tegory and ∗ is a s trong monoidal fuctor, then W beco mes a mono idal V -catego ry . In this case, if C is a comonoid in W , then W ( C, − ) : W → V is a mo no idal V - functor . W e shall use these o bserv atio ns for W ∈ {V ∆(1) , V ∆ , V ∆ + } , with the ob vious action of V and with the monoida l pro ducts describ ed in section 2. Let E i (1 ≤ i ≤ 3) b e categor y and let T : E 1 × E 2 → E 3 H : E op 2 × E 3 → E 1 be a TH-situa tion. E very small catego ry C induces a n ob vious TH-situation T C : E C 1 × E C op 2 → E 3 H C : ( E C op 2 ) op × E 3 → E C 1 Suppo se tha t E i (1 ≤ i ≤ 3) is a monoida l categ ory , T is a monoidal functor and all the functor categor ies in the preceding TH-situation admit a Day conv olution pro duct. Then T C is a monoida l functor . In pa rticular, if A is a monoid in E C op 2 , the functor T C ( − , A ) : E C 1 → E 3 is monoidal. 4. Some monoidal functors Let ( V , ⊗ , I ) b e an ar bitrary mono idal category . W e deno te by V - Cat (r esp. V - CA T ) the catego ry of small (resp. large) V -catego ries a nd by O b the functor sending an V -ca tegory to its set of ob jects. F or a set S , we denote by V - Cat ( S ) the categ ory of small V -categorie s with fixed set of ob jects S . When V is symmetric mo noidal, V - Cat is a s ymmetric monoidal categ ory with monoidal product ⊗ a nd unit I , where I has a single ob ject ∗ and I ( ∗ , ∗ ) = I . A monoidal functor F : V → W b et ween monoida l catego ries induces a functor F : V - Cat → W - Cat . Let A b e a V -categor y . W e deno te by A op the opp os ite of A . Suppo se that V is a closed catego ry . W e write Y X for the in ternal hom of tw o ob jects X , Y of V . Given V -categor ies A and B , we denote by V - M o d ( A , B ) the V -catego ry of V -functors B op ⊗ A → V . Suppos e, in addition, that V is co co mplete. V - M o d ( A , A ) is a biclos e d monoidal V -categ ory , with monoidal pro duct φ ◦ ψ ( a, a ′ ) = x ∈ O b ( A ) Z φ ( a, x ) ⊗ ψ ( x, a ′ ) unit A : A op ⊗ A → V a nd r ight internal hom [ φ, ψ ] r ( a, a ′ ) = Z x ∈ O b ( A ) ψ ( x, a ′ ) φ ( x,a ) Let D : S e t → V - Cat be the discre te V -catego ry functor. W e denote V - Mo d ( DS, DS ) by V - Graph ( S )– this is just the functor ca teg ory V S × S . The ob jects of V - Graph ( S ) are ca lle d V -gr aphs with fixed set of ob jects S . F or X , Y ∈ V - Graph ( S ), one now has X ◦ Y ( a, b ) = a z ∈ S X ( a, z ) ⊗ Y ( z , b ) the unit is I S ( a, b ) = ( I , if a = b ∅ , otherwise and the right internal ho m is [ X , Y ] r ( a, b ) = Y x ∈ S Y ( x, b ) X ( x ,a ) There is an adjunction δ : V ⇄ V - Graph ( S ) : () 6 A. E. ST ANCULESCU where δ X ( a, b ) = ( X , if a = b ∅ , otherwise and X = Q a ∈ Ob ( A ) X ( a, a ). The functor δ is stro ng monoidal, therefore () is monoidal for forma l reasons. The ca tegory V - Cat ( S ) is pre cisely the ca tegory o f monoids in V - Graph ( S ) with r e sp e ct to ◦ , and V - Mo d ( A , A ) is pr ecisely the c ategory of ( A , A )-bimo dules in ( V - Graph ( Ob ( A )) , ◦ ) (in the sens e of categ orical alg ebra). Thus, there is a free-forgetful adjunction F = A ◦ − ◦ A : V - Graph ( Ob ( A )) ⇄ V - M o d ( A , A ) : U One has F X ◦ F Y ∼ = F ( X ◦ A ◦ Y ) for every V -gr aph X and every φ ∈ V - Mo d ( A , A ), which implies that U [ F X , φ ] r ∼ = [ A ◦ X , U φ ] r Moreov er, U ( φ ◦ ψ ) is the co equaliser of the (reflexive) pair U ( φ ) ◦ A ◦ U ( ψ ) / / / / U ( φ ) ◦ U ( ψ ) so that ( a ) the forgetful functor U is monoidal, and ( b ) [ φ, ψ ] r is the equaliser of the pair [ U ( φ ) , U ( ψ )] r / / / / [ U ( φ ) ◦ A , U ( ψ )] r There is a functor C : V - Mo d ( A , A ) × ∆ op + → V - Mo d ( A , A ) given by C ( φ, n ) = U φ ◦ A ◦ n ∼ = φ ◦ A ◦ ( n +1) The functor C ( φ, − ) is (stro ng) monoidal. Let us de no te by ⋆ the Day convolution pro duct on V - Mo d ( A , A ) ∆ op + ; its unit ob ject is ⊔ ∆ + ( − , 0) A . O ne has ( φ ⋆ ψ )( n ) ∼ = a i + j = n φ ( i ) ◦ ψ ( j ) Setting C ( φ )( n ) = C ( φ, n ) defines a monoidal functor C : V - Mo d ( A , A ) → V - M o d ( A , A ) ∆ op + In particular , C ( A ) is a monoid in V - Mo d ( A , A ) ∆ op + . ¿F rom the la st paragr aph of section 3 it follows that, in the adjoint pair − ⋆ ∆ + C ( A ) : V - Mo d ( A , A ) ∆ + ⇄ V - Mo d ( A , A ) : [ C ( A ) , − ] r the left adjoint is monoidal. It can be rea dily seen that C ( A ) is a non-counital comonoid in V - Mo d ( A , A ) ∆ op + . Lemma 4 .1. The functor [ C ( A ) , − ] r is monoidal. Pr o of. T he THC-tra ns po se of the natural ma p ( ⊔ ∆ + (0 , − ) A ) ⋆ ∆ + C ( A ) → A is the unit map. W e shall constr uct a map F φ,ψ : [ C ( A ) , φ ] r ⋆ [ C ( A ) , ψ ] r → [ C ( A ) ⋆ C ( A ) , φ ◦ ψ ] r ¿F rom the pre v ious considerations w e ha ve U ([ C ( A ) , φ ] r ( n )) = [ A ◦ n , U φ ] r . Using this, one can define a natural cup pro duct ⌣ : [ A ◦ m , U φ ] r ◦ [ A ◦ n , U ψ ] r → [ A ◦ ( m + n ) , U ( φ ◦ ψ )] r which is co mpatible with the actions of A and is suitably asso cia tive and unital, so that it induces a suitably asso ciative a nd unital map [ C ( A )( m ) , φ ] r ◦ [ C ( A )( n ) , ψ ] r → [ C ( A )( m + n ) , φ ◦ ψ ] r This map induces the map F φ,ψ . It follows that [ C ( A ) , − ] r is asso ciative and unital.  FORMAL ASPECTS OF GRA Y’S TENSOR P R ODUCTS OF 2-CA TEGORIES 7 Notation 4. 2. W e denote by Y + the comp osite V - Mo d ( A , A ) [ C ( A ) , − ] r − → V - Mo d ( A , A ) ∆ + U ∆ + − → V - Graph ( O b ( A )) ∆ + () ∆ + − → V ∆ + and b y Y the functor i ∗ Y + , where i ∗ : V ∆ + → V ∆ is the restriction functor. It follows fro m lemma 4.1 , the prev io us considera tio ns and 2.2 that Y is mono idal. The functor Y is a familiar one. F or φ ∈ V - Mo d ( A , A ), the cosimplicial ob ject Y ( φ ) • in V is given by Y ( φ ) n =      Q a ∈ Ob ( A ) φ ( a, a ) , if n = 0 Q a 0 ,...,a n ∈ Ob ( A ) φ ( a 0 , a n ) A ( a 0 ,...,a n ) , if n ≥ 1 , where A ( a 0 , ..., a n ) = A ( a 0 , a 1 ) ⊗ ... ⊗ A ( a n − 1 , a n ). Her e ar e so me examples o f coface and co de g eneracy maps, fo r a full description see [4, pages 6 and 7]. d 0 : Y ( φ ) n → Y ( φ ) n +1 is obtained from the diagram Q a 0 ,...,a n ∈ Ob ( A ) φ ( a 0 , a n ) A ( a 0 ,...,a n ) / / ❴ ❴ ❴ ❴ ❴ ❴ pr b 1 ,...,b n +1   Q b 0 ,...,b n +1 ∈ Ob ( A ) φ ( b 0 , b n +1 ) A ( b 0 ,...,b n +1 ) pr b 0 ,...,b n +1   φ ( b 1 , b n +1 ) A ( b 1 ,...,b n +1 ) / / φ ( b 0 , b n +1 ) A ( b 0 ,...,b n +1 ) , where the botto m ho rizontal map is the adjoint trans po se of φ ( b 1 , b n +1 ) A ( b 1 ,...,b n +1 ) ⊗ A ( b 0 , b 1 ) ⊗ A ( b 1 , ..., b n +1 ) → A ( b 0 , b 1 ) ⊗ φ ( b 1 , b n +1 ) → ( A op ⊗ A )(( b 1 , b n +1 ) , ( b 0 , b n +1 )) ⊗ φ ( b 1 , b n +1 ) → φ ( b 0 , b n +1 ) whereas d n +1 is obtained from the diagra m Q a 0 ,...,a n ∈ Ob ( A ) φ ( a 0 , a n ) A ( a 0 ,...,a n ) / / ❴ ❴ ❴ ❴ ❴ ❴ pr b 0 ,...,b n   Q b 0 ,...,b n +1 ∈ Ob ( A ) φ ( b 0 , b n +1 ) A ( b 0 ,...,b n +1 ) pr b 0 ,...,b n +1   φ ( b 0 , b n ) A ( b 0 ,...,b n ) / / φ ( b 0 , b n +1 ) A ( b 0 ,...,b n +1 ) where the botto m ho rizontal map is the adjoint trans po se of φ ( b 0 , b n ) A ( b 0 ,...,b n ) ⊗ A ( b 0 , ..., b n ) ⊗ A ( b n , b n +1 ) → φ ( b 0 , b n ) ⊗ A ( b n , b n +1 ) → ( A op ⊗ A )(( b 0 , b n ) , ( b 0 , b n +1 )) ⊗ φ ( b 0 , b n ) → φ ( b 0 , b n +1 ) Similarly , the co degeneracy s i : Y ( φ ) n +1 → Y ( φ ) n is obtained from the diagram Q a 0 ,...,a n +1 ∈ Ob ( A ) φ ( a 0 , a n ) A ( a 0 ,...,a n +1 ) / / ❴ ❴ ❴ ❴ ❴ ❴ pr b 0 ,...,b i − 1 ,b i ,b i ,...,b n   Q b 0 ,...,b n ∈ Ob ( A ) φ ( b 0 , b n ) A ( b 0 ,...,b n ) pr b 0 ,...,b n   φ ( b 0 , b n ) A ( b 0 ,...,b i ) ⊗A ( b i ,b i ) ⊗A ( b i ,...,b n ) insert id b i / / φ ( b 0 , b n ) A ( b 0 ,...,b n ) Calculation 4.3. In 5.1 w e shall need an understanding of the category ( V - Mo d ( A , A ))- Cat . Step 1. Let ( E , ⊗ , I ) b e an arbitrar y monoida l categor y having sufficient colimits and with monoidal pro duct preserving the existent colimits in each v aria ble s eparately . W e denote by M on ( E ) the catego ry of monoids in E . Let A ∈ M on ( E ). Let A M od A be the catego ry of ( A, A )-bimo dules in E . The catego r ies M on ( A M od A ) and ( A ↓ M on ( E )) are isomorphic as categories above M on ( E ): M on ( A M od A ) ∼ = / / ' ' ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ( A ↓ M on ( E )) w w ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ M on ( E ) Step 2. Let ( E , ⊗ , I ) b e a s in step 1 a nd let S b e a s et. At the b eginning o f this section we defined a str ong monoidal functor δ : E → E - Graph ( S ). Let A b e a mo noid in E . The categor ies A M od A - Cat ( S ) and M on ( δ A M od δ A ) are 8 A. E. ST ANCULESCU isomorphic. Therefore , by s tep 1 the categor ies A M od A - Cat ( S ) and ( δ A ↓ E - Cat ( S )) are isomorphic as categ o ries ab ov e E - Cat ( S ): A M od A - Cat ( S ) ∼ = / / ( ( P P P P P P P P P P P P ( δ A ↓ E - Cat ( S )) v v ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ E - Cat ( S ) Step 3. Let ( V , ⊗ , I ) b e a clo sed catego ry having sufficient colimits. Let A b e a V -categor y and S a set. By step 2 and previous co ns iderations the categ ories ( V - Mo d ( A , A ))- Cat ( S ) and ( δ A ↓ ( V - Graph ( O b ( A )))- Cat ( S )) ar e isomorphic as categories ab ov e ( V - Graph ( O b ( A )))- Cat ( S ): ( V - Mo d ( A , A ))- Cat ( S ) ∼ = / / + + ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ( δ A ↓ ( V - Graph ( Ob ( A )))- Cat ( S )) s s ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ( V - Graph ( Ob ( A )))- Cat ( S ) The right-hand co rner o f the prev io us diagram implies that to g ive an ob ject o f ( V - Mo d ( A , A ))- Cat amo un ts to a set S , a V -catego ry Z with ob ject set S × O b ( A ) and, for all x ∈ S , V -functors u x : A → Z with ob ject maps u x ( a ) = ( x, a ). 5. V -coherent transforma tions with respect to a coa ugmented cosimplicial comono id a nd the Gra y tensor product with respect to a cosimpl icial comonoid Throughout this section ( V , ⊗ , I ) is a co mplete and c o complete closed category . W e write Y X for the in ternal hom o f tw o o b jects X , Y of V . Given tw o small V -categ o ries A , B and a comono id C in V ∆ + , we shall construct a V - category C oh C ( A , B ) whose ob jects ar e the V -functors fro m A to B . O f particular imp ortance will b e the co monoids C of the form i ! C • (2.2). Let C be a comonoid in V ∆ + and A a V -catego ry . W e define C oh C ( A , − ) : V - Cat → V - Cat as the comp osite of the three canonical functors V - Cat ← − V ∆ + ( C, − ) V ∆ + - Cat ← − Y + ( V - Mo d ( A , A ))- Cat ← − hA , −i V - Cat The a rrows p oint to the rig ht in or der to emphasize that each of the three functor s is a r ight adjoint, a s we shall later show. W e need to construct hA , −i . If B is a V -c a tegory , the ob jects of hA , B i are the V -functors A → B a nd hA , B i ( f , g )( a, a ′ ) = B ( f op ⊗ g )( a, a ′ ) = B ( f a, g a ′ ) The co mpo sition maps hA , B i ( f , g ) ◦ hA , B i ( g , h ) → hA , B i ( f , h ) are induced b y the comp os ition maps of B . V ar ying C and A we o btain a functor C oh − ( − , − ) : ( C omon ( V ∆ + ) × V - Cat ) op × V - Cat → V - Cat C oh C ( A , B ) = V ∆ + ( C, Y + hA , B i ) W e call C oh C ( A , B )( f , g ) the ob ject of V -coheren t transformatio ns from f to g wi th resp ect to C . Let C • be a como noid in V ∆ . W e call C oh i ! C • ( A , B )( f , g ) the ob ject of V -coheren t transformations from f to g wi th resp ect to C • . W e shall write C oh C • ( A , B ) instead of C oh i ! C • ( A , B ). Since the adjunction − ∗ i ! C • : V ⇄ V ∆ + : V ∆ + ( i ! C • , − ) splits as V −∗ C • / / V ∆ V ∆ ( C • , − ) o o i ! / / V ∆ + i ∗ o o it follows that C oh C • ( A , B ) = V ∆ ( C • , Y hA , B i ) The previous form ula w a s considered by Cordier a nd P orter [4, Definition 3.1] in the case when V is the category o f simplicial sets and the cosimplicia l s implicial set ∆ is in the plac e o f C • . See also the refer ences there in. W e hav e bo rrow ed the notation C oh fro m them. How ever, the for malism leading to this for mu la is not present in [4]. Also , FORMAL ASPECTS OF GRA Y’S TENSOR P R ODUCTS OF 2-CA TEGORIES 9 in the case of simplicia l sets it is known that ∆ is not a comonoid with r esp ect to ⋆ a nd that C oh ( A , B ) ca nnot be naturally made in to a simplicial categor y; se e [4, page 28] for a discuss ion of the la tter fact. Examples 5.1. ( a ) Given a V -categor y A a nd a como noid C in V , le t us deno te by A C the V -catego ry having the same ob jects as A and having the V -homs A C ( a, a ′ ) = A ( a, a ′ ) C . Then one has C oh C ( I , A ) ∼ = A C (1) and C oh C • ( I , A ) ∼ = A C 0 . ( b ) C oh cstI ( A , B ) coincides with the internal hom V N at ( A , B ) of the standar d closed categor y structure on V - Cat , since C oh cstI ( A , B )( f , g ) is the ob ject of V -natural transformations from f to g . ( c ) C oh sk ( I ) ( A , B )( f , g ) = Q a ∈ Ob ( A ) B ( f a, g a ). More generally , for a como noid C in V , C oh sk ( C ) ( A , B ) = C oh sk ( I ) ( A , B ) C . Remark 5.2. Ther e are v ariants of C oh C • ( A , B ), if one is willing to re place V ∆ with V ∆(1) or V ∆(1) mon . F or example, in the case of V ∆(1) one obtains a functor C oh − ∆(1) ( − , − ) : ( C omon ( V ∆(1) ) × V - Cat ) op × V - Cat → V - Cat In the case of V ∆(1) mon one obtains a V -ca tegory without unit. 5.1. The m ain TH-situation. W e s ha ll no w show that C oh − ( − , − ) is part of a TH-situation − ⊠ − − : V Cat × ( C omon ( V ∆ + ) × V - Cat ) → V - Cat C oh − ( − , − ) : ( C omon ( V ∆ + ) × V - Cat ) op × V - Cat → V - Cat T o construct this TH-situation it suffices to construct, for every co monoid C in V ∆ + , a TH-situation − ⊠ C − : V - Cat × V - Cat → V - Cat C oh C ( − , − ) : V - Cat op × V - Cat → V - Cat such that the adjunction isomorphisms in this TH-situation are natural in C . In turn, to construct the latter TH-situation it suffices to c o nstruct, for every V -catego ry A , a left adjoint − ⊠ C A to C oh C ( A , − ) such that the adjunction is o morphisms are natural in A . W e show that eac h of the three functor s which make up C oh C ( A , − ) ha s a left adjoin t, thus − ⊠ C A will be by definition the compos ite o f these left adjoin ts. • The left adjoin ts to V ∆ + ( C, − ) and Y + are constructed using the general F act 5.3. L et E 1 and E 2 b e two c o c omplete monoida l c ate gories with m onoidal pr o duct s c o c ontinuous in e ach varia ble sep ar ately. We denote by E i - Graph the c ate gory of smal l E i -gr aphs. L et F i : E i - Graph ⇄ E i - Cat : U i b e the fr e e-for getful adjunction [1 7] , i ∈ { 1 , 2 } . L et F : E 1 ⇄ E 2 : G b e an adjoint p air with G monoidal. The functor G : E 2 - Cat → E 1 - Cat has a left adjoint F ′ c onstru cte d in su ch a way that F 2 F ∼ = F ′ F 1 . In p articular, F ′ pr eserves the unit obje ct. F ′ is c onstructe d (fibr ewise) as fol lows. F or A ∈ E 1 - Cat , F ′ A is the c o e qualiser of the r eflexive p air F 2 F U 1 F 1 U 1 A / / / / F 2 F U 1 A One arr ow is obtaine d by apply ing F 2 F U 1 to t he c ounit F 1 U 1 A → A . The other one is obtaine d by substituting X = U 1 A in the adjoint tr ansp ose of t he natu r al m ap F U 1 F 1 X → U 2 F 2 F X , X ∈ E 1 - Graph . • The left adjoint − ♦ A to hA , −i is defined as follows. O b ( C ♦ A ) = O b ( C ) × O b ( A ) and C ♦ A (( c, a ) , ( c ′ , a ′ )) = C ( c, c ′ )( a, a ′ ). T o see that C ♦ A is w ell-defined and indeed a left adjoint one uses calculation 4.3. This finishes the construction of − ⊠ C A . The adjunction isomorphism is clea rly natural in A , hence w e obtain a TH-situation ( − ⋆ ∆ + C ( − ))( F ∆ + ) ′ δ ∆ + ( − ) ♦ − : V ∆ + - Cat × V - Cat → V - Cat Y + h− , −i : V - Cat op × V - Cat → V ∆ + - Cat Using the adjunction − ∗ C : V ⇄ V ∆ + : V ∆ + ( C, − ), where C is a comonoid in V ∆ + , and the fact that TH-situa tions can be changed along adjoin t functors, we obtain a TH-situation − ⊠ C − : V - Cat × V - Cat → V - Cat C oh C ( − , − ) : V - Cat op × V - Cat → V - Cat It is clear that the adjunction isomor phism in this TH-situatio n is natural in C , therefore we obta in the desired TH- situation. Using again the fact that TH-situations can b e changed along adjoin t functors, we obtain a TH-situation − ⊠ − − : V Cat × ( C omon ( V ∆ ) × V - Cat ) → V - Cat C oh − ( − , − ) : ( C omon ( V ∆ ) × V - Cat ) op × V - Cat → V - Cat 10 A. E. ST ANCULESCU F or a comonoid C • in V ∆ one has A ⊠ C • B = A ⊠ i ! C • B . W e ca ll A ⊠ C • B the Gray tensor pro duct of A and B wi th resp ect to C • . W e do not cla im that ⊠ C • is a mo no idal pro duct o n V - Cat , but see prop osition 5 .9. The naming will be justified in 5.3 . Notation 5.4. F or an obje ct X of an arbitr ary monoidal c ate gory E with unit I and having an initial obje ct, we denote by 2 X the E -c ate gory with two obje ct s 0 and 1 and with 2 X (0 , 0) = 2 X (1 , 1) = I , 2 X (1 , 0) = ∅ and 2 X (0 , 1) = X . F or example, in the setting of 5.3 one has 2 F X ∼ = F ′ (2 X ) for every X ∈ E 1 . If φ ∈ V - Mo d ( A , A ) we repr esent 2 φ ♦ A as (0 , a ′ ) φ ( a ′ ,a ′ ) (1 , a ′ ) (0 , a ) φ ( a,a ) A ( a,a ′ ) φ ( a,a ′ ) ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ (1 , a ) A ( a,a ′ ) Examples 5 .5. ( a ) I ⊠ C A ∼ = A and − ⊠ C I ∼ = ( − ⊗ C (1)) ′ . ( b ) By example 5.1( c ), A ⊠ sk ( C ) B ∼ = ( − ⊗ C ) ′ ( A ) ⊠ sk ( I ) B for an arbitrary comonoid C in V . ( c ) 2 X ⊠ C A ∼ = 2 ( A◦ δ X ∗ C ◦A ) ⋆ ∆ + C ( A ) ♦ A . Remark 5. 6 . Let C be a co monoid in V ∆ + with C (1 ) 6 = I . Ex ample 5.5( a ) s hows that there is no right closed category structure on V - Cat with unit I and right internal hom C oh C ( − , − ). The next result, whose proo f is left to the reader, is inspired b y [4, Section 7]. Lemma 5 .7. Ther e ar e two n atur al maps Y + hA , B i ( f , g ) ⋆ Y + hB , C i ( k , l ) → Y + hA , C i ( k f , l g ) which ar e suitably asso ciative and unital. Conse quently, for every c omonoid C in V ∆ + , C oh C ( − , − ) ∈ ( V - Cat ) - CA T in t wo ways. 5.2. The case of sk ( I ) . It is known, more or le s s from [6], that C oh sk ( I ) ( − , − ) is the internal hom o f a close d category structure on V - Cat . W e provide b elow mor e details than in lo c.cit. . Let A , B a nd C b e three V -categor ie s . A pre - bi - V - functor F : ( A , B ) → C consists of the following data: for all a ∈ Ob ( A ) and b ∈ Ob ( B ) there ar e V -functor s F ( a, − ) : B → C and F ( − , b ) : A → C such that F ( a, − )( b ) = F ( − , b )( a ). W e denote by P r e - bi - V - F un ( A , B ; C ) the set of pre-bi- V -functors ( A , B ) → C . W e obta in a functor P r e - bi - V - F un ( − , − ; − ) : ( V - Cat × V - C at ) op × V - Cat → S et . It follows from example 5.1( c ) that Lemma 5 .8. Ther e is a natu r al bije ct ion V - Cat ( A , C oh sk ( I ) ( B , C )) ∼ = P r e - bi - V - F un ( A , B ; C ) Let now A and B b e t wo V -categ o ries and let S = O b ( A ), T = Ob ( B ). W e define A ⊠ sk ( I ) B to be the pushout of the diagram I S ⊗ I T / /   I S ⊗ B   A ⊗ I T / / A ⊠ sk ( I ) B This pushout is calculated in V - Cat ( S × T ). Prop ositio n 5.9. The c ate gory ( V - Cat , ⊠ sk ( I ) , I ) is a close d c ate gory, with internal hom C oh sk ( I ) ( − , − ) . 5.3. The relation with Gra y’ s tensor pro du cts. A co category interv al in V is a co catego ry ob ject in V with ob ject of coo b jects equal to I . W e write s uch a gadget as I 0 = I d 1 / / d 0 / / I 1 p o o i 0 / / c / / i 1 / / I 2 c denotes the co comp osition. T he r e is also the ob vious notion of cogroup o id interv al . Co categor y interv als are pre s erved by functors which pres erve the unit ob ject a nd finite colimits. A co ca tegory int erv a l as abov e is the be ginning of a cosimplicial ob ject in V w hich we denote b y I • . W e shall alw ays use the same FORMAL ASPECTS OF GRA Y’S TENSOR P R ODUCTS OF 2-CA TEGORIES 11 notation for b oth the data for a co ca teg ory interv al and the co simplicial ob ject it gives rise to. W e s hall need to b e explicit a bo ut certain cofa c e and co de g eneracy maps o f I • . The co fa ce maps d i : I 1 → I 2 are d 0 = i 0 , d 1 = c and d 2 = i 1 . The c oface maps d i : I 2 → I 3 are depicted in the diagrams below, in whic h all squares are pushouts: I 1 i 0 / / i 1   I 2 d 3   I 2 d 0 / / I 3 I d 1 / / d 0   I 1 i 0   d 0 i 0   ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ I 1 i 1 / / c   I 2 d 1 ❅ ❅ ❅ ❅ I 2 d 3 / / I 3 I d 0 / / d 1   I 1 i 1   d 3 i 1   ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ I 1 i 0 / / c   I 2 d 2 ❅ ❅ ❅ ❅ I 2 d 0 / / I 3 The coface map s 0 : I 2 → I 1 is the unique map suc h that s 0 i 0 = 1 I 1 and s 0 i 1 = d 1 p . The other coface map s 1 : I 2 → I 1 is the unique map suc h that s 1 i 1 = 1 I 1 and s 1 i 0 = d 0 p . More details about co categ ory in terv als can b e found in [16]. Examples 5 .10. ( a ) The initial co ca tegory in ter v al in S et has I 1 = { 0 , 1 } a nd I 2 = { 0 , 1 , 2 } , with the usual coface maps. The co comp osition c is the map which o mits 1. This is a co g roup oid interv al. Therefore V ha s the initial cogro up o id in terv a l obtained using the functor F : S et → V , F ( S ) = ⊔ S I ; this is precise ly sk ( I ) (2.2). ( b ) The standard co categ ory interv al in Cat has I 1 = [1], the totally o r dered se t { 0 < 1 } , and I 2 = [2], the totally ordered set { 0 < 1 < 2 } . The co comp ositio n is the map which o mits 1 . W e shall denote this co catego ry in ter v al by I • . Applying to I • the free gr oup oid functor, we obtain the standar d co group oid interv al J • in the catego ry Grp d of small g r oup oids. J 1 has tw o o b jects and one arr ow betw een them and J 2 has three ob jects and one a rrow betw een any t wo ob jects. Alternativ ely , J n is the indiscrete/chaotic category on the set I n considered in ( a ). W e shall v ie w J • as living in Cat . The functor F from ( a ) induces a functor V : Cat → V - Cat , left a djoint to the underlying category functor. Ther efore V ( J • ) is a cogroup oid in ter v al in V - Cat . Prop ositio n 5.11. I • is a c omonoid in ( Cat ∆ , ⋆ ) . Pr o of. W e fir st show that I • is a comonoid in ( Cat ∆(1) , ⋆ ). W e denote by 1 the ter mina l catego ry . Since ( I • ⋆ I • ) 0 = 1 × 1, we c ho ose a map f 0 : 1 → 1 × 1 to b e the inv erse of the left (or right) c o nstraint of Cat ev aluated at 1. The ob ject ( I • ⋆ I • ) 1 is the pushout of the diagram 1 × I 1 I d 1 × d 0 ← − 1 × 1 d 1 × I d I − → I 1 × 1 12 A. E. ST ANCULESCU hence it is isomorphic to I 2 . The co fa ce maps ar e D 0 = i 0 ( d 0 × I d 1 ) and D 1 = i 1 ( I d 1 × d 1 ), the co degener acy s 0 being depicted in the diagram below: 1 f 0 / / d 1   d 0   1 × 1 1 × d 0 w w ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ 1 × d 1 w w ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ d 0 × 1 ' ' ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ d 1 × 1 ' ' ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ 1 × I 1 i 1 & & ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ 1 × p   ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ ❂ I 1 × 1 i 0 x x ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ p × 1   ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ I 1 p O O ( I • ⋆ I • ) 1 = I 2 s 0   1 × 1 This forces us to set f 1 = c . W e hav e obtained a map f : I • → I • ⋆ I • . Nex t, using the description of I • ⋆ I • one can see that (( I • ⋆ I • ) ⋆ I • ) 1 is (isomorphic to) the pushout of the diagram I 1 1 d 1 / / d 0 O O I 1 i 1 / / I 2 hence it is I 3 . The coface maps are D 0 = d 3 i 1 d 1 and D 1 = d 0 i 0 d 0 . The ab ove pusho ut can also be calculated as the pushout of the diagram I 2 I 1 i 0 O O 1 d 1 / / d 0 O O I 1 which is ( I • ⋆ ( I • ⋆ I • )) 1 . It follows that ( f ⋆ I • ) 1 = d 2 and ( I • ⋆ f ) 1 = d 1 , therefore f is coa sso ciative. The counit I • → cst 1 is the co degeneracy map. Using no w the presentations of ⋆ from 2.2 one can compute that the category ( I • ⋆ I • ) 2 is a 0 , 2 / / a 1 , 2 / / a 2 , 2 a 0 , 1 / / O O a 1 , 1 O O a 0 , 0 O O (More gener a lly , ( I • ⋆ I • ) n is the functor categ ory I n I 1 .) The cofaces D i : I 2 → ( I • ⋆ I • ) 2 are given by D 0 (0) = a 1 , 1 , D 0 (1) = a 1 , 2 , D 0 (2) = a 2 , 2 , D 1 (0) = a 0 , 0 , D 1 (1) = a 0 , 2 D 1 (2) = a 2 , 2 , D 2 (0) = a 0 , 0 , D 2 (1) = a 0 , 1 , and D 2 (2) = a 1 , 1 . Then f 2 := ( D 0 c, D 2 c ) is given by f 2 (0) = a 0 , 0 , f 2 (1) = a 1 , 1 and f 2 (2) = a 2 , 2 , and so f 2 c = D 1 c . By lemma 5.12 we obtain a map f • : I • → I • ⋆ I • which is coas so ciative. The counit axiom is easy to see.  Let c i : [1] → [ n ] b e the cosimplicial ope rator given by c i (0) = i − 1 and c i (1) = i , where 1 ≤ i ≤ n . W e write res : V ∆ → V ∆(1) for the functor given by restriction a long the inclusion res : ∆(1) ⊂ ∆. Lemma 5. 12. L et A • b e a c o c ate gory obje ct in V , X • is a c osimplicial obje ct in V and f : res ( A • ) → res ( X • ) . F or n ≥ 2 denote by f n : A n → X n the u nique morphism of V su ch that f n c i = c i f 1 . Then (1) Ther e is at most one ¯ f : A • → X • such t hat res ( ¯ f ) = f . FORMAL ASPECTS OF GRA Y’S TENSOR P R ODUCTS OF 2-CA TEGORIES 13 (2) Such an ¯ f exists if and only if f 2 c = d 1 f 1 , and when this is so, ¯ f n = f n . Let us take V = ( Cat , × ) and C • = I • . T o give a 2 -functor 2 1 → C oh I • ( A , B ) is to give t wo 2-functors F, G : A → B and an ob ject o f Cat ∆ ( I • , Y hA , B i ( F, G ) • ). By adjunction, the la tter data amounts to giving a map I • → Y hA , B i ( F, G ) • in Cat ∆ . The c osimplicial ob ject Y hA , B i ( F , G ) • is describ ed below notation 4.2, th us Y hA , B i ( F, G ) n =      Q a ∈ Ob ( A ) B ( F a, Ga ) , if n = 0 Q a 0 ,...,a n ∈ Ob ( A ) B ( F a 0 , Ga n ) A ( a 0 ,...,a n ) , if n ≥ 1 , where A ( a 0 , ..., a n ) = A ( a 0 , a 1 ) × ... × A ( a n − 1 , a n ). The co degenera cy s 0 is g iven by ( u a,b ) 7→ ( u a,a (1 a )). Let us compute some cofaces, first d 0 , d 1 : Y hA , B i ( F, G ) 0 → Y hA , B i ( F, G ) 1 W e hav e, on ob jects, d 0 (( α a )) = ( f 7→ α b ◦ F f ) and d 1 (( α a )) = ( f 7→ Gf ◦ α a ) Next, let’s compute d 0 , d 1 , d 2 : Y hA , B i ( F , G ) 1 → Y hA , B i ( F, G ) 2 W e hav e, on ob jects, d 0 (( u a,b )) = (( f : a 0 → a 1 , g : a 1 → a 2 ) 7→ u a 1 ,a 2 ( g ) ◦ F f ) , d 1 (( u a,b )) = (( f : a 0 → a 1 , g : a 1 → a 2 ) 7→ u a 0 ,a 2 ( g f )) , d 2 (( u a,b )) = (( f : a 0 → a 1 , g : a 1 → a 2 ) 7→ Gg ◦ u a 0 ,a 1 ( f )) W e app eal now to le mma 5.12 to see wha t a map I • → Y hA , B i ( F, G ) • is. T o give a commutativ e diagram 1 / / d 1   d 0   Q a ∈ Ob ( A ) B ( F a, Ga ) d 1   d 0   I 1 p O O f 1 / / Q a 0 ,a 1 ∈ Ob ( A ) B ( F a 0 , Ga 1 ) A ( a 0 ,a 1 ) s 0 O O is to give the data consis ting of ( i ) 1-cells α a : F a → Ga for each ob ject a of A , and ( ii ) a coherence 2- cell α f : Gf ◦ α a → α b ◦ F f filling in the squa re for each 1-cell f : a → b , such that α 1 a : 1 Ga ◦ α a → α a ◦ 1 F a is a n ident ity 2-cell whenev er f = 1 a and for ev ery 2-cell γ : f → g , α g ( Gγ α a ) = ( α b F γ ) α f . A map f 2 : I 2 → Y a 0 ,a 1 ,a 2 ∈ Ob ( A ) B ( F a 0 , Ga 2 ) A ( a 0 ,...,a 2 ) such that f 2 = ( d 0 f 1 , d 2 f 1 ) is given on ob jects by f 2 (0) = (( f : a 0 → a 1 , g : a 1 → a 2 ) 7→ G ( g f ) ◦ α a 0 ) , f 2 (1) = (( f : a 0 → a 1 , g : a 1 → a 2 ) 7→ Gg ◦ α a 1 ◦ F f ) , f 2 (2) = (( f : a 0 → a 1 , g : a 1 → a 2 ) 7→ α a 2 ◦ F ( g f )) T o say that f 2 c = f 1 d 1 on arrows is to say that ( α g F f )( Gg α f ) = α gf . In conclusio n, the 1- cells of the 2-ca tegory C oh I • ( A , B ) are the quasi- natural (also called lax natural) transformations b e tw een 2-functors. A similar arg umen t inv olv ing now the 2 -functor 2 [1] → C oh I • ( A , B ) shows that the 2-cells are the mo difications. Therefore A ⊠ I • B is Gray’s tensor pro duct of 2-ca tegories. The same consider a tions apply to C oh J • ( A , B ). Ac knowledgemen ts. This work would not hav e b een p ossible without the help o f Andr´ e J oy al and Michael Makk ai. I heartily thank the r eferees for their comments a nd suggestions and Michael W arr e n for useful discussions related to the material of this article. 14 A. E. ST ANCULESCU References [1] M. A. Batanin, Coher ent c ate gories with r esp e ct to monads and c oher e nt pr ohomotopy the ory , Cahiers T op ologie G´ eom. Diff´ erentielle Cat ´ eg. 34 (1993), no. 4, 279–304. [2] M. A . 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