The category of 3-computads is not cartesian closed

The category of 3-computads is not cartesian closed Mihaly Makk ai and Marek Za wado wski Departmen t of Mathematics and Statistics, M cGill Univ ersit y , 805 Sherbro ok e St., Mon tr´ eal, PQ, H3A 2K6, Canada Inst ytut Matemat yki, Uniw ersytet W arsza wski ul. S.Banac ha 2, 00-913 W arsza wa, P o land F ebruary 26, 2008 Abstract W e show, using [CJ] and Eckmann-Hilton argument, that the categor y of 3-computads is not car tesian closed. As a corolla ry w e get that neither the category of all computads nor the category of n -computads, for n > 2, do for m lo cally cartesia n closed categorie s, and hence elementary top ose s. 1 In tro duction S.H. Schan u el (unp ublished) made an observ ation, c.f. [CJ], that the categ ory of 2- computads Comp 2 is a presheaf category . W e show b elo w that n either th e category of computads nor the catego ries n -computads, f or n > 2, are lo cally cartesian closed. This is in contrast with a remark in [CJ] on page 453, and an explicit statemen t in [B] claiming th at these categories are presh ea v es categories. Note that s ome inter- esting sub categories of computads, like m any-to-o ne compu tads, do form p resheaf catego ries, c.f. [HMP], [HMZ]. W e thank the anonymous r eferee for commen ts that help ed to clarify the exp o- sition of the example. The diagrams for this pap er w ere prepared with a help of c atmac of Mic hael Barr. 2 Computads Computads were introd uced b y R.Str eet in [S], s ee also [B]. Reca ll that a computad is an ω -catego ry that is leve lwise fr ee. Belo w w e recall one of the definitions. Let nCat b e the cate gory of n -categorie s and n -functors b et w een them, ω Cat b e the catego ry of ω -catego ries and ω -fun ctors b et w een th em. W e hav e th e ob vious truncation functors tr n − 1 : nC at − → ( n − 1 ) Cat By Comp n w e den ote th e cate gory of n -computads, a non-fu ll su b category of the catego ry nCat . By CCat n w e denote the non-full sub category of nCat , whose ob jects are ’computads up to the lev el n − 1’, i.e. an n -fu nctor f : A → B is a morphism in CCat n if and only if tr n − 1 ( f ) : tr n − 1 ( A ) → tr n − 1 ( B ) is a morphism in Comp n − 1 . Clearly CCat n is defined as so on as Comp n − 1 is defi ned. The catego ries Comp n and n -comma category Com n are defin ed b elo w. The cate gories Comp 0 , CCat 0 and Com 0 are equal to S et , the category of sets. W e ha v e an adjun ction 1 Com 0 CCat 0 ✛ U 0 ✲ F 0 with b oth functors b eing the identit y on S et , F 0 ⊣ U 0 . C omp 0 is the image of Com 0 under F 0 . Com 1 is the category of graphs, i.e. an ob ject of Com 1 is a pair of s ets and a pair of functions b et w een them h d, c : E → V i . CCat 1 is simply Cat , the category of all small categories. The forgetful functor U 1 (forgetting comp ositions and identi ties) has a left adjoint F 1 ’the free category (o v er a graph)’ functor Com 1 CCat 1 ✛ U 1 ✲ F 1 W e ha v e a diagram Com 0 CCat 0 ✲ F 0 Com 1 CCat 1 ✲ F 1 ❄ tr ′ 0 ❄ tr 0 Comp 0 ✟ ✟ ✟ ✟ ✟ ✯ F 0 ❍ ❍ ❍ ❍ ❥ ι 0 ❍ ❍ ❍ ❍ ❍ ❥ tr 0 ✟ ✟ ✟ ✟ ✟ ✙ tr 0 where three triangles comm u te, moreo v er the left triangle and the outer square comm ute up to an isomorphism. tr 1 and tr ′ 1 are the ob vious truncation morph isms. Then we define the category of 1-co mpu tads Comp 1 as the essential (non-full) image of the functor F 1 in CCat 1 , i.e. 1-computads are the free ca tegories o v er graphs and computad maps b et w een them are fu nctors sending indets (=in determi- nates=generators) to ind ets. No w supp ose th at we ha v e an adjun ction U n ⊣ F n Com n CCat n ✛ U n ✲ F n Comp n ✟ ✟ ✟ ✟ ✟ ✯ F n ❍ ❍ ❍ ❍ ❥ ι n and Comp n is defined as the the essen tial (non-full) image of th e functor F n in CCat n . W e define the n -parallel p air fun ctor Π n : C omp n S et ✲ suc h that Π n ( A ) = {h a, b i| a, b ∈ A n , d ( a ) = d ( b ) , c ( a ) = c ( b ) } for any n -computad A . The ( n + 1)-comma categ ory Com n + 1 is the catego ry S et ↓ Π n . Th u s an ob ject in Com n + 1 is a p air ( A, h d, c i : X → Π n ( A ), su c h that A is an n -computad X is a set of ( n + 1)-i ndets and h d, c i is a function asso ciating n -domains and n -cod omains. The forgetful fun ctor U n +1 : CCat n + 1 − → Com n + 1 (forgetting comp ositions and id entities at the lev el n + 1) creates limits and satisfies the solution s et condition. Thus it has a left adjoint F n +1 . W e get a diagram 2 Com n CCat n ✲ F n Com n + 1 CCat n + 1 ✲ F n +1 ❄ tr ′ n ❄ tr n Comp n ✟ ✟ ✟ ✟ ✟ ✯ F n ❍ ❍ ❍ ❍ ❥ ι n ❍ ❍ ❍ ❍ ❍ ❥ tr n ✟ ✟ ✟ ✟ ✟ ✙ tr n where three triangles comm u te, moreo v er the left triangle and the outer square comm ute u p to an isomorphism . tr n are the o bvio us tru ncation fun ctors and tr ′ n is a truncation functor that at the leve l n lea v es the indets only . Then we define the category of ( n + 1) -computads Comp n + 1 as the essenti al (non-full) image of th e functor F n +1 in CC at n + 1 , i.e. ( n + 1)-computads are the free ( n + 1)-cat egories o v er ( n + 1)-c omma categories and ( n + 1)-c omputad map s b etw een them are ( n + 1)- functors sending indets to ind ets. The cate gory of computads Comp is a (non- full) sub category of the category of ω -categories and ω -functors ω Cat suc h , that for eac h n , the trun cation o f ob jects and m orphisms to nCat is in Comp n . As F n : Com n → C Cat n is faithful and full on isomorphisms, after r estricting the co domain w e get an equiv alence of categories F n : Com n → Comp n . Notation. If A is a compu tad then A n denotes the set of n -cells of A and | A | n denotes th e set of n -indets of A . The truncation functor tr n : Comp n + 1 − → Comp n has b oth adjoin ts i n ⊣ tr n ⊣ f n Comp n + 1 Comp n ✲ tr n ✛ i n ✛ f n where i n ( A ) = F n +1 ( A, ∅ → Π n ( A )) and f n ( A ) = F n +1 ( A, id Π n ( A ) : Π n ( A ) → Π n ( A )) for A in Comp n . This s ho ws that tr n preserve s limits and colimits. The colimits in C omp n + 1 are calculated in ( n + 1 ) Cat but the limits in Comp n + 1 are m ore in v olv ed. It is more conv enient to describ e them in Com n + 1 and then apply the functor F n +1 . If H : J → Com n + 1 is a fun ctor and P is the limit of its truncation tr n ◦ H to Comp n then Lim H , the limit of H , trun cated to C omp n is P and th e ( n + 1)-indets | Lim H | n +1 of Lim H are as f ollo ws | Lim H | n +1 = {h a i i i ∈J | a i ∈ | H ( i ) | n +1 , h d ( a i ) i i ∈J , h c ( a i ) i i ∈J ∈ P n } The terminal o b ject 1 n in Comp n is quite complicated, for n ≥ 2. How ever the Com 2 part of 1 2 is still easy to describ e. 1 2 has one 0-indet x and one 1-ind et ξ : x → x . T h us the 1-cells can b e identified w ith finite (p ossibly emp ty) strings of of arr o ws: x x ✲ ξ x ✲ ξ . . . x x ✲ ξ x, or simply with elemen ts of ω . Th e set | 1 2 | 2 of 2-indets in 1 2 con tains exactly one indet for ev ery pair of strings. The fi rst elemen t of s uc h a pair is the domain of the indet and the second element of the pair is the co domain of the indet. T h us | 1 2 | 2 can b e iden tified with the set ω × ω . In particular h 0 , 0 i corresp ond to the only ind et from id x to id x ( id x is the id en tit y on x ). The description of all 2-cells in 1 2 is more in v olv ed but we don’t n eed it here. 3 3 The count erexample Lemma 3.1 C omp 3 is not c artesian close d. Pr o of. As it wa s noted in Lemma 4.2 [CJ], the functor Π 2 factorizes as Comp 2 S et ↓ Π 2 (1 2 ) ✲ c Π 2 S et ✲ Σ where c Π 2 ( A ) = Π 2 (! : A → 1 2 ), and Σ( b : B → Π 2 (1 2 )) = B , f or A in Comp 2 and b in S et ↓ Π 2 (1 2 ). Moreo v er, th e category S et ↓ Π 2 , which is equ iv alen t to Comp 3 , is also equiv alent to ( S et ↓ Π 2 (1 2 )) ↓ c Π 2 . No w, as Comp 2 and S et ↓ Π 2 (1 2 ) are cartesian closed categories w ith initial ob jects (in fact b oth catego ries are presheaf top oses) and c Π 2 preserve s the terminal ob ject, by Theorem 4.1 of [CJ ], Comp 3 is a cartesian closed category if and only if c Π 2 preserve s b inary pro d ucts. W e finish the pro of b y sh o wing that c Π 2 do es n ot preserves the b inary p r o ducts. Let A b e a 2-co mputad w ith one 0-cel l x , one 1-ce ll id x the iden tit y on x (no 1-indets). Mo reo v er A has as 2-cell s all cells ge nerated b y the tw o indeterminate 2-cells a 1 , a 2 : id x → id x . Th u s, by Eckmann-Hilton argumen t, a ny 2-cell in A is of form a m 1 ◦ a n 2 , for m, n ∈ ω (if m = n = 0 then a m 1 ◦ a n 2 = id id x ). Let B b e a 2-computad isomorphic to A with indeterminate 2-cells b 1 , b 2 . Let x b e the un iqu e 0-cell in 1 2 , c b e the only ind eterminate 2-cell in 1 2 that h as id x as its domain and codomain and C a sub computad of 1 2 generated b y c . T he unique maps of 2-computads ! : A → 1 2 and ! : B → 1 2 sends a i and b i to c , for i = 1 , 2. Th us they factor through C as α : A → C and β : B → C , r esp ectiv ely . T he 2-computad C do es not play a crucial role in the count erexample but it make s the explanations simpler. Let us describ e the pro d uct A × B in C omp 2 . T he 0-cell and 1-cell s are as in A , B and C . As there is only one 1-c ell id x in A × B , the compatibilit y condition for domain and co domains of 2-indets is trivially satisfied, and the set 2-indets of A × B is just the p r o duct of 2-indets of A and B , i.e. | A × B | 2 = {h a i , b j i| i, j = 1 , 2 } and th e set of all 2-cells of A × B is ( A × B ) 2 = {h a 1 , b 1 i n 1 ◦ h a 1 , b 2 i n 2 ◦ h a 2 , b 1 i n 3 ◦ h a 2 , b 2 i n 2 | n 1 , n 2 , n 3 , n 4 ∈ ω } The p ro jections A A × B ✛ π 1 B ✲ π B are d efined as the only 2-functors s uc h that π A ( a i , b j ) = a i and π A ( a i , b j ) = b j , f or i, j = 1 , 2. T hus w e ha v e a commuting square A B A × B π A   ✠ π B ❅ ❅ ❘ C α ❅ ❅ ❘ β   ✠ 1 2 ❄ ❄ m ❍ ❍ ❍ ❥ ✟ ✟ ✟ ✙ ! ! ( ∗ ) 4 As C is a sub ob ject of the termin al ob ject A × B is A × C B and A × 1 2 B , i.e. b oth inner and outer squares in the ab o v e diagram are p ullbac ks. Since all the 2-cells in A , B , C and A × B are parallel we ha v e Π 2 ( A ) = A 2 × A 2 , Π 2 ( B ) = B 2 × B 2 , Π 2 ( C ) = C 2 × C 2 , and Π 2 ( A × B ) = ( A × B ) 2 × ( A × B ) 2 c Π 2 preserve s the pro d uct of A and B if in the diagram ( ∗∗ ) b elow, whic h is the application of Π 2 to the diagram ( ∗ ) ab o v e, the outer square is a p u llbac k in S et A 2 × A 2 B 2 × B 2 ( A × B ) 2 × ( A × B ) 2 Π 2 ( π A )     ✠ Π 2 ( π B ) ❅ ❅ ❅ ❅ ❘ C 2 × C 2 Π 2 ( α ) ❅ ❅ ❅ ❅ ❘ Π 2 ( β )     ✠ Π 2 (1 2 ) ❄ Π 2 ( m ) ❍ ❍ ❍ ❍ ❍ ❥ ✟ ✟ ✟ ✟ ✟ ✙ Π 2 (!) Π 2 (!) ( ∗∗ ) As Π 2 ( m ) is mon o, the outer squ are in ( ∗∗ ) is a pullbac k in S et if and only if the inner square in ( ∗∗ ) is a p ullbac k in S et . W e ha v e Π 2 ( π A ) = ( π A ) 2 × ( π A ) 2 , Π 2 ( π B ) = ( π B ) 2 × ( π B ) 2 , Π 2 ( α ) = α 2 × α 2 , and Π 2 ( β ) = β 2 × β 2 . Hence the inn er squ are in ( ∗∗ ) is a pullback if and only if th e squ are ( ∗ ∗ ∗ ) b elo w A 2 B 2 ( A × B ) 2 ( π A ) 2   ✠ ( π B ) 2 ❅ ❅ ❘ ( C ) 2 α 2 ❅ ❅ ❘ β 2   ✠ ( ∗ ∗ ∗ ) is a pullb ack. But ( ∗ ∗ ∗ ) is not a pullbac k in S et . The tw o 2-cells h a 1 , b 1 i ◦ h a 2 , b 2 i , and h a 1 , b 2 i ◦ h a 2 , b 1 i in A × B are different since they are comp ositions of differen t indets. On the other hand ( π A ) 2 (( a 1 , b 1 ) ◦ ( a 2 , b 2 )) = a 1 ◦ a 2 = ( π A ) 2 (( a 1 , b 2 ) ◦ ( a 2 , b 1 )) and ( π B ) 2 (( a 1 , b 1 ) ◦ ( a 2 , b 2 )) = b 1 ◦ b 2 = b 2 ◦ b 1 = ( π B ) 2 (( a 1 , b 2 ) ◦ ( a 2 , b 1 )) i.e. they agree on b oth pro jections and hence ( ∗ ∗ ∗ ) is not a pu llb ac k. Thus c Π 2 do es not p reserv e bin ary pr o ducts, as required. ✷ Theorem 3.2 The c ate g ory of c omputads Comp and the c ate gories of n -c omputa ds Comp n , for n > 2 , ar e not lo c al ly c artesian close d. 5 Pr o of. The slice categories Comp ↓ 1 3 , as well as Comp n ↓ 1 3 , for n > 2, are equiv alen t to Comp 3 , w here 1 3 is the terminal ob ject in Comp 3 lifted (by adding suitable iden tities) to the categ ory of appropriate computads. As, b y Lemma 3.1, Comp n ↓ 1 3 is not cartesian closed we get the theorem. ✷ R emark. In p articular th e categ ories men tioned in the ab o v e theorem are not presheaf (or even elemen tary) top oses. References [B] M.Batanin, Computads for finitary monads on globular sets . Con temp o- rary Mathematics, vol 230, (1998), p p. 37-57. [CJ] A. Carb oni, P .T. Johnstone Conne cte d limits, familial r epr esentability and Artin glueing . Math. Str u ct. in Comp Science, v ol 5. (1995), p p. 441-4 59. [HMZ] V. Harnik, M. Makk ai, M. Zaw adowski, Multitopic sets ar e the same as many-to-one c omputads . Preprint 2002. [HMP] C. Hermida, M. Makk ai, J. Po wer, On we ak higher dimensional c ate gories, I P arts 1,2,3, J. 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