Comparison of the definitions of Abelian 2-categories

COMP ARISON OF THE DEFINITIONS OF ABELIAN 2-CA TEGORIES HIRO YUKI NAKAOKA Abstract. In the efforts to define a 2-categorical analog of an ab elian cat- egory , tw o (or three) n otions of “ab elian 2-catego r ies” are de fined in [4] and [2]. One is the r elatively exact 2-c ate gory defined i n [4], and the other(s) is the (2-)ab elian G p d -c ate gory defined by Dupont [2]. W e compare these no- tions, using the argumen ts in [4] and [2] . Si nce they pr oceed indep enden tly in their o wn wa y , in differen t settings and terminologies, it will be worth whil e to coll ect and unify them. In this paper, by comparing their definitions and argumen ts, we show the relationship among these cl ass es of 2-categories. 1. intr od uction Motiveted by [5], we defined a g eneral cla ss o f 2 -categories ‘ r elatively exact 2- c ate gories ’ in [4] (origina lly written a s our ma s ter’s thesis in 200 6 ), so as to make the 2-categor ical homolog ic al a lgebra work well in an abstract setting. A relatively ex a ct 2-ca tegory is a g eneralization of SCG (= the 2-categ ory of symmetric catego rical groups), and defined a s a 2-catego rical analo g of an ab elian category . category 2-catego ry general theory ab elian c a tegory relatively exa ct 2-categ o ry example Ab SCG On the other ha nd, with a similar motiv ation, Dupo n t defined t wo cla sses o f 2- categries ‘ 2-ab elia n Gp d -c ate gory ’ and ‘ ab elian Gp d -c ate gory ’ in [2 ]. Thus there ar e three differen t class es o f 2-c a tegories • (Relativ ely exact 2-categ ory) • (2-ab elian Gp d-category) • (abelia n Gp d-categor y) defined as 2-dimensional a nalogs of ab elian catego ries. So it w ill b e necessar y to make explicit the rela tions. W e co mpa re these notions, using the arg umen ts in [4] and [2]. Since they pro ceed independently in their own wa y , in different settings a nd terminologies , it will b e worth while to co llect and unify them. In this pap er, by co mparing their definitions and a rgumen ts, we s ho w the rela - tionship a mong three cla sses of 2-categ o ries mentioned ab ov e. In Theor em 5.3, we show there a re implications for these notions (2-Ab elian Gp d) ⇒ (Relatively ex a ct) ⇒ (Abe lian Gpd) , except fo r so me minor differences (see Theorem 5.3). The author wi shes to thank Dr. Mathieu Dupont, f or p oin ting out the author’s misunder- standing of Definition 165 in [ 2 ]. 1 2 HIR OYUKI NAKAOKA 2. Preliminaries Let S denote a 2-ca tegory (in the strict sense). W e use the following notation. S 0 , S 1 , S 2 : class o f 0- cells, 1-ce lls , and 2 -cells in S , resp ectiv ely . S 1 ( A, B ) : 1-cells from A to B , where A, B ∈ S 0 . S 2 ( f , g ) : 2-cells from f to g , where f , g ∈ S 1 ( A, B ) for certain A, B ∈ S 0 . S ( A, B ) : Hom-categor y b etw een A and B (i.e. Ob( S ( A, B )) = S 1 ( A, B ), S ( A, B )( f , g ) = S 2 ( f , g )). In diag rams, − → repres e n ts a 1-c ell, = ⇒ repr esen ts a 2-ce ll, ◦ repr e sen ts a hor i- zontal comp osition, a nd · repr esen ts a vertical comp osition. W e use capital letters A, B , . . . for 0-cells , small letter s f , g , . . . for 1-cells, a nd Greek symbols α, β , . . . for 2-cells. The co mposition of A f − → B and B g − → C is deno ted by g ◦ f , co n versely to [4]. Similarly for the co mposition of 2-cells. In the following arg umen ts, a n y 2- cell in a 2-catego ry is inv ertible. This helps us to av oid b eing fussy a bout the directions o f 2-cells, a nd we use the word ‘dual’ simply to r ev erse the directions of 1-cells. F or example, c okernel is the dua l notion of kernel , and pul lb ack is dual to pushout . As for the definitions of (co-)kernels, pullbacks, and pusho uts in a 2-categ ory , see [2] o r [4]. (The definitions in [4] and [2] agree.) 3. Rela tivel y exact 2-ca tegor y Let SCG denote the 2-catego ry of sma ll symmetric catego rical groups (= s ym- metric 2 - groups). This is denoted by 2-SGp in [2]. 0-c ells are sy mmetric categ orical groups, 1-cells ar e s ymmetric monoida l functors, and 2 -cells are monoida l tr ansfor- mations (cf. [5] or [4]). F or any symmetric monoida l functor f : A → B , let f I denote the unit isomor- phism f I : f (0 A ) ∼ = − → 0 B , where 0 A and 0 B are r espectively the unit o f A a nd B , with resp ect to ⊗ . Definition 3 . 1. (Definition 3.7 in [4]) A 2-ca tegory S is said to b e lo c al ly SCG if the following co nditions are satisfied: (LS1) F or every A, B ∈ S 0 , there is a given functor ⊗ A,B : S ( A, B ) × S ( A, B ) → S ( A, B ), a nd an ob ject 0 A,B ∈ Ob( S ( A, B )) such that ( S ( A, B ) , ⊗ A,B , 0 A,B ) b e- comes a symmetric c ategorical gro up, and the following natura lit y conditions are satisfied: 0 A,B ◦ 0 B ,C = 0 A,C ( ∀ A, B , C ∈ S 0 ) (LS2) Hom = S ( − , − ) : S op × S → SCG is a 2- functor (in the strict sence). Moreov er , for any A, B , C ∈ S 0 , ( − ◦ 0 A,B ) I = id 0 A,C ∈ S 2 (0 A,C , 0 A,C ) (1) (0 B ,C ◦ − ) I = id 0 A,C ∈ S 2 (0 A,C , 0 A,C ) . (2) are sa tisfied. A B C 0 A,B 3 3 g g g g g g g 0 B,C + + W W W W W W W 0 A,C 8 8 id   (Remark that ( − ◦ 0 A,B ) and (0 B ,C ◦ − ) are s ymmetric mo noidal functors.) COMP ARISON OF ABELIAN 2-CA TEGORIES 3 (LS3) There is a 0-c e ll 0 ∈ S 0 called zer o obje ct , which satisfies the following conditions: (ls3-1) F o r any f : 0 → A in S , there ex is ts a unique 2-cell θ f ∈ S 2 ( f , 0 0 ,A ). (ls3-2) F o r any f : A → 0 in S , there ex is ts a unique 2-cell τ f ∈ S 2 ( f , 0 A, 0 ). (LS3+) S (0 , 0) is the zero categ orical group. (LS4) F o r any A, B ∈ S 0 , their pro duct and copro duct exist. Caution 3.2. In [4], zer o obje ct w a s also a ssumed to s atisfy (LS3+). On the other hand, the definition of zer o o b ject in [2] only r equires (ls3-1) and(ls3-2 ). In fact, condition (LS3+) is no t used e s sen tially in [4]. So in the following, w e mainly consider lic ally SCG 2-ca teg ories without condition (LS3 +). Definition 3 .3. (Definition 3.7 in [4]) Let S b e a lo cally SCG 2-categ ory . S is said to b e r elatively exact if the following conditio ns are satisfied: (RE1) F or an y 1-cell f , its k er nel and co k ernel exist. (RE2) An y 1-ce ll f is faithful if and only if f = ker(cok( f )). (RE3) An y 1-ce ll f is cofaithful if and only if f = cok(ker( f )). (F or the definitions of (fully) (co -) faithfulness, see [4] or [3].) Remark 3.4 . F or a n y 1-ce ll f : A → B , its kernel is defined as the triplet (Ker( f ) , ker ( f ) , ε f ) Ker( f ) A B ker ( f ) / / f / / 0 & & ε f K S , universal among tho se ( K , k , ε ) K A B k / / f / / 0 $ $ ε K S . F or the precis e definition, se e [4] or [2]. Dually , the cokernel of f is the universal triplet (Cok( f ) , cok( f ) , π f ) A B Cok( f ) f / / cok( f ) / / 0 ' ' π f K S . 4. (2-)Abelian Gp d -ca tegor y (2-)Abe lian Gp d-catego ries, defined in [2], a r e Gp d ∗ -categor ies s atisfying certain conditions. By de finitio n, a Gp d ∗ -categor y is a categ o ry C enriched by the category Gpd ∗ of small pointed group oids (Prop osition 70 in [2]). F o r any A, B ∈ Ob( C ), the distinguished p oint in C ( A, B ) is denoted b y 0 A,B or simply b y 0. In [2], it is remar k ed that any Gp d ∗ -categor y C is equiv alent to a strictly describ e d one, and thus C is ass umed to be s trictly describ ed, namely , it satisfies the fo llo wing: (SD1) F or any s equence of morphisms A f − → B g − → C h − → D in C , h ◦ ( g ◦ f ) = ( h ◦ g ) ◦ f 4 HIR OYUKI NAKAOKA is s atisfied. (SD2) F or any f : A → B in C , f ◦ id A = f id B ◦ f = f are sa tisfied. (SD3) F or any f : A → B a nd any ob jects A ′ , B ′ in C , f ◦ 0 A ′ ,A = 0 A ′ ,B 0 B ,B ′ ◦ f = 0 A,B ′ are sa tisfied. (SD4) F or any A B f # # g ; ; α   and a n y ob jects A ′ , B ′ in C , α ◦ 0 A ′ ,A = id 0 A ′ ,B 0 B ,B ′ ◦ α = id 0 A,B ′ are sa tisfied. Remark 4.1. A Gp d ∗ -categor y C is regar ded as a 2- category in the following, and we us e 2-categ orical terminologies, e.g. ‘0-ce ll’ for an ob ject, ‘1- cell’ for an arrow. Definition 4.2 . (Definition 1 65 in [2]) An ab elian Gp d -c ate gory is a Gpd ∗ -categor y C with zer o ob ject, finite (co -)products and (co-)kernels, satisfying the following conditions: (A G1) Every 0-mono morphic 1-cell f satisfies f = ker( co k( f )). (A G2) Every 0-epimor phic 1-cell f sa tisfies f = co k(k er( f )). (A G3) F ully 0-faithful 1-cells and 0- mo nomorphic 1-cells are sta ble under pushout. (A G4) F ully 0-cofaithful 1-ce lls and 0 -epimorphic 1-cells a re stable under pullback. Definition 4. 3. (Definition 179 a nd 18 3 in [2]) A 2-ab elian Gp d -c ate gory is a Gpd ∗ -categor y C with zero ob ject, finite (co -)products a nd (co -)k ernels , satisfying the following co nditions: (2AG1) If f is a 0 -faithful 1-c ell, then f = ker(cok( f )). (2AG2) If f is a 0 -cofaithful 1-cell, then f = cok (k er( f )). (2AG3) Any fully 0-faithful 1- c ell is canonically the ro ot of its copip. (2AG4) Any fully 0-cofaithful 1 -cell is canonically the coro ot of its pip. F or the definitions of (co -)roo ts and (co -)pips, see [2]. W e do not require them explicitly in the following arguments. W e introduce the res t of the notions app earing in the above definitions. The definition of 0-mo nomorphic 1-cells is the following. 0-epimorphicity is defined dua lly . Definition 4 .4. (Definition 118 in [2 ]) A 1- c e ll f : A → B is 0-monomorphic if, for any 1- cell a : X → A and any 2 -cell β : f ◦ a = ⇒ 0 compatible with π f (of Remar k 3.4), there ex is ts a unique α : a = ⇒ 0 such that f ◦ α = β . The definitions of (fully) 0-faithful 1-cells ar e the following. (F ully) 0-c o faithful 1-cells a re defined dually . COMP ARISON OF ABELIAN 2-CA TEGORIES 5 Definition 4.5 . (Definition 78, 8 0 in [2]) Le t C b e a Gp d ∗ -categor y , and f : A → B be a 1-c ell in C . (i) f is 0-faithful if for a n y X A 0 # # 0 ; ; α   in C , f ◦ α = id 0 ⇒ α = id 0 is s atisfied. (ii) f is ful ly 0-c ofaithful if for a n y 1 -cell a : X → A and a n y 2-cell X B f ◦ α # # 0 ; ; α   , there exists a unique 2-ce ll α : a = ⇒ 0 s uc h that β = f ◦ α . F act 4.6. In [2], it is shown tha t any 2-ab elian Gpd -category C admits a weak enrichmen t by SCG, i.e., C is pr e additive , in the terminology of [2]. F or the general definition o f a pr eadditiv e Gp d-catego ry , see [2]. W e o nly consider the case where C is strictly describ ed. (In this ca se, the na tur al transfor mations app earing in Definition 218 in [2] ar e identities) Definition 4.7 . A strictly describ ed Gp d-categor y C is pr e addi t ive if it satisfies the following: (o) F or any 0 -cells A, B in C , Hom-ca tegory C ( A, B ) is equipp ed with a s tr ucture of a symmetric ca tegorical group ( C ( A, B ) , ⊗ , 0). (a1) F or an y 1-cell A f − → B a nd any 0- cell C in C , the comp osition b y f − ◦ f : C ( B , C ) → C ( A, C ) is s ymmetric monoidal. (a2) The dual of (a1). (b1) F o r any 1-c e lls A f − → B g − → C a nd any 0-cell D in C , we hav e C ( C , D ) C ( B , D ) C ( A, D ) −◦ g / / −◦ ( g ◦ f )   9 9 9 9 9 9 9 −◦ f           as monoidal functors . (b2) The dual o f (b1 ). (c) F or any 1 -cells A f − → B and C g − → D , we hav e C ( B , C ) C ( A, C ) C ( B , D ) C ( A, D ) −◦ f / / g ◦−   g ◦−   −◦ f / /  as monoidal functors . 6 HIR OYUKI NAKAOKA (d) F or an y 0-cells A and B in C , we have C ( A, B ) C ( A, B )   id B ◦− & & −◦ id A 8 8 id / / (e1) F o r any 0- cell X a nd any ℓ, k : X → A , ( − ◦ f ) ℓ,k : ( ℓ ⊗ k ) ◦ f = ⇒ ( ℓ ◦ f ) ⊗ ( k ◦ f ) ( ∀ f : A → B ) is na tural in f . X A B ℓ / / k / / f / / (e2) The dual of (e1). (f1) F or any X , ( − ◦ f ) I : 0 X,A ◦ f = ⇒ 0 X,B ( ∀ f : A → B ) is na tural in f . (f2) The dual of (f1). Here, since ( − ◦ f ) is monoida l, ( − ◦ f ) ℓ,k denotes the s tructure is omorphism ( − ◦ f ) ℓ,k : ( ℓ ⊗ k ) ◦ f = ⇒ ( ℓ ◦ f ) ⊗ ( k ◦ f ) natural in ℓ, k : X → A . Similarly , ( − ◦ f ) I denotes the unit isomorphism. 5. Comp arison Lemma 5.1. If C is a s trictly describ ed preadditive Gp d-category , then Hom = C ( − , − ) : C op × C → SCG is a 2-functor . Pr o of. F or the definitio n o f a 2- functor, see Definition 7.2.1 in [1]. It can be eas ily shown that, to show the lemma, it suffices to show the following co nditions: (i) F or any 1 -cells A ′ f − → A and B g − → B ′ , g ◦ − ◦ f : C ( A, B ) → C ( A ′ , B ′ ) is a symmetric mo noidal functor. (ii) F or any 2 -cells A ′ A f # # f ′ ; ; α   and B ′ B g # # g ′ ; ; β   , β ◦ − ◦ α : g ◦ − ◦ f = ⇒ g ′ ◦ − ◦ f ′ is a monoida l transfor ma tion. (iii) F or any 1 -cells A ′′ f ′ − → A ′ f − → A and B g − → B ′ g ′ − → B ′′ , we hav e C ( A, B ) C ( A ′ , B ′ ) C ( A ′′ , B ′′ ) g ◦−◦ f / / ( g ′ ◦ g ) ◦−◦ ( f ′ ◦ f )   9 9 9 9 9 9 9 g ′ ◦−◦ f ′           COMP ARISON OF ABELIAN 2-CA TEGORIES 7 as monoidal functors . (iv) F or any 0-cells A a nd B in C , we hav e C ( A, B ) C ( A, B )  id ◦−◦ id ( ( id 6 6 as monoidal functors . (iv) follows from (d). (i) follows from (a1 ), (a2 ) (and (c)). (iii) follows from (b1), (b2) (and (c)). (ii) follows from (e1), (e2), (f1), (f2).  Lemma 5.2 . Let f : A → B be a ny 1 -cell in a relatively exact 2-categ ory . Then the following ar e satisfied. (i) f is faithful if a nd only if it is 0-faithful, if and o nly if it is 0-monomor phic. (ii) f is fully faithful if and o nly if it is fully 0-faithful. Pr o of. (i) By Co rollary 3 .24 in [4 ], f is faithful if and o nly if it is 0-faithful. As r e- marked after Definition 118 in [2], any 0 -monomorphic 1-cell is faithful. Conv er sely , if f is faithful, then f satisfies f = ker(cok( f )), and beco mes 0-monomo rphic b y Lemma 3 .19 in [4]. (ii) This is nothing o ther than Lemma 3.22 (2 ) in [4].  Theorem 5.3 . There are the implications among the conditions on 2- categories (2-Ab elian Gp d) ⇒ (Relatively exa ct ) ⇒ (Abe lia n Gp d) . More pr ecisely , we have: (i) Any strictly describ ed 2-ab elian Gp d-categ ory is a relatively exact 2-ca teg ory without condition (LS3 +). (ii) Any r elativ ely exact 2-ca tegory without co ndition (LS3+) is an ab elian Gp d- category no t necessa rily strictly describ ed. Pr o of. First remark that each of these 2-ca tegories is a Gp d ∗ -categor y with zero ob ject, finite (co-)pro ducts a nd (co - )k ernels. (i) Let C be a 2 - abelian Gp d-categ ory . C satisfies (LS1 ), as a particula r ca se of (SD3). By Lemma 5.1, Hom = C ( − , − ) : C op × C → SCG is a 2-functor. Moreov er , (1) and (2) in (LS2) follows from (SD4). T hus C satisfies (LS2). By Pr o position 180 in [2], any 1-cell in C is 0-faithful if and only if it is faithful. Thus (RE 2) follows from (2A G1). Dually , (RE3) follows fr o m (2AG2). (ii) Let S b e a relatively e x act 2-catego ry . By the duality , it suffices to show (A G1) and (AG3). By Le mma 5 .2, we hav e equiv alences of the notions faithful = 0-faithful = 0 -monomorphic fully faithful = fully 0 -faithful for 1- cells in S . Thus (AG1) follows from (RE2), a nd (AG3) follows from the duals of P ropo sition 3 .32 and Pro p osition 5 .12 in [4]. Remark also that (SD1) a nd (SD2) are satisfied, but (SD3) and (SD4) ar e not satisfied in genera l. So S is not ne c essarily stric tly describ ed as a Gp d ∗ -categor y .  8 HIR OYUKI NAKAOKA References [1] F. Borceux, Handb o ok of c ate goric al algabr a I , Basic category theory . Encyclopedia of Math- ematics and i ts Appli cations, 50 . Cam bri dge Universit y Pr ess, Cambridge, 1994. xvi+345 pp. [2] M. Dupont, Ab elian c atego ries in dimension 2 , ar Xiv:0809.1760 . [3] M. Dup on t, E . M. Vi tale, Pr op er factorization systems in 2-c atego rie s , Journal of Pure and Applied Algebra 179 (2003) 65–86. [4] H. Nak aok a, Cohomolo g y the ory in 2-c ate gories , Theory and Applications of Categories 20 (2008) 542–604. [5] A. del R ´ ıo, J. Mart ´ ınez-Moreno, E. M . V itale, Chain c omplexes of symmetric c ate goric al gr oups , Journal of Pure and Appli ed Algebra 196 (2004), 279-312. Gradua te School of Ma them a tical Sciences, The University of Tokyo 3-8-1 Komaba, Meguro, Tokyo, 153-89 14 Ja p a n E-mail addr ess : deutsche@ms .u-tokyo.ac.jp