On the functoriality of cohomology of categories

a Xi :ma h/0411478 1 [ma h.CT] 22 No 2004 ON THE FUNCTORIALITY OF COHOMOLOGY OF CATEGORIES FERNANDO MURO Abs ac . In his pape we show ha he Baues-Wi sching complex used o de ine cohomology o ca ego ies is a 2- unc o om a ce ain 2-ca ego y o na u al sys ems o abelian g oups o he 2-ca ego y o chain complexes, chain homomo phism and ela i e homo opy classes o chain homo opies. As a consequence we de i e (co)localiza ion heo ems o his cohomology. 1. In oduc ion Baues-Wi sching cohomology o a small ca ego y Cwi h coe icien s in a na u al sys em Don Cwas de ined in [2] as he cohomology o a ce ain cochain complex F∗(C, D). This cohomology gene alizes some o he cohomologies p e iously known, as o example • he Hochschild-Mi chell cohomology o Cwi h coe icien s in a unc o D:Cop ×C→Ab ([10]), • he cohomology o he classi ying space BCwi h local coe icien s D, • he Mac Lane cohomology o a ing ([8], [7]). The Baues-Wi sching complex F∗(C, D) as well as i s cohomology H∗(C, D) a e known o be unc o s on a ce ain ca ego y Na o pai s (C, D). I is also known ha equi alences o ca ego ies induce homo opy equi alences in Baues-Wi sching complexes and isomo phisms in cohomology g oups, howe e his does no ollow immedia ely om he ac ha F∗and H∗a e unc o s in Na . Mo e p ecisely, i is no known he beha iou o F∗(C, D) and H∗(C, D) wi h espec o na u al ans o ma ions be ween unc o s in he i s a iable. The goal o his pape is o shed some ligh on ha issue. We de ine a new 2-ca ego y Na Fo pai s (C, D) con aining Na and p o e ha F∗is in ac a 2- unc o in Na F(Theo em 5.1). The 2-mo phisms in he ca ego y o cochain complexes will be homo opy classes o homo opies ela i e o he bounda y o he cylinde . The ca ego y ob ained om Na Fby aking se s o connec ed componen s on mo phism ca ego ies u ns ou o be a quo ien o Na (P oposi ion 5.2) and he cohomology unc o H∗ ac o s h ough his quo ien ca ego y (Co olla y 5.3). As an applica ion o hese esul s we ob ain localiza ion and colocaliza ion he- o ems o Baues-Wi sching cohomology (Theo ems 6.5 and 6.12). We also gi e some examples o how hese (co)localiza ion heo ems can be used o ca y ou compu a ions in cohomology o ca ego ies. The au ho was pa ially suppo ed by he MCyT g an BFM2001-3195-C02-01 and he MECD FPU ellowship AP2000-3330. 1 2 FERNANDO MURO Pi ash ili and Waldhausen de ined in [12] he homology o a small ca ego y C wi h coe icien s in a unc o D:Cop ×C→Ab by using a complex F∗(C, D) which is simila , and in some sense dual, o he Baues-Wi sching complex. They also p o ed ha his homology ex ends Mac Lane homology o ings ([8]), which is isomo phic o opological Hochschild homology in he sense o [3]. We claim ha he de ini ion o he Pi ash ili-Waldhausen homology H∗(C, D) can be ex ended o na u al sys ems Das coe icien objec s and he unc o ial p ope ies o F∗and H∗a e simila o hose desc ibed he e o F∗and H∗, in pa icula (co)localiza ion heo ems should hold. 2. No a ion and con en ions Compelled by he necessa ily in ica e no a ion o his pape , we ha e decided o include a he beginning a sec ion o ix he meaning o some symbols. Addi ional no a ion will also appea in he de elopmen o he pape , bu i will no con adic in any case ha in oduced he e. symbols meaning C,D,Esmall ca ego ies. X, Y objec s in hose ca ego ies. , g, h, k, σ mo phisms in hose ca ego ies. ϕ, ψ, ξ, ζ unc o s be ween hose ca ego ies. α, β, γ, ε, , s na u al ans o ma ions. →a ow o mo phisms in o dina y ca ego ies, 1-mo phisms in 2-ca ego ies, unc o s, and 2- unc o s. ⇒a ow o 2-mo phisms and na u al ans o ma ions. 1 iden i y mo phism o iden i y 2-mo phism, a subsc ip will cla i y he meaning in ambiguous cases. Ab he ca ego y o objec s: abelian g oups, mo phisms: homomo phisms. Ca he ca ego y o objec s: small ca ego ies, mo phisms: unc o s. Ca 2 he s anda d 2-ca ego y o objec s: small ca ego ies, 1-mo phisms: unc o s, 2-mo phisms: na u al ans o ma ions. D, E, G na u al sys ems, see De ini ion 4.1. ccochain in cohomology o ca ego ies. A∗, B∗, C∗cochain complexes o abelian g oups. d he di e en ial in all cochain complexes. p, q, h, g aded mo phisms be ween g aded abelian g oups. These symbols can be al e ed by adding supe sc ip s o subsc ip s. In 2-ca ego ies he wo d “mo phism” will be synonym o “1-mo phism”. All ca ego ies can be ega ded as 2-ca ego ies wi h only he i ial 2-mo phisms. ON THE FUNCTORIALITY OF COHOMOLOGY OF CATEGORIES 3 The symbol •s ands o an unspeci ied objec in an a bi a y ca ego y. I can appea se e al imes in a diag am, howe e in gene al i will s and o a di e en objec each ime. The composi e o mo phisms, say • → • g → •, o unc o s will be indica ed by jux aposi ion g , as well as he e ical composi ion o 2-mo phisms o na u al ans o ma ions βα:ϕ⇒ξas in he diag am ϕ α  • ϕ  ψ// ξ ??•, α  β  o equi alen ly ψ. β  ξ We will use he symbol ∗ o he ho izon al composi ion o 2-mo phisms o na - u al ans o ma ions β∗α:ξϕ ⇒ζψ as in he ollowing diag am • ϕ %% ψ 99• α  ξ %% ζ 99• β  3. Fac o iza ion ca ego ies Fac o iza ion ca ego ies a e he sou ce o he coe icien objec s o he Baues- Wi sching cohomology o small ca ego ies, see De ini ion 4.1. A ho ough s udy o hei p ope ies is essen ial o s udy in dep h he unc o iali y o he Baues- Wi sching complex in Sec ion 5. De ini ion 3.1. The ac o iza ion ca ego y FCo a small ca ego y Chas objec s: mo phisms in C, mo phisms: (h, k): →ga e pai s o mo phisms in Csuch ha k h =g, ha is commu a i e diag ams in C •k//• • OO • g OO h oo and composi ion is de ined by (h′, k′)(h, k) = (hh′, k′k). One can easily check ha ac o iza ion ca ego ies de ine a unc o F:Ca −→ Ca . This unc o is de ined on mo phisms as ollows: a unc o ϕ:C→Dis sen o ano he one F(ϕ): FC−→ FD, which is gi en on objec s: by F(ϕ)( ) = ϕ( ), on mo phisms: by F(ϕ)(h, k) = (ϕ(h), ϕ(k)). No ice ha he unc o Fp ese es p oduc s, he e o e we can conside he 2-ca ego y Ca Fob ained om Ca 2by applying he unc o F o mo phism ca ego ies. Le us make explici he s uc u e o Ca F: 4 FERNANDO MURO (3.A) objec s: a e small ca ego ies; 1-mo phisms: α:C→Da e ac ually na u al ans o ma ions α:ϕ⇒ψ be ween unc o s ϕ, ψ :C→D, and composi ion βα in Ca Fis ho izon al composi ion β∗αo na u al ans o ma ions; 2-mo phisms: (ε, γ): α⇒βa e na u al ans o ma ions such ha γαε =β, ha is commu a i e diag ams o na u al ans o ma ions ψγ+3ζ ϕ α KS ξ β KS ε ks e ical composi ion o 2-mo phisms is gi en by (ε′, γ′)(ε, γ) = (εε′, γ′γ), and he ho izon al composi ion o (ε, γ) and (ε′, γ′) as in he ollowing diag am in Ca F C α && β 88D (ε,γ)  α′ %% β′ 99E (ε′,γ′)  is (ε′, γ′)∗(ε, γ) = (ε′∗ε, γ′∗γ). The e is a unique 2- unc o Ca −→ Ca 2 which is he iden i y on objec s and 1-mo phisms. Mo eo e , he e is also a unique 2- unc o Ca −→ Ca F which is he iden i y on objec s and sends a unc o ϕ:C→D o he iden i y na u al ans o ma ion 1ϕ:ϕ⇒ϕ ega ded as a mo phism 1ϕ:C→Din Ca F. P oposi ion 3.2. The e is de ined a 2- unc o F:Ca F→Ca 2 i ing in o a commu a i e diag am Ca F//  Ca  Ca FF//Ca 2 whe e he e ical a ows a e he 2- unc o s p e iously de ined. P oo . The new 2- unc o Fis de ined in he ollowing way, we use he no a ion in De ini ion 3.1 and (3.A): on objec s: FCis he ac o iza ion ca ego y, on 1-mo phisms: he unc o F(α): FC→ FDis de ined on objec s: gi en an objec in FC, which is a mo phism :X→Y in C,F(α)( ) = αYϕ( ) = ψ( )αX; on mo phisms: F(α)(h, k) = (ϕ(h), ψ(k)). ON THE FUNCTORIALITY OF COHOMOLOGY OF CATEGORIES 5 on 2-mo phisms: F(ε, γ): F(α)⇒ F(β) is he na u al ans o ma ion which e alua ed on as abo e is he mo phism F(ε, γ) = (εX, γY) in FD. I is a s aigh o wa d exe cise o check ha his de ini ion is consis en and Fis indeed a 2- unc o . Mo eo e , he diag am in he s a emen commu es because all 2- unc o s a e he iden i y on objec s, F(1ϕ)( ) = ϕ( ) = F(ϕ)( ) and F(1ϕ)(h, k) = (ϕ(h), ϕ(k)) = F(ϕ)(h, k).  4. Baues-Wi sching cohomology o ca ego ies De ini ion 4.1. Recall om [2] ha a na u al sys em on Cis a unc o D:FC→ Ab. The Baues-Wi sching complex F∗(C, D) o a small ca ego y Cwi h coe i- cien s in a na u al sys em Don Cis a cochain complex o abelian g oups concen- a ed in non-nega i e dimensions. In dimension n his complex is gi en by he ollowing p oduc indexed by all sequences o mo phisms o leng h n−1 in C Fn(C, D) = Y •σ1 ←···σn ←• D(σ1···σn). In his o mula we assume ha a sequence o leng h 0 is an objec Xin Cwhich we also iden i y wi h he iden i y mo phism 1X. The coo dina e o c∈Fn(C, D) in •σ1 ← · · · σn ← • will be deno ed by c(σ1,...,σn). The di e en ial dis de ined as d(c)(σ1,...,σn+1) = D(1, σ1)c(σ2,...,σn+1) + n X i=1 (−1)ic(σ1,...,σiσi+1, . . . , σn+1) +(−1)n+1D(σn+1,1)c(σ1,...,σn). o e an n-cochain c o n≥1, and d(c)(σ) = D(1, σ)c(X)−D(σ, 1)c(Y) o n= 0 and σ:X→Y. The cohomology o Cwi h coe icien s in Dis he cohomology o he complex F∗(C, D), i is deno ed by H∗(C, D). Baues and Wi sching no iced ha H∗and F∗a e unc o s in he ca ego y Na de ined as ollows: objec s: a e pai s (C, D) whe e Dis a na u al sys em on C, mo phisms: (ϕ, ): (C, D)→(D, E) a e pai s gi en by a unc o ϕ:D→C and a na u al ans o ma ion :DF(ϕ)⇒E, and composi ion is gi en by he o mula (ψ, s)(ϕ, ) = (ϕψ, s( ∗1F(ψ))). Le Cochain be he ca ego y o cochain complexes o abelian g oups and cochain homomo phisms. As a unc o F∗:Na −→ Cochain is de ined as ollows on objec s: F∗(C, D) is he Baues-Wi sching complex; on mo phisms: Fn(ϕ, )(c)(σ1,...,σn) = σ(c(ϕ(σ1), . . . , ϕ(σn))), whe e σ=σ1···σn; and Hn=HnF∗:Na −→ Ab, n ∈Z. 6 FERNANDO MURO 5. The Baues-Wi sching complex as a 2- unc o This sec ion is he co e o he pape . I s main goal is o ex end F∗ o a 2- unc o om an adequa e 2-ca ego y Na Fwi h he same objec s as Na o he ollowing 2-ca ego y Cochain2: objec s: cochain complexes o abelian g oups; 1-mo phisms: cochain homomo phisms, ha is g aded homomo phisms p:A∗→B∗o deg ee 0 such ha dp =pd; 2-mo phisms: [h]: p⇒qa e ela i e homo opy classes o homo opies be- ween pand q, ha is [h] is ep esen ed by a deg ee −1 homomo phism h:A∗→B∗such ha dh +hd =−p+qand [h] = [h′] i he e exis s :A∗→B∗o deg ee −2 such ha d − d =−h+h′; e ical composi ion o 2-mo phisms is gi en by [h′][h] = [h′+h], and he ho izon al composi ion o [h] and [h′] in he ollowing diag am A∗ p '' q 77B∗ h  p′ '' q′ 77C∗ h′  is [h′]∗[h] = [h′p+q′h] = [p′h+h′q]; one can use he deg ee −2 homomo - phism h′h:A∗→C∗ o check he las equali y. No ice ha mo phism ca ego ies in Cochain2a e in ac g oupoids. Mo eo e , he e is a unique 2- unc o (5.A) ı:Cochain −→ Cochain2 which is he iden i y on objec s and 1-mo phisms. Le us de ine he 2-ca ego y Na F: (5.B) objec s: a e pai s (C, D) whe e Dis a na u al sys em on C; 1-mo phisms: (α, ): (C, D)→(D, E) a e pai s gi en by a na u al ans- o ma ion α:ϕ⇒ψbe ween unc o s ϕ, ψ:D→C, o equi alen ly a mo phism α:D→Cin Ca F, see (3.A), and a na u al ans o ma ion :DF(α)⇒E, whe e Fis he unc o de ined in P oposi ion 3.2, and composi ion is de ined as (β, s)(α, ) = (α∗β, s( ∗1F(β))); 2-mo phisms: (ε, γ): (α, )⇒(β, s) a e 2-mo phisms (ε, γ): α⇒βin Ca Fsuch ha =s(1D∗F(ε, γ)), ha is he ollowing diag am o na u al ans o ma ions commu es (5.C) DF(α) !) L L L L L L L L L L L L L L 1D∗F(ε,γ)  E DF(β) s 5= ON THE FUNCTORIALITY OF COHOMOLOGY OF CATEGORIES 7 e ical and ho izon al composi ions o 2-mo phisms in Na Fa e de ined as in Ca F, ha is (ε′, γ′)(ε, γ) = (εε′, γ′γ) and gi en a diag am in Na F (C, D) (α, ) (( (β,s) 66(D, E) (ε,γ)  (α′, ′) '' (β′,s′) 77(E, G) (ε′,γ′)  he ho izon al composi ion (ε′, γ′)∗(ε, γ) = (ε∗ε′, γ ∗γ′) in Na Fcoincides wi h he ho izon al composi ion o he ollowing diag am in Ca F Cyy α β D (ε,γ)  xx α′ β′ E (ε′,γ′)  I is edious bu s aigh o wa d o check ha Na Fis indeed a well-de ined 2-ca ego y. Mo eo e , he e is a unique 2- unc o :Na −→ Na F which is he iden i y on objec s and sends a mo phism (ϕ, ) o (1ϕ, ). This makes sense because o he commu a i i y o he diag am in P oposi ion 3.2. Theo em 5.1. The e is de ined a 2- unc o F∗:Na F→Cochain2 i ing in o a commu a i e diag am Na F∗ //   Cochain ı  Na FF∗//Cochain2 P oo . The new 2- unc o F∗is de ined as ollows, we use he no a ion in (3.B) and (5.B): on objec s: F∗(C, D) is he Baues-Wi sching complex; on 1-mo phisms: F∗(α, )(c)(σ1,...,σn) = σ(1D∗ F(1ϕ, α))σc(ϕ(σ1),...,ϕ(σn)), he e σ=σ1···σnand he o mula makes sense because c(ϕ(σ1),...,ϕ(σn)) ∈D(ϕ(σ)) = (DF(1ϕ))(σ); on 2-mo phisms: F∗(ε, γ) = [h(ε,γ)]: F∗(α, )→F∗(β, s) whe e o an (n+ 1)-dimensional cochain ci n > 0h(ε,γ)(c) is de ined as h(ε,γ)(c)(σ1,...,σn) = sσ(1D∗ F(1ξ, γα))σPn i=0(−1)ic(ϕ(σ1),...,ϕ(σi), εXi, ξ(σi+1),...,ξ(σn)), whe e Xiis he sou ce o σiand/o he a ge o σi+1, no ice ha c(ϕ(σ1),...,ϕ(σi), εXi, ξ(σi+1),...,ξ(σn)) ∈(DF(ε))(σ); and i n= 0 h(ε,γ)(c)(X) = s1X(1D∗ F(1ξ, γα))1Xc(εX). 8 FERNANDO MURO A edious bu s aigh o wa d compu a ion shows ha indeed dh(ε,γ)+h(ε,γ)d=−F∗(α, ) + F∗(β, s). Fo his, essen ially, one only needs o use he na u ali y p ope y o na u al ans- o ma ions and he commu a i i y o (5.C). I is easy o see ha F∗p ese es composi ion o 1-mo phisms. In o de o check ha F∗p ese es e ical composi ion o 2-mo phisms we conside a diag am in Na F (C, D) (α, ) !! (α′, ′)// (β,s) == (D, E) (ε,γ)  (ε′,γ′)  whe e ψγ+3ψ′γ′ +3ζ ϕ α KS ϕ′ α′ KS ε ksξ β KS ε′ ks is a commu a i e diag am o na u al ans o ma ions be ween unc o s ϕ, ϕ′, ψ, ψ′, ξ, ζ :D→C. We de ine a deg ee −2 homomo phism (ε′,γ′);(ε,γ):F∗(C, D)−→ F∗(D, E) in he ollowing way, i cis an (n+ 2)-cochain wi h n > 0 hen (ε′,γ′);(ε,γ)(c)(σ1,...,σn) = sσ(1D∗ F(1ξ, γ′γα))σPn i=0 Pn j=i(−1)i+jc(ϕ(σ1),..., ϕ(σi), εXi, ϕ′(σi+1),...,ϕ′(σj), ε′ Xj, ξ(σj+1),...,ξ(σn)) and o n= 0 (ε′,γ′);(ε,γ)(c)(X) = s1X(1D∗ F(1ξ, γ′γα))1Xc(εX, ε′ X). I is ha d bu s aigh o wa d o check ha d (ε′,γ′);(ε,γ)− (ε′,γ′);(ε,γ)d=−h(ε,γ)−h(ε′,γ′)+h(εε′,γ′γ), he e o e F∗(ε′, γ′)F∗(ε, γ) = F∗(εε′, γ′γ). Le us see ha F∗p ese es ho izon al composi ion o 2-mo phisms. Conside a diag am in Na F (C, D) (α, ) (( (β,s) 66(D, E) (ε,γ)  (α′, ′) '' (β′,s′) 77(E, G) (ε′,γ′)  ON THE FUNCTORIALITY OF COHOMOLOGY OF CATEGORIES 9 He e ψγ+3ζ ϕ α KS ξ β KS ε ks and ψ′γ′ +3ζ′ ϕ′ α′ KS ξ′ β′ KS ε′ ks a e commu a i e diag ams o na u al ans o ma ions be ween unc o s ϕ, ψ, ξ, ζ :D→Cand ϕ′, ψ′, ξ′, ζ′:E→D. We de ine a deg ee −2 homomo phism ′ (ε′,γ′);(ε,γ):F∗(C, D)−→ F∗(E, G) o e an (n+ 2)-cochain cwi h n > 0 as ′ (ε′,γ′);(ε,γ)(c)(σ1,...,σn) = s′ σ(s∗1F(β′))σ(1D∗ F(1ξξ′,(γ∗γ′)(α∗α′)))σPn i=0 Pn j=i(−1)i+jc(ϕϕ′(σ1),..., ϕϕ′(σi), ϕ(ε′ Xi), ϕξ′(σi+1),...,ϕξ′(σj), εξ′(Xj), ξξ′(σj+1),...,ξξ′(σn)) and o n= 0 ′ (ε′,γ′);(ε,γ)(c)(X) = s′ 1X(s∗1F(β′))1X(1D∗F(1ξξ′,(γ∗γ′)(α∗α′)))1Xc(ϕ(ε′ X), εξ′(X)). A e a labo ious compu a ion one can check ha d ′ (ε′,γ′);(ε,γ)− ′ (ε′,γ′);(ε,γ)d=−h(ε′,γ′)F∗(α, )−F∗(β′, s′)h(ε,γ)+h(ε∗ε′,γ∗γ′), hence F∗(ε′, γ′)∗F∗(ε, γ) = F∗(ε∗ε′, γ ∗γ′). The commu a i i y o he diag am in he s a emen ollows easily om he com- mu a i i y o he diag am in P oposi ion 3.2.  The se π0Co connec ed componen s o a small ca ego y Cis o med by equi - alence classes {X}o objec s in C. Two objec s X, Y a e equi alen {X}={Y}i he e exis s a sequence o (non-composable) mo phisms in Cconnec ing hem X→ • ← · · · → • ← Y. This de ines a p oduc -p ese ing unc o om small ca ego ies o se s π0:Ca −→ Se wi h π0(ϕ){X}={ϕ(X)}. Mo eo e , one can ob ain an o dina y ca ego y M0 om a 2-ca ego y Mby aking π0on mo phism ca ego ies and also an o dina y unc o ρ0:M0→N0 om a 2- unc o ρ:M→N. I Mis a ca ego y ega ded as a 2-ca ego y wi h only he i ial 2-mo phisms hen M0=M. The homo opy ca ego y o cochain complexes Cochain/≃coincides wi h Cochain0 2and he 2- unc o ı:Cochain →Cochain2in (5.A) induces he na u al p ojec ion ı0:Cochain →Cochain/≃on o he quo ien ca ego y. By Theo em 5.1 he e is a commu a i e diag am o unc o s (5.D) Na F∗ // 0  Cochain ı0  Na 0 F(F∗)0//Cochain/≃ P oposi ion 5.2. The unc o 0:Na →Na 0 Fis ull.