Directional derivatives and higher order chain rules for abelian functor calculus

47 Pages·2016·0.29 MB·English

DIRECTIONAL DERIVATIVES AND HIGHER ORDER CHAIN RULES FOR
ABELIAN FUNCTOR CALCULUS
KRISTINEBAUER,BRENDAJOHNSON,CHRISTINAOSBORNE,EMILYRIEHL,ANDAMELIATEBBE
Abstract. Inthispaper,weconsiderabelianfunctorcalculus,thecalculusoffunctorsof
abeliancategoriesestablishedbythesecondauthorandMcCarthy.Wecarefullyconstruct
acategoryofabeliancategoriesandsuitablyhomotopicallydefinedfunctors,andshowthat
thiscategory,equippedwiththedirectionalderivative,isacartesiandifferentialcategory
inthesenseofBlute,Cockett,andSeely.Thisprovidesanabstractframeworkthatmakes
certainanalogiesbetweenclassicalandfunctorcalculusexplicit.InspiredbyHuang,Mar-
cantognini,andYoung’schainruleforhigherorderdirectionalderivativesoffunctions,we
defineahigherorderdirectionalderivativeforfunctorsofabeliancategories.Weshowthat
ourhigherorderdirectionalderivativeisrelatedtotheiteratedpartialdirectionalderiva-
tivesofthesecondauthorandMcCarthybyaFaa` diBrunostyleformula. Weobtaina
higherorderchainruleforourdirectionalderivativesusingafeatureofthecartesiandif-
ferentialcategorystructure,andwiththisprovideaformulationforthenthlayersofthe
TaylortowerofacompositionoffunctorsF◦Gintermsofthederivativesanddirectional
derivativesofFandG,reminiscentofsimilarformulationsforfunctorsofspacesorspec-
trabyAroneandChing. Throughout,weprovideexplicitchainhomotopyequivalences
thattightenpreviouslyestablishedquasi-isomorphismsforpropertiesofabelianfunctor
calculus.
Contents
1. Introduction 1
2. Crosseffectsforfunctors 7
3. Acategoricalcontextforabelianfunctorcalculus 10
4. TheTaylortowerinabelianfunctorcalculus 13
5. Linearapproximations 16
6. Thefirstdirectionalderivative 22
7. HigherorderdirectionalderivativesandtheFaa` diBrunoformula 28
8. Thehigherorderchainrule 33
9. Derivatives 37
AppendixA. Ageneralbicomplexretraction 39
AppendixB. ProofofProposition5.7 43
References 46
1. Introduction
1.1. Chain Rules. When studying the calculus of functions, the chain rule for the first
derivativeofacompositionoffunctionsisgivenbythefamiliarformula
(f ◦g)(cid:48) =(f(cid:48)◦g)·g(cid:48).
Date:October4,2016.
1
2 BAUER,JOHNSON,OSBORNE,RIEHL,ANDTEBBE
Byrepeatedlyapplyingthisformula,onecanderivetheclassicalFaa`diBrunoformulafor
thenthderivativeofthecomposite f ◦g:
(cid:88) n! (cid:32)g(cid:48)(x)(cid:33)k1 (cid:32)g(n)(x)(cid:33)kn
(f ◦g)(n)(x)= f(k)(g(x)) ...
k !k !...k ! 1! n!
1 2 n
wherethesumistakenoverallnon-negativeintegersolutionsto
k +2k +...+nk =n
1 2 n
andk = k +...+k . Analternativehigherorderchainrulecanbeobtainedbyusingthe
1 n
directionalderivativeof f atxalongv:
∇f(v;x)=lim1(cid:2)f(x+tv)− f(x)(cid:3).
t→0 t
Huang,Marcantognini,andYoung[HMY]definehigherorderdirectionalderivatives∆ f,
n
functionsof(n+1)-variableswith∆ f =∇f,withwhichtheyobtainaconciseformulation
1
forthenthderivativeofacomposition
(1.1) (f ◦g)(n)(x)=∆ f(g(n)(x),...,g(cid:48)(x);g(x)),
n
whengisafunctionofasinglevariable,or
(1.2) ∆ (f ◦g)=∆ f(∆ g,...,∆ g;g),
n n n 1
inthegeneralcase.
Forhomotopyfunctors(suchasfunctorsfromspectratospectraorspacestospacesthat
preserve weak equivalences), Goodwillie’s calculus of functors builds a Taylor tower of
“polynomial”functorsandnaturaltransformations:
F →···→ P F → P F →···→ P F
n n−1 0
where each P F is an n-excisive functor that approximates F in a range of homotopy
n
groupsdependingonthefunctor F andtheconnectivityoftheinput[G]. Thehomotopy
fiberofthemapP F → P F isthenthlayerofthetower,D F. Forhomotopyfunctors
n n−1 n
from spectra to spectra, the layers are of the form DnF (cid:39) ∂nF ∧hΣn X∧n, where ∂nF is a
spectrum with an action of the the nth symmetric group, Σ . This spectrum is called the
n
nthderivativeofF.
Severalversionsofchainrulesforthederivativesoffunctorcalculushavebeendevel-
oped.Notably,AroneandChing[AC]derivedachainruleforthederivatives∂ Fusingthe
n
factthatforfunctorsofspacesorspectra,thesymmetricsequence{∂ F}isamoduleover
n
the operad formed by the derivatives of the identity functor of spaces. This generalized
earlier work of Klein and Rognes [KR] that established a chain rule for first derivatives.
Forfunctorsofspectra,achainruleforthederivativesisgivenbyChing[C]. Forfunctors
ofspaces, Yeakelhasdevelopedanalternativemethodforderivingthesechainrulesthat
doesnotrequirepassagetospectra[Y].
1.2. Chain Rules in Abelian Functor Calculus. The chain rule results of the preced-
ing paragraph use Goodwillie’s original formulation of the Taylor tower. The second
author and McCarthy [JM2] defined Taylor towers for not necessarily additive functors
betweenabeliancategoriesusingtheclassicalcrosseffectfunctorsofEilenbergandMac
Lane[EM]. Thisapproachhasbeengeneralizedandappliestoawidevarietyofcontexts
[BJM],[B&c]. In[JM2],JohnsonandMcCarthydefinedanotionofdirectionalderivative
DIRECTIONALDERIVATIVESANDHIGHERORDERCHAINRULESFORABELIANFUNCTORCALCULUS 3
∇F(V;X)forafunctor F valuedinanabeliancategoryandprovedtheanalogofHuang,
Marcantognini,andYoung’schainruleindegreeone[JM2,Proposition5.6]:
∇(F◦G)(cid:39)∇F(∇G;G),
underthehypothesisthatthefunctorGpreservesthezeroobject.
The present work was motivated by the goal of proving an analog of the higher or-
derdirectionalderivativechainruleofHuang, Marcantognini, andYoungfortheabelian
functorcalculusofJohnsonandMcCarthy. Achievingthisgoalrequiredfirstdealingwith
the questionof howto define ahigher orderdirectional derivative inthis context. Toget
a sense of the choices involved, consider the functor ∇F(V;X). This is a functor of two
variables,sothefirstchoicetobemadeindefiningasecondorderdirectionalderivativeis
whetheroneshoulddifferentiatewithrespecttooneofthevariables(apartialderivative)
orbothvariablessimultaneously(atotalderivative).
Forthefirstpossibility,onenotesthat∇F(V;X)isalreadyalinearfunctorwithrespect
toitsfirstvariable,sothatdifferentiatingwithrespecttothatvariableyieldsnothingnew.
So, for the partial derivative approach, we can restrict our attention to taking the direc-
tionalderivativeof∇F(V;X)withrespecttoX toobtainasecondorderpartialdirectional
derivative ∇2F that is a functor of 3 variables. Iterating this process yields functors ∇nF
of (n + 1)-variables defined in [JM2], which we refer to as iterated partial directional
derivatives.
Ifwefollowthetotalderivativeapproachbytreating∇F asafunctorwhosedomainis
a product category B×B, then we obtain as a second order total directional derivative a
functor∇×2F = ∇(∇F)whosesourcecategoryisB4. But,thisleadstosomeredundancy,
as we end up differentiating in some directions repeatedly. The “correct” approach, at
least if one wants to define the analog of the Huang-Marcantognini-Young higher order
directionalderivatives,liessomewhereinbetween.
To define our second order directional derivative ∆ F of F, we restrict ∇×2F along a
2
diagonal functor L : B3 → B4 that takes the triple of objects (V ,V ,X) to the ordered
2 2 1
pairoforderedpairs((V ,V ),(V ,X)). Withthisdefinitionweareabletoprovethatfora
2 1 1
pairofcomposablefunctorsF andG
∆ (F◦G)(V ,V ;X)(cid:39)∆ F(∆ G(V ,V ;X),∆ G(V ;X);G(X)).
2 2 1 2 2 2 1 1 1
Inductively,wedefinethenthorderdirectionalderivative∆ F andprovethedesiredana-
n
logoftheHuang-Marcantognini-Youngchainruleequation(1.2):
Theorem8.1. Foracomposablepair F: B (cid:32) AandG: C (cid:32) Boffunctorsofabelian
categories,thereisachainhomotopyequivalencebetweenthenthdirectionalderivatives
∆ (F◦G)(V ,...,V ;X)(cid:39)∆ F(∆ G(V ,...,V ;X),...,∆ G(V ;X);G(X)).
n n 1 n n n 1 1 1
Ofcourse,this“in-between”definitionofthehigherorderdirectionalderivativesraises
anaturalquestion–howisitrelatedtothemoreobviouschoicesfordefininghigherorder
directional derivatives? In the case of ∇×nF, the functor ∆ F is obtained by restricting
n
alongadiagonalfunctorL asinthedegree2case. Thecaseof∇nF provesmoreinterest-
n
ing. Weshowthat∆ F and∇nF satisfyaFaa` diBruno-typerelationship:
n
Theorem 7.7. For a functor F between abelian categories, there is a chain homotopy
equivalence
(cid:77) n! (cid:32)1(cid:33)k1 (cid:32)1(cid:33)kn
∆nF(Vn,...,V1;X)(cid:39) k !k !···k ! 1! ··· n! ∇k1+···+knF(Vnkn,...,V1k1;X)
1 2 n
wherethesumisovernon-negativeintegersolutionstotheequationk +2k +···+nk =n.
1 2 n
4 BAUER,JOHNSON,OSBORNE,RIEHL,ANDTEBBE
A more concise formulation of this result, given in §7, makes use of the fact that the
Faa` di Bruno coefficients count the number of partitions of a set of n elements. Using
theseresults,wedefineordinary(non-directional)derivativesforfunctorsofmodulesover
acommutativeringRandproveachainruleforthesederivativeswhichisanalogoustothe
Huang-Marcantognini-Youngrulefromequation(1.1):
Corollary 9.4. For a composable pair of functors F: ChB → ChA and G: Mod →
R
ChBoffunctorsofabeliancategories,thenthderivative
dn
(F◦G)(X)
dRn
ischainhomotopyequivalentto
(cid:77) n! (cid:32)1(cid:33)k1···(cid:32)1(cid:33)kn∇k1+···+knF(cid:32) dkn G(X),..., dk1 F(X);G(X)(cid:33)
k1!k2!···kn! 1! n! dRkn dRk1
wherethesumisovernon-negativeintegersolutionstotheequationk +2k +···+nk =n.
1 2 n
1.3. Acategorialcontextforabelianfunctorcalculus. Althoughouroriginalmotivation
forthisprojectwastoprovehigherorderchainrules,wefoundthatasignificantpartofthe
challenge in doing so involved placing these chain rules and the abelian functor calculus
itself in an appropriate categorical context. This was developed in two stages, using the
conceptsofKleislicategoriesandcartesiandifferentialcategories.
ForafunctorF : B → Abetweentwoabeliancategories,theabelianfunctorcalculus
constructsadegreenapproximationP F :B→ChA,whereChAisthecategoryofchain
n
complexesinAconcentratedinnon-negativedegrees. Foracomposablepairoffunctors
F : B → A and G : C → B, this results in a pair of functors P F : B → ChA and
n
P G :C→ChBwhichnolongerappeartobecomposable. In[JM2],theauthorsformthe
n
compositionP F◦P Gby“prolonging”P F toafunctorChB → Ch(ChA),composing
n n n
theprolongationwith P G andtakingthetotalcomplexoftheresultingdoublecomplex.
n
Weobserveinsection3thatthisprocessispreciselythecompositionintheKleislicategory
associatedtoaparticularpseudomonadChactingonthe2-categoryofabeliancategories,
anduseAbCat todenotethatKleislicategory.
Ch
Infunctorcalculus,theuseoftermssuchas“calculus”and“derivatives”hasbeenjus-
tifiedbypointingoutstrongformalresemblancestotheclassicalconstructionsfromanal-
ysis and undergraduate calculus. Although these analogies are compelling, one wonders
ifthereisadeeperjustification. Inthecourseofinvestigatingthehigherorderchainrule
for directional derivatives, we discovered that the properties of the directional derivative
forabelianfunctorcalculuscorrespondexactlytotheaxiomsdefiningthecartesiandiffer-
entialcategoriesofBlute,Cockett,andSeely[BCS]. Thisconceptcapturesthenotionof
differentiationinawidevarietyofcontexts(thecategoryofrealvectorspacesandsmooth
maps equipped with the usual differential operator is a standard example of a cartesian
differentialcategory)andprovidesanexplanationforsomeoftheformalresemblances.
Weprove
Corollary6.6. ThehomotopycategoryHoAbCat isacartesiandifferentialcategory.
Ch
In this statement, HoAbCat is the homotopy category of AbCat obtained by in-
Ch Ch
verting pointwise chain homotopy equivalences. The directional derivative ∇ gives this
categoryitscartesiandifferentialcategorystructure. Fittingcalculusintothiscontextled
toadditionalinsightstemmingfromthefactthatacartesiandifferentialcategoryisalsoan
example of a tangent category, as defined by Cockett and Cruttwell [CC]. Tangent cate-
goriesarecharacterizedbytheexistenceofanendofunctorT satisfyingessentialproperties
DIRECTIONALDERIVATIVESANDHIGHERORDERCHAINRULESFORABELIANFUNCTORCALCULUS 5
oftangentbundlesformanifolds. OurfirstproofofthechainruleofTheorem8.1didnot
make use of the tangent category structure on HoAbCat , but the proof presented here,
Ch
whichdoesusetheendofunctorT,providesaconceptualsimplification. Asthefirst-order
chainrulecanbeseentoencodethefunctorialityofthederivative,thenthorderchainrule
can be extracted from the functoriality of the iterated endofunctor Tn, which is an easily
derivedconsequenceofourCorollary6.6.
Proving that HoAbCat is a cartesian differential category required reworking many
Ch
oftheconstructionsandresultsforabelianfunctorcalculusestablishedin[JM2]. Driving
thesechallengeswasthefactthatchainrulesintheabeliancontextdonotcomeintheform
ofisomorphisms. Forexample,applyingthedefinitionof∇(seeDefinition6.1or6.2)to
F◦Gproducesachaincomplexthatisnotisomorphicto∇F(∇G,G)ingeneral. Instead,
when G is reduced (G(0) (cid:27) 0), [JM2] prove that there is a quasi-isomorphism between
∇(F◦G)and∇F(∇G,G).
The axioms of a cartesian differential category require that ∇(F ◦G) and ∇F(∇G,G)
be isomorphic. This suggests that we attempt to realize AbCat as a cartesian differen-
Ch
tial category by inverting quasi-isomorphisms. But because the functors we consider be-
tweenabeliancategoriesarefarfrombeingexact,whenweinvertquasi-isomorphismsin
AbCat ,thecompositionofmorphismsisnolongerwell-defined.However,thesomewhat
Ch
delicately defined composition in our Kleisli category preserves chain homotopy equiva-
lences,sowedefineHoAbCat tobethecategorythatinvertstheseinstead. Becauseof
Ch
thisweneededtoupgrademanydefinitionsandresultsof[JM2]fromquasi-isomorphisms
tochainhomotopyequivalences.
Revisingresultsof[JM2]toworkuptochainhomotopyequivalenceratherthanquasi-
isomorphism was a relatively straightforward process, with the notable exception of the
chain rule itself. That required proving the following proposition, which refines Lemma
5.7of[JM2].
Proposition5.7. Foranycomposablepairoffunctors F: B → ChAandG: C → ChB
withGreduced,thereisachainhomotopyequivalence
D (F◦G)(cid:39) D F◦D G.
1 1 1
Toprovethis,weproveatechnicalgeneralresultthatmightbeofindependentinterest
(ormightalreadybeintheliterature):
TheoremA.1. LetC beafirst-quadrantbicomplexsothateveryrowexceptthezeroth
•,•
rowC iscontractible. ThenthenaturalinclusionC (cid:44)→ Tot(C) isachainhomotopy
0,• 0,• •
equivalence.
We include a proof of this lemma in Appendix A. The proof defines an explicit chain
homotopyonthetotalcomplexthatdoesnotcomefromachainhomotopyofbicomplexes,
asdefinedin[W,5.7.3]. TheproofofProposition5.7appearsisinAppendixB.
CombiningProposition5.7withsomeotherresults,weprove
Theorem6.5. ForanypairoffunctorsF: B→ChAandG: C→ChB,thereisachain
homotopyequivalence
∇(F◦G)(V;X)(cid:39)∇F(∇G(V;X);G(X)).
This improves on Proposition 5.6 of [JM2] in two ways – by replacing the quasi-
isomorphismwithachainhomotopyequivalence,andbyremovingtheconditionthatGbe
areducedfunctor.
6 BAUER,JOHNSON,OSBORNE,RIEHL,ANDTEBBE
1.4. Organizationofthepaper. Insection2,wedefineandreviewpropertiesofthecross
effectfunctors. ThesearethebuildingblocksfortheTaylortowersofabelianfunctorcal-
culus, includingthelinearizationanddirectionalderivativefunctors. Section3isusedto
definetheKleisliandhomotopycategoriesinwhichtheconstructionsandmainresultsof
thepapertakeplace.Insection4,weshowhowtheTaylortowersofabelianfunctorcalcu-
lusarebuiltandestablishtheirfundamentalproperties. Thistreatmentdiffersfromthatof
[JM2]intwoimportantways. Asdiscussedinsection1.3,weneedthefundamentalprop-
ertiesofthetowertoholduptochainhomotopyequivalenceinsteadofquasi-isomorphism.
Inaddition,wehavestreamlinedthemeansbywhichtermsintheabelianfunctorcalculus
toweraredefined. WedosobyidentifyingacomonadC onthecategoryofallfunctors
n
betweenafixedpairofabeliancategories,anddefining P F directlyasaresolutionof F
n
bythiscomonad. In[JM2],thisapproachwasonlyusedforthecategoryofreducedfunc-
tors. An extra step involving a mapping cone was used to extend this construction to all
functors.
Insection5,westartworkingoutthemainingredientsforourchainrules–wedefine
the linearization functor D and derive its essential properties. We use these to define
1
thedirectionalderivativeforabelianfunctorcalculusinsection6, andprovethatitgives
HoAbCat the structure of a cartesian differential category. In section 7, we define the
Ch
iteratedpartialdirectionalderivativesandthehigherorderdirectionalderivativesandprove
that they are related by the Faa` di Bruno formula of Theorem 7.7. We prove the analog
ofthehigherorderchainruleof[HMY]insection8,usingthetangentcategorystructure
ofHoAbCat . Finally,insection9,wediscusshowthedirectionalderivativesarerelated
Ch
toanotionofordinaryderivativesforabelianfunctorcalculusandusethemainresultof
section7toprovetheFaa` diBruno-styleformulaforthesederivativesinCorollary9.4.
Conventions. Thispaperintroducesanumberofconstructionsonfunctors(e.g.,thehigher
orderdirectionalderivatives)andinvestigatestheirbehaviorwithrespecttocomposition,
productconstructions,andsoforth. Aconvenientcontexttodescribetheseoperationsin-
volves a large category whose objects are abelian categories and whose morphisms are
functorsbetweenthem; thepreciseconstructionofthiscategory, whichissomewhatdel-
icate, is given in §3. There likely exist more than a set’s worth of functors between any
fixedpairofnon-smallabeliancategories,soifourconstructionsonfunctorsaretobein-
terpretedglobally,theyneedtotakeplaceinsomeextensionoftheusualZFCaxiomsfor
settheory: e.g.,byassumingthereexistsahierarchyofinaccessiblecardinals. See[S]for
afriendlydiscussionofthemyriadpossiblechoices.
Acknowledgments. TheauthorsthanktheBanffInternationalResearchStationforhost-
ing the second Women in Topology workshop, which brought us together for a week in
whichmanyoftheresultsinthispaperwereproven,andthePacificInstitutefortheMath-
ematicalSciences, whichprovidedtravelsupportforustoworktogetherinCalgary. We
also thank Maria Basterra and Kathryn Hess for their roles in organizing the Women in
Topology workshop. The second author is grateful for support from the Union College
Faculty Research Fund. The fourth author is grateful for support from the National Sci-
enceFoundationthroughDMS-1551129.
TslilClingmansuggestedthatchaincomplexesmightdefineamonadonabeliancate-
goriesandRobinCockettpointedoutthatnon-reducedfunctorsmayfailtopreservechain
complexes. Thisiswhythemonadof§3ispseudoandnotstrict. GeoffreyCruttwellex-
plainedthebenefitsofthinkingofacartesiandifferentialcategoryasatangentcategory,a
DIRECTIONALDERIVATIVESANDHIGHERORDERCHAINRULESFORABELIANFUNCTORCALCULUS 7
perspective which led to a simplified proof of Theorem 8.1. We thank Randy McCarthy
forhelpfulconversationsinstartingthisproject,andfortheideasthatinspiredit.
2. Crosseffectsforfunctors
Classically, the term cross effect was used to describe the combined effects of two or
moreforces,ortodescribethedifferencebetweenthequantities f(x+y)and f(x)+ f(y).
Forreducedfunctionssatisfying f(0)=0,theconditionthat f(x+y)= f(x)+ f(y)forall
xandyisequivalenttolinearity. Thus,thecrosseffectfunction,definedby
cr f(x,y)= f(x+y)− f(x)− f(y),
2
measures the failure of a reduced function f to be linear; this failure is called the devia-
tionbyEilenbergandMacLane[EM]. Inthissection,wewillstudyananalogousnotion
for functors from a pointed category to an abelian category. The cross effects were first
extended to functors of additive categories by Eilenberg and Mac Lane [EM], and these
becamethefundamentalbuildingblocksofabelianfunctorcalculusasdevelopedbyJohn-
son and McCarthy [JM2]. We recall the definition of the cross effect and summarize the
propertiesthatwewillneedforcrosseffectsfunctors.
For the duration of this section, let B be a category with a basepoint, i.e., an initial
object(cid:63)whichisalsoterminal,andfinitecoproducts,denotedby∨. LetAbeanabelian
categorywithzeroobject0andbiproductsdenoted⊕.
Definition 2.1 ([EM]). The nth cross effect of a functor F: B → A is the n-variable
functorcr F: Bn →Adefinedrecursivelyby
n
F(X)(cid:27) F((cid:63))⊕cr F(X)
1
cr F(X ∨X )(cid:27)cr F(X )⊕cr F(X )⊕cr F(X ,X )
1 1 2 1 1 1 2 2 1 2
andingeneral,
cr F(X ∨X ,X ,...,X )(cid:27)cr F(X ,X ,...,X )⊕cr F(X ,X ,...,X )
n−1 1 2 3 n n−1 1 3 n n−1 2 3 n
⊕cr F(X ,X ,...,X ).
n 1 2 n
Despite the asymmetry in this definition, the nth cross effect is symmetric in its n-
variables[JM2,Proposition1.2].
Remark2.2. Inacategorywithabasepointandfinitecoproducts,eachcoproductinclusion
X (cid:44)→ X∨Y isasplitmonomorphism;therelevanceoftheretractionisthatsplitmonomor-
phismsarepreservedbyanyfunctor. Inanabeliancategory,splitmonomorphismsextend
tosplitshortexactsequences. Inparticular,itmakessensetodefinethecrosseffectsasthe
directsumcomplementsofDefinition2.1.EilenbergandMacLanedefinedcrosseffectsas
theimagesofcertainhomomorphisms,butthedirectsumcomplementdefinitionismore
usefulforthetypesofcomputationswewishtodo.
Example2.3. FixanobjectAinanabeliancategoryAandconsiderthefunctorF: A→
AdefinedbyF(X):= A⊕X. SinceF(0)(cid:27) A,itfollowsthatcr F (cid:27)id. Now
1
X⊕Y⊕cr F(X,Y)(cid:27)cr F(X)⊕cr F(Y)⊕cr F(X⊕Y)(cid:27)cr F(X⊕Y)(cid:27) X⊕Y
2 1 1 2 1
impliesthatcr F(X,Y)(cid:27)0,soallhighercrosseffectsmustalsovanish.
2
Definition2.1isfunctorialinthesensethatanaturaltransformationF →Ginducesa
naturally-definedmapcr F →cr G. Topreciselydescribethisfunctoriality,write:
n n
• Fun(B,A)forthecategoryofallfunctorsfromBtoA;
8 BAUER,JOHNSON,OSBORNE,RIEHL,ANDTEBBE
• Fun (B,A) for the category of all strictly reduced functors from B to A, that is,
∗
functorsforwhichF((cid:63))(cid:27)0;and
• Fun (Bn,A)forthecategoryofallstrictlymulti-reducedfunctorsfromBntoA,that
∗
is,functorsforwhichF(X ,...,X )(cid:27)0ifanyX =(cid:63).
1 n i
Aneasyexercise[JM2,Proposition1.2]showsthatcr F isanobjectinFun (Bn,A).
n ∗
Lemma2.4. Foranyn≥1andanyF: B→A,cr F (cid:27)cr cr F.
n n 1
Proof. TheisomorphismF((cid:63))(cid:27) F((cid:63))⊕cr F((cid:63))impliesthatcr F((cid:63))(cid:27)0. Now
1 1
cr F(X)(cid:27)cr F((cid:63))⊕cr (cr F(X))(cid:27)cr2F(X)
1 1 1 1 1
provesthatcr F (cid:27) cr2F. ThegeneralcasenowfollowsimmediatelyfromDefinition2.1
1 1
byinduction. (cid:3)
Importantly,thecrosseffectfunctorsadmitadjoints:
Proposition2.5. Theinclusionofthecategoryofstrictlyreducedfunctorsintothecategory
of not-necessarily reduced functors admits a right adjoint, namely the first cross effect
functor:
Fun (B,A) ⊥ Fun(B,A).
∗
cr1
ThecomponentofthecounitofthisadjunctionatafunctorG ∈ Fun(B,A)isthenatural
inclusioncr G (cid:44)→G.
1
Proof. Theuniversalpropertyoftheadjunctionsassertsthatif F isreducedandG isnot
necessarilyreducedthenanynaturalmapα: F →Gfactorsuniquelythroughtheinclusion
cr G (cid:44)→G.Let!denotethenaturaltransformationtothebasepointobjectinB,!: id→(cid:63),
1
andconsiderthediagram:
cr FX :=ker (cid:27) FX F! F(cid:63)
1
∃! αX α(cid:63)
cr GX :=ker GX G(cid:63)
1
G!
The right-hand square commutes by naturality of α. The kernels of the right-hand hori-
zontalmapsdefinethefirstcrosseffects. Bycommutativityoftheright-handsquare,there
existsauniquemapcr FX → cr GX sothattheleft-handsquarecommutes. Butbecause
1 1
F(cid:63) = 0, the top left-hand map is an isomorphism, which gives us the desired unique
factorization. (cid:3)
Remark 2.6. Afullsubcategoryiscoreflectivewhentheinclusionadmitsarightadjoint.
Proposition 2.5 asserts that Fun (B,A) defines a coreflective subcategory of Fun(B,A).
∗
Because the left adjoint is full and faithful the unit is necessarily an isomorphism. By
abstract nonsense, the inclusion of a coreflective subcategory is necessarily comonadic,
thatis,Fun (B,A)isthecategoryofcoalgebrasfortheidempotentcomonadcr actingon
∗ 1
Fun(B,A). Thisformalismgivesacharacterizationofthereducedfunctors: namely F is
reducedifandonlyifthenaturalmapcr F (cid:44)→ F isanisomorphism.
1
JohnsonandMcCarthyobservethatwhentheconstructionofthenthcrosseffectisre-
strictedtoreducedfunctors,itisrightadjointtopre-compositionwiththediagonalfunctor
[JM2, Example 1.8]. Proposition 2.5 allows us to extend this adjunction to non-reduced
functors:
DIRECTIONALDERIVATIVESANDHIGHERORDERCHAINRULESFORABELIANFUNCTORCALCULUS 9
Corollary2.7. Thereisanadjunction
∆∗
Fun(B,A) ⊥ Fun (Bn,A)
∗
crn
betweenthenthcrosseffectfunctorandthefunctorgivenbyprecomposingwiththediag-
onal∆: B→Bn,inducingacomonadC onFun(B,A)definedby
n
C F(X):=cr F(X,...,X).
n n
Proof. By Lemma 2.4 cr (cid:27) cr cr for n ≥ 1. Thus, the adjunction ∆∗ (cid:97) cr is the
n n 1 n
compositeoftheadjunctions
∆∗
Fun(B,A) ⊥ Fun (B,A) ⊥ Fun (Bn,A)
∗ ∗
cr1 crn
ofProposition2.5withtheadjunctionof[JM2,Example1.8]. (cid:3)
The functor categories Fun(B,A) and Fun (Bn,A) are abelian, with kernels and di-
∗
rectsumsandsoforthdefinedobjectwiseinA. Inparticular, thesecategorieshaveshort
exact sequences. A functor from one abelian category to another which preserves exact
sequencesiscalledexact.
Proposition 2.8. For each n ≥ 1, the functors cr : Fun(B,A) → Fun (Bn,A) and
n ∗
C : Fun(B,A)→Fun(B,A)areexact.
n
Proof. Let 0 → F → G → H → 0 be a short exact sequence of functors. Consider the
diagrambelow
(cid:47)(cid:47) (cid:47)(cid:47) (cid:47)(cid:47) (cid:47)(cid:47)
0 cr F cr G cr H 0
1 1 1
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
(cid:47)(cid:47) (cid:47)(cid:47) (cid:47)(cid:47) (cid:47)(cid:47)
0 F G H 0
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
(cid:47)(cid:47) (cid:47)(cid:47) (cid:47)(cid:47) (cid:47)(cid:47)
0 F((cid:63)) G((cid:63)) H((cid:63)) 0.
Since the columns are exact, and the bottom two rows are exact, the 3 × 3 lemma [W,
1.3.2]guaranteesthatthetoprowisexact. Thenassumingthatcr isexact,wecanapply
n
thesamelemmatothediagram
(cid:47)(cid:47) (cid:47)(cid:47)
crn+1F(A1,A2,B2,...,Bn) crn+1G(A1,A2,B2,...,Bn) crn+1H(A1,A2,B2,...,Bn)
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
(cid:47)(cid:47) (cid:47)(cid:47)
cr F(A ∨A ,B ,...,B ) cr G(A ∨A ,B ,...,B ) cr H(A ∨A ,B ,...,B )
n 1 2 2 n n 1 2 2 n n 1 2 2 n
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
cr F(A ,B ,...,B ) (cid:47)(cid:47) cr G(A ,B ,...,B ) (cid:47)(cid:47) cr H(A ,B ,...,B )
n 1 2 n n 1 2 n n 1 2 n
⊕cr F(A ,B ,...,B ) ⊕cr G(A ,B ,...,B ) ⊕cr H(A ,B ,...,B )
n 2 2 n n 2 2 n n 2 2 n
toconcludethatcrn+1 isexact. Theprooffollowsbyinduction. TheresultforCn follows
immediately. (cid:3)
10 BAUER,JOHNSON,OSBORNE,RIEHL,ANDTEBBE
3. Acategoricalcontextforabelianfunctorcalculus
Abelianfunctorcalculus—alsocalledadditiveordiscretefunctorcalculus[BJM]—
considers arbitrary functors valued in an abelian category. The linear approximation de-
finedin[JM2]satisfiesauniversalproperty“uptohomotopy.” Forthistomakesense,the
target abelian category must be replaced by some sort of homotopical category in which
strict universal properties (asserting that certain diagrams commute on the nose) can be
replaced by weak ones (where the commutativity is up to some sort of homotopy rela-
tion). ForanabeliancategoryA, letChAdenotethecategoryofchaincomplexesonA
concentratedinnon-negativedegrees. Ingeneral,thelinearapproximationofF: B → A
definesafunctor D F: B → ChA; aprecisedefinitionofthisisgiveninDefinition5.1.
1
More generally, [JM2] define a linear approximation D F: B → ChA for any functor
1
F: B→ChAsothatwhenthisconstructionisappliedtoafunctorconcentratedindegree
zeroitrecoverstheconstructionforF: B→A.
Now consider a composable pair of functors G: C → B and F: B → A between
abelian categories.1 Their linear approximations D G: C → ChB and D F: B → ChA
1 1
are not obviously composable. The standard trick, which appears in [JM2, Lemma 5.7],
is to prolong the second functor, applying D F degreewise to a chain complex in B to
1
produceachaincomplexinchaincomplexesinA,andthenconvertthisdoublecomplex
intoachaincomplexinAbymeansofthetotalization. Thatis,thecompositeD F◦D G
1 1
isdefinedtobethefunctor
D F◦D G: C D1G ChB Ch(D1F) ChChA Tot ChA.
1 1
If G is reduced, then [JM2, Lemma 5.7] (a stronger version of which appears as Propo-
sition 5.7 below) proves that this composite is quasi-isomorphic to D (F ◦G), i.e., that
1
D is“functorialuptoquasi-isomorphism”withrespecttothecompositionstructurejust
1
introduced.
Tokeeptrackofwhichfunctorsshouldbecomposableandwhicharenot(andtoavoid
aproliferationofCh’s)itisconvenienttoregardbothFandD FasfunctorsfromBtoA.
1
ThissortofbookkeepingiseffortlesslyachievedbythecategoricalformalismofaKleisli
categoryforamonadthatwenowintroduce.
Observation 3.1. There is a pseudomonad2 Ch(−) acting on the (large) 2-category of
abeliancategories,arbitraryfunctorsbetweenthem,andnaturaltransformations. Herewe
arenotinterestedinthe2-dimensionalaspects,soweinsteaddescribethequotientmonad
Ch(−) acting on the 1-category AbCat of abelian categories and isomorphism classes of
functors:
• The monad carries an abelian category A to the category ChA of non-negatively
gradedchaincomplexesinA.
• ThemonadcarriesafunctorF: B→AtoitsprolongationChF: ChB→ChA.Be-
causeF mightnotpreservezeromaps,thedefinitionofitsprolongationissomewhat
delicate. Making use of the Dold-Kan equivalence between non-negatively graded
1AsinSection2, itsufficestoassumethatthedomainofG isapointedcategorywithfinitecoproducts,
butwhenconsideringcompositefunctorsitislinguisticallyconvenienttosupposethatallofthecategoriesare
abelian.
2Incategorytheory,theprefix“pseudo”isattachedtostructureswhichholduptospecifiedcoherentiso-
morphism. Apseudomonadisgivenbythesameunderlying1-categoricaldataasamonad,butwithstructure
diagramswhichcommuteonlyuptospecifiedcoherentnaturalisomorphisms.Foramoredetaileddefinition,see
[Ma].