Homology of path complexes and hypergraphs Topology and its Applications

Contents lists available atScienceDirect

Topology and its Applications

www.elsevier.com/locate/topol

Homology of path complexes and hypergraphs

Alexander Grigor’yan^a, Rolando Jimenez^b,∗, Yuri Muranov^c, Shing-Tung Yau^d

a DepartmentofMathematics,UniversityofBielefeld,33501Bielefeld,Germany

bInstitutodeMatematicas,UNAM,UnidadOaxaca,68000Oaxaca,Mexico

cFacultyofMathematicsandComputerScience,UniversityofWarmiaandMazury,10-710Olsztyn, Poland

dDepartmentofMathematics,HarvardUniversity,Cambridge,MA02138,USA

a r t i cl e i n f o a b s t r a c t

Articlehistory:

Received22May2019

Receivedinrevisedform30August 2019

Accepted31August2019 Availableonline5September2019

MSC:

05C65 05C25 55U05 18G60 55N35 55U10

Keywords:

Pathcomplex

Pathhomologyofhypergraph Graphhomology

Homotopyofpathcomplexes Simplicialcomplex

Thepathcomplexanditshomologyweredefinedinpreviouspapersoftheauthors.

Thenotionofapath complexisanaturaldiscretegeneralizationofthenotionof a simplicial complex.Thetheory ofpath complexes containshomotopyinvariant homologytheoryofdigraphsand(nondirected)graphs.

Inthepaper westudythehomologytheoryof pathcomplexes. Inparticular,we describefunctorialpropertiesofpathscomplexes,introducethenotionofhomotopy forpath complexesandprovethehomotopyinvarianceofpathhomology groups.

Weprovealsoseveraltheoremsthataresimilartotheresultsofclassicalhomology theoryofsimplicialcomplexes.Thenweapplythisapproachtoconstructhomology theoriesonvariouscategoriesofhypergraphs.Wedescribebasicpropertiesofthese homologytheories andrelationsbetweenthem.Asaparticularcase,theseresults givenewhomologytheoriesonthecategoryofsimplicialcomplexes.

©2019ElsevierB.V.Allrightsreserved.

1. Introduction

In this paper we study functorial and homotopy properties of path complexes thatwere introduced in previouspapersoftheauthorsasanaturaldiscretegeneralizationofthenotionofasimplicialcomplex.Now we systematically describe properties of path complexes and provide new definitions, including notionof homotopy,andprovetheoremsthataresimilartotheresultsofsimplicialhomologytheory.Asanapplication weconstructhomologytheoriesofvariouscategories ofhypergraphs(see[6]).

* Correspondingauthor.

E-mailaddress:[email protected](R. Jimenez).

https://doi.org/10.1016/j.topol.2019.106877 0166-8641/©2019ElsevierB.V.Allrightsreserved.

Notethattheparticularcaseofthetheory,thepathhomologytheoryofdigraphsand(nondirected)graphs wasinvestigatedin[8],[9],[10],and[11].Forthecaseof(nondirected)graphs,thepathhomologycoincides withthegraphhomologydefinedin[3] and[4] whichiscloselyconnectedwiththeA-homotopytheory.See [1], [2],and [3]. TheKünnethformulaefor theCartesianproduct andfor thejoinofpath complexeswere provedin[7] and [12].

In Section 2, we recall the notion of the path complex on a finite set V and the definition of path homology.Wedescribealsofunctorialpropertiesofpathhomology.

InSection3,weintroducethenotionofhomotopyforpathcomplexesandprovethehomotopyinvariance of pathhomologygroups.

In Section 4, we introduce the notion of a sub-complex of apath complex and corresponding relative homologygroups.Weconstructalsoseveralnaturalhomologyexactsequences.

In Section5, we apply obtainedresults to construct homologytheories on various categories of hyper- graphs and describe theirs properties. Weprovide also several examples of explicit computations of path homologygroupsofhypergraphs.

Acknowledgments

ThefirstandthethirdauthorswerepartiallysupportedbySFB1283ofGermanResearchCouncil.The second author waspartially supportedbythe CONACyT Grant 284621.The fourthauthor was partially supportedbythegrant“GeometryandTopology ofComplexNetworks”,no.FA-9550-13-1-0097.

2. Pathcomplexesonfinitesets

In this section we recall the notion of the path complex on afinite set V and define the path homol- ogy theory of such complexes. Then we introduceanotion of morphism forpath complexes and describe functorialpropertiesofpathhomologygroups.

Let V be anarbitrarynon-emptyfinite set.Wecall avertex any elementv ∈V.Anelementary n-path onasetV isasequencei0. . . in ofn≥0 verticesfromV.

ForaunitarycommutativeringK,considerafreeK-moduleΛ_n= Λ_n(V) generatedoverKbyelements e_i0...in wherei₀...i_nisanelementaryn-path.TheelementsofΛ_nforn≥0 arecalledn-pathsonV.Setalso Λ₋₁=K andΛ₋₂= 0.Eachn-pathv∈Λ_n forn≥0 hasauniquerepresentationoftheform

v=

i0,...,in∈V

v^i0...inei0...in,

where v^i0...in∈K.

Forn≥1,definetheboundary operator

∂: Λn →Λ_n−1 as alinearoperatorthatactsonelementarypathsby

∂ei0...in= n s=0

(−1)^se_i

0...is...in, (2.1)

where the hat i_s means omission of the index i_s. For n = 0,−1 we define ∂ : Λ₀ → Λ₋₁ = K as the augmentationhomomorphismεgivenby

ε

k_pi_p

=

k_p, k_p∈K, i_p∈V,

anddefine ∂: Λ₋₁→Λ₋₂= 0 tobezero.

Itisaneasyexercisetocheckthat∂²= 0 [7] and,hence,Λ_∗={Λ_n}isachaincomplex.

Anelementarypathi0...iniscallednon-regularifik−1=ik forsomek= 1,. . . ,n,andregularotherwise.

Forn≥2,letIn =In(V)⊂Λn(V) beasubmodulegeneratedbynon-regularpaths.Forn=−2,−1,0,1 we putIn = 0. Then forn≥ −1, therestrictionof ∂ to In satisfies thecondition∂² = 0 :In →In−2 and henceachaincomplexI_∗ isdefined.Thusweobtainaquotientchaincomplex

R∗=R∗(V) := Λ_∗(V)/I_∗(V),

andwecontinueto denoteby∂ theinduceddifferential. Itisclear,thatforn≥0 wehaveisomorphisms Rn∼= span

ei0...ip:i0...ip is regular .

TheelementsofRp arecalledregular p-paths.

LetV,V be twofiniteset.Anymap f :V →V inducesamorphismofchaincomplexes f_∗: Λ_∗(V)→Λ_∗(V)

givenonthebasisofΛn(V) forn≥0 bytherule

f_∗(ei0...in) =e_{f(i0)...f(in)} andf_∗ is an isomorphism for n=−1,−2. (2.2) Sincef_∗(In(V))⊂In(V),themorphismf_∗ inducesamorphismofchaincomplexes ofregularpaths

R∗(V)→ R∗(V) (2.3)

whichwecontinuetodenote f_∗.

Definition 2.1. [7] A path complex over aset V is acollection P = P(V) of elementary paths on V such that:

• i∈P foranyi∈V,

• ifi0...in∈P theni0...in−1∈P andi1...in∈P.

For n ≥0, theset of all n-paths from P is denotedby P_n. The (n−1)-paths i₀...i_n−1 and i₁...i_n are calledthetruncated pathsofthen-pathi₀...i_n.TheelementsofP arecalledallowedelementarypaths,and theelementarypathsthatare notinP are callednon-allowed. NotethatP0=V. Theelementsof P1 are callededges ofP.Itfollowsimmediately fromDefinition2.1thatforn-pathi0...in ∈P(n≥1),all 1-paths ik−1ik are edges.

Definition2.2. Wesay,thatamapf :V →V induces amorphismofpathcomplexes P andP if,forany pathv∈P,thepathf_∗(v) definedin(2.2) liesinP. Wedenote thismorphismas

f_•= (f, f_∗) : (V, P)→(V, P).

Let(V,P),(V,P),(W,S) be pathcomplexes and f: V →V, g: V →W be maps ofsets thatdefine themorphisms

f_•= (f, f_∗) : (V, P)→(V, P) andg_•= (g, g_∗) : (V, P)→(W, S)

of pathcomplexes.Thenwedefine thecompositiong_•f_• as g_•f_•: = (gf, g_∗f_∗).

Themap gf:V →W definesamorphismofpathcomplexes

(gf)_•= (gf,(gf)_∗) : (V, P)→(W, S) thatevidentlycoincideswiththemorphismg_•f_•.

InthiswayweobtainacategoryPwhoseobjectsarepathcomplexesandwhosemorphismsaremorphisms of pathcomplexes.

For anyintegern≥0,define theK-moduleAn(P) thatisspanned byalltheelementaryn-paths from P:

An=An(P) =ei0...in|i0...in∈Pn,

andweputA−1=K,A−2= 0.TheelementsofAn arecalledallowedn-pathsofthepathcomplexP.Thus we haveanaturalinclusionofmodulesAn(P)⊂Λ_n(V) forn≥1 andAn(P)= Λ_n(V) forn=−2,−1,0.

Note thattheset ofallpaths ontheset V givesapath complexwhichwedenote byPV. Weshallcall PV afullpath complex ontheset V.ForthispathcomplexwehaveAn(PV)= Λn(V).

Forsomepathcomplexes itcanhappenthat∂An⊂ An−1 (see[9],[7],[11]),butinthegeneralcasethis is nottrue.

Define asubmoduleΩ_n(P)⊂ An(P) as follows. For n =−2,−1,0,1 we putΩ_n(P)=An(P), and for n≥2 weput

Ω_n= Ω_n(P) ={v∈ An|∂v∈ An−1}.

It is clear that∂(Ωn)⊂ Ωn−1, since ∂² = 0. The elements of Ωn are called ∂-invariant n-paths, and we obtainareduced chaincomplex:

0←K←Ω0←...←Ωn−1←Ωn←Ωn+1←... (2.4) where theboundarymaps areinducedby∂.Thecorrespondingnon-reduced chaincomplexhastheform

0←Ω₀←...←Ω_n−1←Ω_n ←Ω_n+1←... (2.5) Homologygroupsof(2.5) arereferredtoasthepathhomologygroupsofthepathcomplexPandaredenoted byHn(P),n≥0.Thehomologygroupsof(2.4) arecalled thereduced pathhomology groups ofP andare denotedbyH_n(P),n≥ −1.

Now weintroduceregularpath homology groupsofapathcomplexP.Wehaveacommutativediagram of inclusionsofK-modules

An(P) −→ Λn(V)

↑ ↑

An(P)∩I_n −→ I_n whichinduces homomorphismsofK-modules

Rn(P) =An(P)/{An(P)∩I_n} →Λ_n(V)/I_n =Rn(V) forn≥ −2.Denotetheimageofthis homomorphismas

R^regn (P)⊂ Rn(V).

Note,thatRn(P)∼=R^regn (P).

Define asubmoduleΩ^reg_n (P)⊂ R^regn (P) asfollows. For n=−2,−1,0 we putΩ^reg_n (P)=R^regn (P),and forn≥1 weput

Ω^reg_n = Ω^reg_n (P) =

v∈ R^regn (P)|∂v∈ R^reg_n−1(P) where∂: Rn(V)→ Rn−1(V) .

The elements of Ω^reg_n are called ∂-invariant regular n-paths, and we obtain a reduced regular chain complex:

0←K←Ω^reg₀ ←...←Ω^reg_n−1←Ω^reg_n ←Ω^reg_n+1←... (2.6) wheretheboundarymapsareinducedby∂.Thecorrespondingnon-reduced complexhastheform

0←Ω^reg₀ ←...←Ω^reg_n−1←Ω^reg_n ←Ω^reg_n+1←... (2.7) Homology groupsof(2.7) are referred to as theregularpath homology groups of thepath complexP and aredenotedbyH_n^reg(P),n≥0.Thehomologygroupsof(2.6) arecalled thereduced regularpathhomology groups ofP and aredenotedbyH_n^reg(P),n≥ −1.

Somepropertiesoftheintroducedchaincomplexes,variousspecialtypesofpathcomplexes,andexamples of computinghomology groupsaregiven in[12] and[7].From now onwe shallconsider only non-reduced chaincomplexesandnon-reducedhomologygroupsifotherwiseisnotstate.

LetC∗ denotethecategorywhose objectsare chaincomplexesand whosemorphismsarechainmaps.

Nowwediscussmorphismsofchaincomplexeswhichareinducedbymorphismsofpathcomplexes. The proofsoffollowing statementsfollow immediatelyfromdefinitions.

Lemma2.3.(i)LetΛ_∗ andΛ_∗bechaincomplexeswiththedifferentials∂ and∂,respectively,andf_∗: Λ_∗→ Λ_∗ beamorphism. LetthesubmodulesAn⊂Λn andAn⊂Λ_n begiven inasuchwaythat f_∗(An)⊂ An.

DefinethemodulesΩn andΩ_n in suchaway

Ω_n ={v∈ An|∂v∈ An−1} and Ω_n=

v∈ An|∂v∈ An−1

. (2.8)

(i)Then

∂(Ωn)⊂Ω_n−1, ∂(Ω_n)⊂Ω_n−1 (2.9) andf_∗ inducesamorphismof chaincomplexesΩ_∗→Ω_∗ whichwe continuetodenoteby f_∗.

(ii)If f_∗: Λ_∗→Λ_∗ is amonomorphism,then f_∗: Ω_∗ →Ω_∗ isamonomorphism,too.

Proposition 2.4. Any morphismf_•: (V,P)→ (V,P) of path complexes induces a morphism f_∗ of chain complexes

Ω_∗(f_•) =f_∗: Ω_∗(P)→Ω_∗(P) (2.10) andamorphismof regularchaincomplexes

Ω^reg_∗ (f_•) =f_∗: Ω^reg_∗ (P)→Ω^reg_∗ (P) (2.11) and, consequently,homomorphisms

f_∗:H_∗(P)→H_∗(P), f_∗:H_∗^reg(P)→H_∗^reg(P), (2.12) of homology andregularhomologygroups.

Corollary 2.5. Wehave functors Ω_∗ andΩ^reg_∗ from the category P of path complexesto thecategory C∗ of chain complexes.

Proposition 2.6. Foranypath complex P wehave amorphismof chain complexes

Ω_∗(P)→Ω^reg_∗ (P) and hencethehomomorphismsof homology groups

f_∗:H_∗(P)→H_∗^reg(P).

Now wegiveyetonedefinitionweshallneedinthenextsection(see[12] and [7]).

Definition 2.7.ApathcomplexP iscalledregular ifallthepathsi0...in∈P areregular.

It is possible to give aweakerdefinition ofa morphismof path complexes thatinduces amorphism of regularchaincomplexes[7].

Definition 2.8.[7] We say,thatamap f :V →V ofsets provides aweakmorphism ofpath complexesP and P if,foranypathv∈P,thepathf_∗(v) definedin(2.2) liesinP oritisanirregularpathinthefull path complexPV.Inthiscaseweshallwrite

f_◦: (V, P)→(V, P).

Let

f_◦: (V, P)→(V, P) andg_◦: (V, P)→(W, S)

beaweakmorphismofpathcomplexes.Similarlytothecaseofmorphismofpathcomplexes,thecomposition g_◦f_◦ isdefined. Thus we obtainacategory PW whose objectsare path complexes and whose morphisms are weakmorphismsofpathcomplexes.

Proposition 2.9.Any weakmorphismf_◦: (V,P)→(V,P)of pathcomplexesinducesamorphismof chain complexes

Ω^reg_∗ (f_◦) : Ω^reg_∗ (P)→Ω^reg_∗ (P) (2.13) and, consequently,ahomomorphism

H_∗^reg(P)→H_∗^reg(P) (2.14)

of regularhomology groups.

Proof. LetI_n ⊂Λ_n(V) beasubmodulegeneratedbynon-regularpaths.Theresultfollowsfromthenatural isomorphism

An(P)/{An(P)∩I_n}−→ A^∼= n(P∪I_n)/{An(P∪I_n)∩I_n} andfrom Lemma2.3as above.

Consideracommutativediagram

An(P) →^f∗ An(P∪I_n) ← An(P)

↓ ↓ ↓

An(P)/{An(P)∩In} → A^f∗ n(P∪I_n)/{An(P∪I_n)∩I_n} ← A^∼= n(P)/{An(P)∩I_n}

|| ↓∼= ||

Rn(P) →^f∗ Rn(P) = Rn(P)

↓ ↓ ↓

R^regn (P) →^f∗ R^regn (P) = R^regn (P)

↓ ↓

R∗(V) → R∗(V)

where the bottom vertical arrows are natural inclusions, and the morphism f induces the morphism of quotients

f_∗:R∗(V)→ R∗(V).

Nowthemorphism(2.13) followsfromLemma2.3andtherestoftheclaimisstandard. 2

Corollary2.10.We havethat Ω^reg_∗ isafunctorfrom thecategoryPW of pathcomplexestothecategory C∗

of chaincomplexes.

3. Homotopy theoryforpathcomplexes

Inthis sectionwe constructahomotopytheory forpathcomplexes and provethe homotopyinvariance ofhomologygroupsintroducedabove.

LetI={0,1}beasetwithtwoelements.ForanysetV ={0,. . . ,n},letV×IbetheCartesianproduct.

LetV beacopyof theset V andwe denotesuch aset byV ={0,. . . ,n}wherei ∈V corresponds to i∈V.ThenwecanidentifyV×IwithV∪Vinsuchawaythat(i,0) correspondstoiand(i,1) corresponds toi foranyi∈V.ThusV isidentifiedwith V × {0}⊂V ×I andV isidentified withV × {1}⊂V ×I.

The naturalisomorphismV ∼=V defines apath complexP on theset V by thefollowing condition:

i₀. . . i_n∈P iffi₀. . . i_n ∈P.

DefineapathcomplexP×I asapathcomplexonV ×I=V ∪V by

P×I={w|w∈P} ∪ {w|w ∈P} ∪ {w=i0. . . iki_k. . . i_n|i0. . . ikik+1. . . in ∈P} (3.1) where0≤k≤n.It followsfrom(3.1) that wehavenaturalmorphisms

i_•: (V, P)→(V ×I, P×I) and

j_•: (V, P)→(V ×I, P×I)

whichareinducedbynaturalinclusionsi:V →V ×I=V ∪V and j:V →V ×I=V ∪V.

Now,we definethenotionofhomotopyinthecategory ofpathcomplexes. LetP beapathcomplexon thesetV andSbeapathcomplexonthesetW.Notethatanymapf: V →W definesnaturallyaunique

map f: V →W, and similarly,any morphism f_•: (V,P)→(W,S) definesnaturallya uniquemorphism f_•: (V,P)→(W,S).

Definition 3.1.i) We call two morphisms f_•,g_•: (V,P) → (W,S) of path complexes one step homotopic (and write f_•1g_•)ifthereexists amorphismF_•: (V ×I,P×I)→(W,S) ofpath complexessuchthat at leastoneof thetwo followingconditionsaresatisfied.

1. F_•|(V,P)=f_•, F_•|(V,P)=g_•. 2. F_•|(V,P)=g_•, F_•|(V,P)=f_•.

ii) We call two morphisms f_•,g_•: (V,P) → (W,S) of path complexes homotopic and write f_• g_• if there existsasequenceofmorphisms

fi•: (V, P)→(W, S) suchthatf_•=f_0•1f_1•1· · · 1f_n•=g_•.

iii) Twopathcomplexes (V,P) and(W,S) arehomotopyequivalent ifthereexist morphisms f_•: (V, P)→(W, S), g_•: (W, S)→(V, P)

suchthat

f_•g_•IdW•, g_•f_•IdV•

where Id_V :V →V and Id_W:W → W are theidentity morphisms. Inthis case, we shall write (V,P) (W,S) andshallcallthemorphismsf_•, g_• homotopyinverses toeachother.

It followsdirectlyfrom Definition3.1,thattherelation“tobe homotopic”is anequivalencerelationon thesetofmorphismsbetweentwo pathcomplexes, andhomotopyequivalenceisanequivalencerelationon the setof pathcomplexes. Moreover,wewill denotebyP thecategorywhose objectsarepath complexes and morphismsaretheclasses ofhomotopic morphismsofpathcomplexes.

LetV be aset.Forn≥0,define ahomomorphism

τ: Λn(V)→Λn+1(V ×I) onelementaryn-pathsv=ei0...in∈Λn(V) by

τ(v) = n k=0

(−1)^ke_{i0...ikik...i}

n, (3.2)

an extending to Λn(V) by K-linearity. Recall that we consider non-reduced complexes, hence Λi = 0 for i ≤ −1 and we define τ = 0 : Λ₋1(V) → Λ0(V ×I). Recall that we have a natural isomorphism Λn(V)→^∼= Λn(V) of submodules of Λn(V ×I). It is given on thebasis elements by ei0...in → e_i0...i

n and extending by linearity to Λn(V). We shall denote by v ∈ Λn(V) ⊂Λn(V ×I) the image of theelement v∈Λ_n(V)⊂Λ_n(V ×I)

Lemma 3.2.Forn≥0andany path v∈Λ_n(V) wehave

∂τ(v) +τ(∂v) =v−v. (3.3)

Proof. Itissufficienttoprovethestatementforbasiselements v=e_i0...in.Forn= 0 andv=e_i0 ∈Λ₀(V), wehave

∂τ(ei0) =∂(e_i0i0) =e_i0−ei0, τ(∂v) =τ(0) = 0,

andthecondition(3.3) issatisfied.Considerapathv=ei0...in∈Λn(V) withn≥1.Wehave

∂(τ(v)) =∂ n k=0

(−1)^ke_i0...iki k...i_n−1i_n

= n k=0

(−1)^k k m=0

(−1)^me_i

0...im...iki_k...i_n

+

n m=k

(−1)^m+1e_i

0...iki_k...i_m...i_n

=

0≤m≤k≤n

(−1)^k+me_i

0...im...iki_k...i_n+

0≤k≤m≤n

(−1)^k+m+1e_i

0...iki_k...i_m...i_n

and

τ(∂v) =τ n m=0

(−1)^me_i

0...im...in

= n m=0

(−1)^m

m−1

k=0

(−1)^ke_i

0...iki_k...i_m...i_n

+

n k=m+1

(−1)^k−1e_i

0...im...iki_k...i_n

=

0≤k<m≤n

(−1)^k+me_i0...iki

k...i_m...i_n+

0≤m<k≤n

(−1)^k+m−1e_{i0...im...iki} k...i_n

Hence

∂τ(v) +τ(∂v) =

0≤k≤n

(−1)^k+ke_i

0...iki_k...i_n+

0≤k≤n

(−1)^k+k+1e_i

0...iki_k...i_n

=

0≤k≤n

e_i0...ik−1i

k...i_n−

0≤k≤n

e_i0...iki k+1...i_n

=e_i

0...i_n+

1≤k≤n

e_i0...ik−1i k...i_n−

⎛

⎝

0≤k≤n−1

e_i0...iki

k+1...i_n+e_i0...in

⎞

⎠

=e_i

0...i_n−e_i0...in+

⎛

⎝

0≤k−1≤n−1

e_i0...ik−1i

k...i_n−

0≤k≤n−1

e_i0...iki k+1...i_n

⎞

⎠

=ei₀...i_n−ei0...in=v−v. 2

Remark3.3. Itis easy to transferresultsof Lemma3.2 to the regularpaths. Themodule Rn(V) has the basis{e_i0...in|i₀. . . i_n is regular path onV}.Thenwecandefine

τ:Rn(V)→ Rn+1(V ×I) (3.4)

bythesameformulae asforτ: Λn(V)→Λn+1(V ×I).

Theorem3.4. (i)Let

f_•g_•: (V, P)→(W, S)

be homotopic morphisms of path complexes. Then thesemorphismsinduce chain homotopic morphisms of reduced andnon-reduced chain complexes

f_∗g_∗: Ω_∗(P)→Ω_∗(S) and f_∗ g_∗: Ω^reg_∗ (P)→Ω^reg_∗ (S) and hencethesamehomomorphism ofcorresponding homologygroups,respectively.

(ii)Ifthepathcomplexes(V,P)and(W,S)arehomotopyequivalent,thentheyhaveisomorphichomology groups.Furthermore,ifthehomotopyequivalenceisprovidedbyhomotopyinversemorphismsf_•andg_• (as in(iii)ofDefinition3.1)thentheinducedhomomorphismsf_∗andg_∗providemutuallyinverseisomorphisms of reduced and non-reduced homologygroups of(V,P)and(W,S).

Proof. Atfirstwe provethestatementfor non-regular andnon-reduced chaincomplexes. It issufficiently to consideronlythefirstcaseofone-stephomotopyF_• betweenf_• andg_• asin(i) case 1ofDefinition3.1.

ByDefinition3.1wehave

F_•i_•=f_•: Ω_∗(P)→Ω_∗(S) and

F_•j_•=g_•: Ω_∗(P)→Ω_∗(S) whichwe identifynaturallywiththemorphism

g_•: Ω_∗(P)→Ω_∗(S).

ByProposition2.4,morphismsf_• and g_• inducemorphismsofchaincomplexes f_∗, g_∗: Ω_∗(P)→Ω_∗(S),

and F_• induces amorphism

F_∗: Ω_∗(P×I)→Ω_∗(S) suchthat

F_∗|Ω_∗(P)=f_∗, F_∗|Ω_∗(P)=g_∗.

In order to prove thatf_∗ and g_∗ induce the samehomomorphism H_∗(P) → H_∗(S), it suffices by [14, Theorem 2.1,p.40] toconstructachainhomotopy

L_n: Ω_n(P)→Ω_n+1(S) suchthat

∂Ln+Ln−1∂=g_∗−f_∗ Forn≥0,defineahomomorphism

τ:An(P)→ An+1(P×I)

onelementary n-pathsv =e_i0...in ∈ An(P) by formulae(3.2) and extend it to An(P) by K-linearity. We putalsoAi= 0 for i=−1 anddefine τ= 0 : A−1(P)→ A0(P×I).Weprovenowthatifv∈Ωn(P) then τ(v)∈Ωn+1(P×I).Letv∈Ωn(P) thatisv∈ An(P)⊂Ωn(V) and∂v∈ An−1(P)⊂Ωn−1(V).Hence,by (3.2) anddefinitionofA∗(P×I),wehaveτ(v)∈ An+1(P×I) andτ(∂v)∈ An(P×I).ByLemma3.2,we have

∂τ(v) =−τ(∂v) +v−v

wheretheright summandslieinAn(P×I).Itfollows thatifv∈Ω_n(P) thenτ(v)∈Ω_n+1(P×I).

Forn≥0,definethehomomorphism

Ln(v) : =F_∗(τ(v)) : Ωn(P)→Ωn+1(S) for all v∈Ωn(P). Weobtain

(∂Ln+Ln−1∂)(v) =∂(F_∗(τ(v))) +F_∗(τ(∂v)) (by definition L_∗)

=F_∗(∂τ(v)) +F_∗(τ(∂v)) (sinceF_∗ is a chain map) 3.1)

=F_∗(∂τ(v) +τ(∂v)) (sinceF_∗is a homomorphism)

=F_∗(v−v) (by Lemma3.2)

=g_∗(v)−f_∗(v) (by definition ofF).

Thus the case (i) for reduced non-regular chain complexes is proved. The proof of (ii) in this case is standard.

Forthecaseofnon-reduced non-regularchaincomplexestheproof issimilar. RecallthatKis aringof coefficients. In this case f_∗|K =g_∗|K = F_∗|K is the identity map, and define τ:K → Ω0(P×I) as the trivialhomomorphism.

Inthecaseofreducedregularchaincomplexestheproofissimilar.ItisnecessarytousetheRemark3.3 insteadofLemma3.2. 2

Nowconsider two full pathcomplexes PV andPW.Inthis caseanymap ofsets f:V →W defines the morphismf_•: (V,PV)→(W,PW) ofpathcomplexes.

Proposition3.5. Any twomorphisms

f_•, g_•: (V, PV)→(W, PW)

of full path complexesare onestephomotopic and, hence,any full path complex is homotopyequivalent to thefull path complex (∗,P_∗) ontheone pointset∗.The similar statementis true foranyregularfull path complex.

Proof. Define themap F:V ×I=V ∪V →W byF|V =f, F|V =g.Thismap definesamorphismof pathcomplexes

F_•: (V ×I, P_V ×I)→(W, P_W)

which satisfies evidently the conditions on one-step homotopy from Definition 3.1. The Proposition is proved. 2

Corollary 3.6.Forany full path complex(V,P_V) wehave Hn(PV) =H_n^reg(PV) =

K, forn= 0, 0, forn≥1.

Theproofis trivial.

Now wereturntoarbitrarypathcomplexes.

Definition 3.7.i) We call two weak morphisms f_◦,g_◦: (V,P) → (W,S) of path complexes weak one step homotopic (and write f_◦ ∼1 g_◦) if there exists a weak morphism F_◦: (V ×I,P ×I) → (W,S) of path complexes suchthatat leastoneofthetwofollowing conditionsaresatisfied.

1. F_◦|(V,P)=f_◦, F_◦|(V,P)=g_◦. 2. F_◦|(V,P)=g_◦, F_◦|(V,P)=f_◦.

ii) Wecall two weak morphisms f_◦,g_◦: (V,P)→ (W,S) ofpath complexes weak homotopic and write f_◦∼g_◦ ifthereexistsasequenceofweakmorphisms

fi◦: (V, P)→(W, S) suchthatf_◦=f0◦∼1f1◦∼1· · · ∼1fn◦=g_◦.

iii) Twopathcomplexes (V,P) and(W,S) areweakhomotopyequivalent ifthereexist weakmorphisms f_◦: (V, P)→(W, S), g_◦: (W, S)→(V, P)

suchthat

f_◦g_◦∼IdW◦, g_◦f_◦∼IdV◦

where IdV :V →V and IdW:W → W are theidentity morphisms. Inthis case, we shall write (V,P)∼ (W,S) andshallcalltheweakmorphismsf_◦,g_◦ weakhomotopy inversesto eachother.

ItfollowsdirectlyfromDefinition3.7,thattherelation“tobeweakhomotopic”isanequivalencerelation onthesetofweakmorphismsbetweentwopathcomplexes,andweakhomotopyequivalenceisanequivalence relation on the set of path complexes. Moreover, we will denote by PW the category whose objects are path complexesandmorphismsaretheclasses ofweakhomotopic weakmorphismsofpathcomplexes.

Theorem 3.8.(i) Let

f_◦∼g_◦: (V, P)→(W, S)

be weakhomotopic morphismsof path complexes.Thenthese morphismsinducethechain homotopic mor- phisms of regularchaincomplexes

f_∗g_∗: Ω^reg_∗ (P)→Ω^reg_∗ (S) and hencethesamehomomorphism ofcorresponding homologygroups.

(ii) If the path complexes (V,P) and (W,S) are weak homotopy equivalent, then they have isomorphic regular homology groups. Furthermore, if the weak homotopy equivalence is provided by homotopy inverse morphisms f_◦ and g_◦ then theinduced homomorphisms f_∗ and g_∗ provide mutually inverse isomorphisms of reduced andnon-reduced homologygroups of(V,P)and(W,S).

Proof. Theproofissimilar totheproofofTheorem 3.4. 2 4. Relative pathhomologygroups

In this section we introducerelative path homology groups and construct several exactsequences with thesegroups.

Let(W,S) beapathcomplexandV ⊂W.

Definition4.1. Apathcomplex(V,P) overasetV isapath subcomplex ofthepathcomplex(W,S) ifany elementarypathp∈P liesinS.Wewriteinthiscase(V,P)⊂(W,S) or,tosimplifynotations,P ⊂S.

For a subcomplex P ⊂ S, the inclusion morphism i_•: (V,P) → (W,S) induces a monomorphism i_∗: Λ_∗(V)→Λ_∗(W) ofchaincomplexessuchthati_∗(An(P))⊂ An(S).ByLemma3.2,thisimpliesthati_• induces amonomorphism ofchain complexesi_∗: Ω_∗(P)→ Ω_∗(S). Thuswe obtainashort exactsequence of(reducedandnon-reduced)chaincomplexes

0−→Ω_∗(P)−→Ω_∗(S)−→Ω_∗(S)/Ω_∗(P)−→0. (4.1) For the reduced chain complexes, the homomorphism i_∗ in dimension −1 is the identity homomorphism K →K.Hencethe factor-complexΩ_∗(S)/Ω_∗(P) willbe thesamefor thereduced andnon-reduced chain complexes. We define homology groups H_∗(S,P) = H_∗(Ω_∗(S)/Ω_∗(P)), that are called the relative path homology groups.

Thesamelineofargumentsshowthatwehavealsoashortexactsequenceof(reducedandnon-reduced) regularchaincomplexes

0−→Ω^reg_∗ (P)−→Ω^reg_∗ (S)−→Ω^reg_∗ (S)/Ω^reg_∗ (P)−→0. (4.2) Proposition 4.2. Let (V,P) be a path subcomplex of (W,S). There are long exact sequences of homology groups

0←H₀(S, P)←H₀(S)←H₀(P)←H₁(S, P)←H₁(P)←. . . and

0←H₀^reg(S, P)←H₀^reg(S)←H₀^reg(P)←H₁^reg(S, P)←H₁^reg(P)←. . . andsimilarlyforreduced homologygroups.

Proof. Follows from(4.1) and(4.2). 2 5. Thepathhomologyofhypergraphs

Inthissectionweapplytheaboveresultstoconstructahomologytheoryonthecategoryofhypergraphs.

Thehomologytheorybasedonthetheoryofpathcomplexesfortheparticularcaseofdigraphsand(nondi- rected)graphswasconstructedin[9],[10], [11]. Asbefore, wefixacommutativeringK withaunityas a ringofthecoefficients.

Definition5.1.[5] (i)Afinitehypergraph isapairG= (V,E) whereV isanon-emptyset ofvertices andE isafamily{e₁,. . . ,e_k}ofnon-emptyandnon-ordered subsetsofV suchthat