KATSUHIKO KURIBAYASHI
Abstract. A quasi-schemoid is a small category with a particular partition of the set of morphisms. We define a homotopy relation on the category of quasi- schemoids and study the fundamental properties of the homotopy. In par- ticular, the homotopy set of self-homotopy equivalences on a quasi-schemoid is introduced as a homotopy invariant. The main theorem asserts that the homotopy invariant for the quasi-schemoid induced by a finite groupoid is isomorphic to the group of autofunctors on the groupoid.
1. Introduction
The category Cat of small categories is a 2-category whose 2-morphisms are natural transformations. Hoff [2] and Lee [6] have introduced a notion of the strong homotopy onCatusing 2-morphisms; see [3, 5, 7]. Thus if subjects we investigate have the structures of small categories, we may develop homotopy theory for them with the underlying small categories.
Association schemes play crucial roles in the study algebraic combinatorial the- ory, design and coding theory; see [8] and references contained therein. Then it is important to consider their classification problem. One might make rough clas- sification of the association schemes relying on abstract homotopy theory. Since association schemes are regarded as complete graphs and hence objects inCat, the completeness allows us to deduce that every association scheme is contractible in the sense of the strong homotopy. In fact, each association scheme is equivalence to the trivial category as a category. Thus we need an appropriate category rather thanCatto develop homotopy theory for combinatorial objects such as association schemes.
Matuso and the author [4] have proposed the notion ofquasi-schemoidsgeneraliz- ing that of association schemes from small categorical points of view. An important point here is that the categoryCatis embedded into the 2-categoryqASmdof quasi- schemoids; see Theorem 3.9 below. In this paper, we define a homotopy relation on the categoryqASmdextending that due to Hoff, Lee and Minian and study funda- mental properties of the homotopy. In particular, the group (of homotopy classes) of self-homotopy equivalences on a quasi-schemoid is investigated. It is expected that the homotopy relation captures appropriate properties of quasi-schemoids includ- ing association schemes, which play important roles in homotopical classification of such objects.
2010 Mathematics Subject Classification: 18D35, 05E30
Key words and phrases.Association scheme, small category, schemoids, homotopy.
This research was partially supported by a Grant-in-Aid for Scientific Research HOUGA 25610002 from Japan Society for the Promotion of Science.
Department of Mathematical Sciences, Faculty of Science, Shinshu University, Matsumoto, Nagano 390-8621, Japan e-mail:[email protected]
1
Though association schemoids and their category ASmd are introduced in [4], we do not develop homotopy theory inASmdin this paper; see Appendix.
This manuscript is organized as follows. In Section 2, we recall the definition of the quasi-schemoid with examples. Section 3 explains a homotopy relation which we use in the category of quasi-schenoids. Section 4 is devoted to describing rigidity properties of the homotopy for association schemes and groupoids. In particular, our main theorem (Theorem 4.7) asserts that the group of self-homotopy equiva- lences on an quas-schemoid, which a finite groupoid induces, is isomorphic to the group of the autofunctors on the given groupoid.
2. A brief recollection on quasi-schemoids
We begin by recalling the definition of the association scheme. LetX be a finite set and S a partition of X ×X, namely a subset of the power set 2X×X with X×X =qσ∈Sσ, which contains the subset 1X :={(x, x)|x∈X}as an element.
Assume further that for each g ∈ S, the subset g∗ := {(y, x) | (x, y) ∈ g} is in S. Then the pair (X, S) is called anassociation schemeif for alle, f, g ∈S, there exists an integerpgef such that for any (x, z)∈g
pgef =]{y∈X |(x, y)∈eand (y, z)∈f}. Observe thatpgef is independent of the choice of (x, z)∈g.
Let G be a finite group. Define a subset Gf of G×G for f ∈ G by Gf :=
{(k, l)| k−1l = f}. Then we have an association scheme S(G) = (G,[G]), where [G] = {Gf}f∈G. Moreover, the correspondence S( ) induces a functor form the categoryGrof finite groups to the categoryASof association schemes in the sense of Hanaki [1]; see also [10, Section 5.5].
We here recall the definition of a quasi-schemoid.
Definition 2.1. ([4, Definition 2.1]) LetC be a small category; that is, the class of the objects of the categoryC is a set. LetS:={σl}l∈I be a partition of the set mor(C) of all morphisms in C. We call the pair (C, S) aquasi-schemoid if the set S satisfies the condition that for a tripleσ, τ, µ∈S and for any morphismsf,g in µ, as a set
(πστµ )−1(f)∼= (πµστ)−1(g),
whereπστµ :π−στ1(µ)→µis the map defined to be the restriction of the concatenation mapπστ :σ×ob(C)τ:={(f, g)∈σ×τ|s(f) =t(g)} →mor(C).
We denote bypµστ the cardinality of the set (πµστ)−1(f).
For an association scheme (X, S), we define a quasi-schemoid(X, S) by the pair (C, V) for whichob(C) =X, HomC(y, x) ={(x, y)} ⊂X×X andV =S, where the composite of morphisms (z, x) and (x, y) is defined by (z, x)◦(x, y) = (z, y).
For a groupoid H, we have a quas-schemoid S(eH) = (He, S) , where ob(He) = mor(H) and
HomHe(g, h) =
{{(h, g)} if t(h) =t(g)
∅ otherwise.
The partitionS ={Gf}f∈mor(H) is defined byGf ={(k, l)|k−1l =f}. We refer the reader to [4, Section 2] for examples of quasi-schemoids and more details.
Let (C, S) and (E, S0) be quasi-schemoids. It is readily seen that (C × E, S×S0) is a quasi-schemoid, whereS×S0 ={σ×τ | σ∈S, τ ∈S0} ⊂mor(C)×mor(E).
In what follows, we write (C, S)×(E, S0) for the product.
Definition 2.2. Let (C, S) and (E, S0) be quasi-schemoids. A functor F :C → E is amorphismof quasi-schemoids if for anyσin S,F(σ)⊂τ for someτ inS0. We then writeF : (C, S)→(E, S0) for the morphism.
We denote byqASmdthe category of quasi-schemoids and their morphisms. Let C be a small category and K(C) = (C, S) the discrete quasi-schemoid associated withC; that is, the partitionS is defined byS ={{f}}f∈mor(C). We then have a pair of adjoints U :qASmdoo //Cat:K for whichU is the forgetful functor andK is a fully faithful embedding functor which is the left adjoint toU; see [4, Remark 3.1] and [4, Diagram (6.1)].
The correspondences S( ) ande j mentioned above give rise to functors. With such functors, we obtain a commutative diagram of categories
(2.1) Gpd S( )e //qASmd
U //
Cat,
oo K
Gr
ı
OO
S( ) //AS
OO
whereGpddenotes the category of groupoids andı:Gr→Gpdis the natural fully faithful embedding; see [4, Section 3, Diagram (6.1)]. Observe that the composite U◦S( ) ise notthe usual embedding fromGpdtoCat.
The homotopy category ofCatwith respect to weak equivalences in the sense of Thomason is equivalent to that of topological spaces [9]. Moreover, the result [4, Theorem 3.2 (ii)] asserts thatSe( ) is faithful and hence so areS( ) and. Thus a quasi-schemoid is regarded asgeneralized spacesand asgenaralized groupsin some sense.
3. Strong homotopy
We extend the notion of the strong homotopy inCatin the sense of Hoff [3] and Lee [6] to that inqASmd. Let [1] be the category consisting of two objects 0 and 1 and the only one non-trivial morphism u: 0 →1. We write I for the discrete schemoidsK([1]).
Definition 3.1. LetF, G: (C, S)→(D, S0) be morphisms between the schemoids (C, S) and (D, S0) in qASmd. We write H : F ⇒ G if there exists a morphism H : (C, S)×I →(D, S0) inqASmd such thatH◦ε0=F and H◦ε1=G, where (C, S)×Iis the products of the quasi-schemoids mentioned above andεi : (C, S)→ (C, S)×Iis the morphism of the schemoids defined byεi(a) = (a, i) for an object ainC andεi(f) = (f,1i) for a morphismf in C.
A morphismF isequivalentto G, denotedF ∼G, ifH :F ⇒Gor H :G⇒F for someH : (C, S)×I→(D, S0) in qASmd.
Remark 3.2. Suppose that there exists a morphism H : (C, S)×I→(D, S0) such that H : F ⇒ G. Then for any morphism f ∈ mor(C), we have a commutative diagram
H(s(f),0)H(1s(f),u)//
H(f,u)
((R
RR RR RR RR RR
F(f)=H(f,10)
H(s(f),1)
H(f,11)=G(f)
H(t(f),0)
H(1t(f),u)//H(s(f),1)
in the underlying categoryD. Here we use the same notation as in Definition 3.1.
SinceH is a morphism of quasi-schemoids, it follows thatH(g, u) andH(h, u) are in the same element of S0 for anyg and hin the same element of S. We observe that, ineachsquare for a given morphismf, morphismsH(1s(f), u) andH(1t(f), u) are in the same element ofS0 if 1s(f) and 1t(f)are in the same element ofS as the case of an association scheme (X, S). As for the diagonal arrows, in order to define the homotopy inqASmd, we need to verify that the arrowH(f, u) in a square and H(g, u) inother squares are in the same element of S0 ifg in the same element of S as that containingf.
In what follows, we will define a homotopy assigning objects and morphisms in Dto those inC ×Ias in the square above.
Let F : (C, S) → (D, S0) be a morphism of quasi-schemoids. Then for any f :i→j inmor(C), we have a commutative diagram
F(i) F(1i) //
F(f)
&&
LL LL LL LL L
F(f)
F(i)
F(f)
F(j)
F(1j) //F(j)
in the underlying categoryD. If 1i and 1j are in the same element of S,F(1i) and F(1j) are in the same element ofS0. This implies thatF ∼F.
Definition 3.3. For morphisms F, G : (C, S) → (D, S0), F is homotopic to G, denotedF 'G, if there exists a finite sequence of morphismsF =F0, F1, ..., Fn=G such that Fk ∼Fk+1 for any k = 0, ..., n. We say thatF : (C, S)→ (D, S0) is a homotopy equivalence if there exists a morphism G : (D, S0) → (C, S) such that F G'1 andGF '1.
An usual argument deduces the following result.
Proposition 3.4. The homotopy relation'in the categoryqASmddefined in Def- inition 3.3 is an equivalence relation which is preserved by compositions of mor- phisms.
We denote by'S the homotopy relation, which is called the strong homotopy, in the categoryCatdue to Hoff [3], Lee [6] and Minian [7]. The relation is defined in the same way as in Definitions 3.1 and 3.3.
Proposition 3.5. Let F, G : (C, S) → (D, S0) be morphisms in qASmd. Then U(F)'S U(G) if F 'G. Assume further that (C, S) =K(C), namely a discrete schemoids. ThenF 'Gif and only ifU(F)'S U(G).
Proof. LetHbe a homotopy betweenFandG. SinceU((C, S)×I) =U((C, S))×[1], it follows that U(H) is a homotopy betweenU(F) andU(G). We have the former half of the results. Suppose that (C, S) =K(C). The forgetful functor U gives rise to a natural bijection
U : HomqASmd(K(C)×I,(E, S0)) = HomqASmd(K(C ×[1]),(E, S0))
∼=
→ HomCat(C ×[1], U((E, S0))).
This implies thatL:C ×[1]→U((E, S0)) is a homotopy betweenU(F) andU(G) ifU−1(L) is a homotopy betweenF andG. We have the result.
Let aut((C, S)) denote the monoid of self-homotopy equivalences on (C, S) in qASmd; that is, the composition of the equivalences gives rise to the product in the monoid. Then the monoid structure gives rise to a group structure on the set of equivalence classes
haut((C, S)) := aut((C, S))/'.
We observe that the grouphaut((C, S)) is a homotopy invariant for quasi-schemoids.
Proposition 3.5 enables us to deduce that the functorU induces a map Ue : [(C, S),(D, S0)] := HomqASmd((C, S),(E, S0))/' −→HomCat(C,E)/'S
which is a bijection if (C, S) is a discrete quasi-schemoid. Especially, the homomor- phism of groups
Ue :haut((C, S))−→haut(C),
is an isomorphism if (C, S) = K(C). Moreover, the composition of morphisms in qASmd gives rise to a left haut((D, S))-set structure and a right haut((C, S))-set structure on the homotopy set [(C, S),(D, S0)]. This follows from Proposition 3.4.
LetB :Cat→Topbe the functor which sends a small category to its classifying space. A natural transformation between functors F and G induces a homotopy between BF and BG. This enables us to conclude that B◦U induces a group homomorphism
ρ:haut((C, S))−→ E(BC),
whereE(X) denotes the homotopy set of self-homotopy equivalences on a spaceX. We here give an example of a contractible quasi-schemoid. Let C be a small category in whichσ:={φij : i→j}i,j∈ob(C) is the set of non-identity morphisms and the composite is given byφjk◦φij =φik. Let 1 be the set of all identity maps inC. Then it follows that (C, S={σ,1}) is a quasi-schemoids. In fact, it is readily seen that pσ1σ = 1, pσσ1 = 1, pσ11 = 0, p111 = 0, p11σ = 0, p1σ1 = 0 and p1σσ = 0.
Moreover, we see that a mapθ: (πσσσ)−1(φij)→ob(C) defined byθ((φkj, φik)) =k is bijective.
Proposition 3.6. The schemoid (C, S ={σ,1}) mentioned above is contractible;
that is, it is homotopy equivalent to the trivial schemoid(•,1).
Proof. Let 0 be an object ofC. We define a morphisms: (•,1)→(C, S) inqASmd bys(•) = 0. Letp: (C, S)→(•,1) be the trivial morphism. We define a homotopy H : (C, S)×I→(C, S) by
k φk0 //
φk0
>
>>
>>
>>
>
φkl
0
id0
l
φl0
//0
for anyφkl. Observe thatφk0 andφl0 are inσ for anykand l. Thus we see that
1C ∼sp. We have the result.
The following proposition gives a sufficient condition for a self-morphism between a quasi-schemoid (C, S) to be injective on the set of morphismsmor(C).
Proposition 3.7. Let F: (C, S)→(C, S)be a morphism of quasi-schemoid (C, S) which homotopy equivalent to the identity functor. Suppose that 1 = {1x}x∈ob(C)
is a subset of an element in the partition S and F(f) is an identity for some
non-identity element f ∈mor(C). Then there exist elements σ andτ such that τ contains a non-identity element andpσστ6= 0 orpστ σ6= 0.
Proof. By assumption, we have a sequence of morphisms F =F0 ∼ F1 ∼ · · · ∼ Fn−1 ∼Fn = 1C. Since F(f) is an identity, we see that Fl(f) is an identity and Fl+1(f) is not an identity for some l. Then the homotopy H which induces the relationFi ∼Fi+1 gives rise to a commutative diagram
sFl(f) φ //
Fl(f)=1
sFl+1(f)
Fl+1(f)
tFl(f)
φ0
//tFl+1(f)
or sFl(f)
Fl(f)=1
sFl+1(f)
oo φ
Fl+1(f)
tFl(f) tFl+1(f).
φ0
oo
Since 1 is a subset of an element in S, it follows that φ and φ0 are in the same elementσin the partitionS; see Remark 3.2. We choose an elementτ inS which contains the morphismFl+1(f). It turns out thatpσστ 6= 0 orpστ σ6= 0.
Remark 3.8. Let us consider a quasi-schemoid (C, S) whose underlying categoryC is define by the diagram
a β
''O
OO OO OO O
x ε //
αooooo77 oo o
γOOOOO'' OO
O y with βα=ε=δγ;
b δ
77p
pp pp pp pp
and whose partition S ={σ1, σ2, σ3,1} of mor(Cl) is given by σ1 = {α, γ}, σ2 = {β, δ}, σ3 ={ε} and 1 ={1x,1y,1a,1b}. A direct computation enables us deduce that pσστ = 0 and pστ σ = 0 for σ, τ ∈ S if τ 6= 1. Then Proposition 3.7 implies that every self homotopy-equivalence on (C, S) is an isomorphism and hence the quas-schemoid (C, S) is not contractible in qASmd but the underlying category U(C, S) =C is contractible in Catbecause C has the initial (terminal) object; see [5, (3.7) Proposition].
We conclude this section after describing a 2-category structure onqASmd.
Let Im be a discrete schemoid of the form K([m]). For morphisms F and G from (C, S) to (D, S0), if there exists an non-negative integer m and a morphism φ : (C, S)×Im → (E, S0) such that φ◦ε0 = F and φ◦εm = G, then we write φ:F ⇒mGor (C, S)
F ++
G
33
m φ (D, S0) when emphasizing the source and target of the functors. We call such a morphismφa homotopy fromF toG. Observe that there exists a homotopyφ:F ⇒mGif and only ifφ0:F ⇒F1, ...,φm:Fm−1⇒G for some functorsFiand morphismsφj; see Definition 3.1. Then we identifyφwith the compositeφm◦ · · · ◦φ0.
Theorem 3.9. The category qASmdof quas-schemoids admits a2-category struc- ture whose 2-morphisms are homotopies mentioned above and for which the fully faithful embeddingK:Cat→qASmdis a functor of 2-categories.
Proof. We see that the homsetA((C, S),(D, S0)) := HomqASmd((C, S),(D, S0)) is a category whose objects are morphisms from (C, S) to (D, S0) in qASmdand mor- phisms are homotopies. We observe that the composite ψ◦φ:F ⇒m+n Lof two
homotopiesφ: F ⇒m Gand ψ:G⇒n L to be the vertical composite of natural transformations. Moreover the interchanging low inCat enables us to deduce that the horizontal composition of the homotopies
(C, S)
F1
++
F2
33
m κ (D, S0) and (D, S0)
G1
++
G2
33
n ν (E, S00)
gives rise to a functor∗:A((D, S0),(E, S00))×A((C, S),(D, S0))→ A((C, S),(E, S00)).
In fact, the compositeν∗κis defined to be the vertical composite (νF2)◦(G1κ) of natural transformations, which coincides with the vertical composite (G2κ)◦(νF1).
Suppose that ν : G1 ⇒1 G2 is a homotopy in the sense of Definition 3.1. Since F2 preserves the partition, it follows from Remark 3.2 that νF2 : G1F2 ⇒ G2F2 is a well-defined homotopy in qASmd. Thus, in general, νF2 is the composite of homotopies in the sense of Definition 3.1. The same argument yields that G1κ is the composite of homotopies and hence so is ν ∗κ. It turns out that∗ is well
defined.
4. Rigidity of the homotopy for the trivial association schemes and groupoids
We first investigate the structure of the group of self-homotopy equivalences on a trivial association scheme.
Lemma 4.1. Let (X, S) be an association scheme with the trivial partition S = {1, σ}. Then every self-homotopy equivalence on (X, S)is an isomorphism.
Proof. The assertion is trivial if]X= 1. Assume that]X ≥2. We have a sequence of morphisms GF ∼ F1 ∼ · · · ∼ Fn ∼ 1C, where G is a homotopy inverse of F. Then there exists an integer l such that Fl+1 is injective and hence bijective on X but not Fl. Suppose that Fl(i) = x = Fl(j) for distinct elements i and j of X. Since Fl(φij) = 1x and Fl is a morphism of schemoids, it follows that Fl(f) = 1x for any f ∈ mor((X, S)). In fact, we see that Fl(φij ◦φt(f)i◦f) = Fl(φij)◦Fl(φt(f)i)◦Fl(f) = 1x◦1z◦1y for somez andy inX. Thenx=z=x.
LetH be a homotopy betweenFl andFl+1, sayH :Fl⇒Fl+1. We choose an object j0 with Fl+1(j0) =x. Then for a mapf :i0 →j0 which is not the identity, the homotopyH gives a commutative diagram
x
φxFl+1 (i0)
//
1x
Fl+1(i0)
Fl+1(f)
x
φxx=1x //x.
We see thatφxFl+1(i0)is in 1∈S and henceFl+1(i0) =x, which is a contradiction.
This completes the proof.
Remark4.2. An association scheme with the trivial partition is not contractible in general.
Lemma 4.3. Let (X, S) be an association scheme with the trivial partition S = {1, σ}andF, G:(X, S)→(X, S)self-homotopy equivalences. Suppose that]X≥ 3 andF ∼G. Then F=G.
Proof. In order to prove the lemma, it suffices to show that if there exists a ho- motopy H : F ⇒G, then F =G. The homotopy gives rise to the commutative diagram
F(i) φF(i)G(i)//
φF(i)G(j)
%%J
JJ JJ JJ JJ J
F(φij)
G(i)
G(φij)
F(j)
φF(j)G(j)//G(j), whereφij = (j, i)∈X×X.
Suppose thatF is different fromG. Assume further that there exists an object isuch thatF(i) =G(i). SinceF 6=G, it follows thatF(j)6=G(j) for somej. We see thatH(1i, u) =φF(i)G(i)= 1i ∈1 andH(1j, u) =φF(j)G(j)∈mor(C)\1, which is a contradiction; see Remark 3.2. This implies thatF(j)6=G(j) for anyj.
If there exists an element (i, j) ∈/ 1 such that F(i) = G(j), then H(φij, u) = φF(i)G(j) is in 1 and hence so is φF(k)G(l) for any (k, l) ∈/ 1. This yields that F(k) =G(l) for any (k, l)∈/ 1. Since]X ≥2, it follows thatG(1) =F(0) =G(2), which is a contradiction. In fact, by Lemma 4.1 the morphismGis an isomorphism.
In consequence, we see thatF(i)6=G(j) for anyiand j inX. Thus,F(0)6=G(i) for any i. The fact enables us to deduce that G is not surjective, which is a
contradiction. This completes the proof.
Theorem 4.4. Let(X, S)be an association scheme with the trivial partition. Then the group haut((X, S)) is isomorphic to the permutation group of order ]X if ]X ≥3. If]X= 2, thenhaut((X, S))is trivial.
Proof. The result for the case where]X≥3 follows from Lemmas 4.1 and 4.3.
Suppose that ]X = 2. Let Gbe the only non-identity isomorphism on(X, S).
Then we define a homotopyH : 1⇒Gby 0 φ01 //
φ01
>
>>
>>
>>
>
id0
1
id1
0 φ01 //1,
1 φ10 //
φ01
>
>>
>>
>>
>
id1
0
id0
1 φ10 //0,
0 φ01 //
id0
>
>>
>>
>>
>
φ01
1
φ01
1 φ10 //0,
1 φ10 //
id1
?
??
??
??
?
φ10
0
φ01
0
φ01
//1.
In each square, upper and lower horizontal arrows are in the same element of S.
In the first two squares, the diagonals are in the same element of S. The same condition holds for the second two squares. This implies that H is well defined;
that is,H is in a morphism inqASmd; see Remark 3.2. We have the result.
We consider the group of self-homotopy equivalences on the quasi-schemoid which comes from afinitegroupoid via the functorS( ).e
Theorem 4.5. For a finite groupoidG, every self-homotopy equivalence on a quasi- schemoid of the form S(e G) is an isomorphism and hence S(eG) is not contractible inqASmdif]mor(G)≥2.
Proof. Let F : S(e G) → S(e G) be a self-homotopy equivalence. In order to prove the theorem, it suffices to show that F is injective on ob(S(e G)) = mor(G). By assumption, there exists a homotopy inverse G of F. Then we have GF ' 1C. Suppose that F is not injective on the set ob(eS(G)). Since the composite GF is
not injective onob(S(e G)), it follows from Proposition 3.7 that there exist elements σ and τ such that τ contains a non-identity element and pσστ 6= 0 or pστ σ 6= 0.
Suppose thatpστ σ6= 0,σ=Glandτ=Gk. Then we see that there exist morphisms (f, g) : g → f and (h, g) : g → h in Gl and (h, f) : f → h in Gk. Therefore, it follows thath−1g =l, f−1g=l andh−1f =kand henceτ =G1x, wherex=t(l).
Since G1x ={(m, m) |m∈ mor(G), s(m) =x}, each element in τ is the identity, which is a contradiction. The same argument above is applicable to the case where
pσστ6= 0. We have the result.
Example4.6. For a non-trivial finite group, the schemoidU S(G) is contractible in Catbut notS(G) in qASmd.
The following result exhibits rigidity of the homotopy for the quasi-schemoid which a finite groupoid induces; see Section 2.
Theorem 4.7. Let G be a finite groupoid. Then the functor S( )e induces an isomorphism of groups
Se∗:Aut(G)→∼= haut(S(eG)), whereAut(G)denotes the group of autofunctors onG.
Corollary 4.8. LetG be a finite group. Thenhaut(S(G))∼=Aut(G)as a group.
Proof. The commutativity of the diagram (2.1) deduces the result.
Example4.9. SinceS(Z/2) is the trivial schmeme, it follows from Theorem 4.4 that haut(S(Z/2)) is trivial. On the other hand, Corollary 4.8 implies thathaut(S(Z/2)) is isomorphic to the groupAut(Z/2) which is trivial.
We need a lemma to prove Theorem 4.7.
Lemma 4.10. Let G and H be groupoids, which are not necessarily finite. Let φ, ψ :S(e G)→S(e H) be morphisms of quasi-schemoids. Then φ'ψ if and only if ψ(j)−1φ(i) =ψ(l)−1φ(k)for any (j, i)and(l, k)in mor(Ge)with j−1i=l−1k.
Proof. We recall that in the categoryS(e G),f = (j, i) is a unique morphism fromi to j. Suppose that there exists a homotopyL:φ⇒ψbetween morphisms φand ψfrom S(eG) toS(e H). Then for any morphismf :i→j and g:k→l inS(eG), we have commutative diagrams inS(e H)
φ(i) L(1i,u) //
L(f,u)
$$I
II II II II
φ(f)
ψ(i)
ψ(f)
φ(j) L(1j,u)//ψ(j)
and φ(k) L(1k,u)//
L(g,u)
$$I
II II II II I
φ(g)
ψ(k)
ψ(g)
φ(l) L(1l,u) //ψ(l).
Observe that 1i = (i, i) ∈ G1s(i) for any i and that L(f, u) = (ψ(j), φ(i)). By definition, morphisms f and g are in the same element Gh of S if and only if j−1i=h=l−1k. Thus ifj−1i=h=l−1k, thenL(f, u) andL(g, u) are in the same elementHh0 for someh0∈mor(H). Therefore, we have the necessary condition for φ'ψ.
Suppose thatψ(j)−1φ(i) =ψ(l)−1φ(k) for any (j, i) and (l, k) inmor(S(e G)) with j−1i=l−1k. Then the mapL:S(e G)×I→S(e H) defined by the squares above is
a well-defined homotopy. We haveL:φ⇒ψ.
Remark 4.11. Letφ, ψ : S(e G)→ S(e H) be morphisms of schemoids which are in- duced by groupoids. Then the proof of Lemma 4.10 implies thatφ∼ψif and only ifφ'φ.
Proof of Theorem 4.7. We show that the homomorphismSe∗:Aut(G)→haut(S(e G)) defined by Se∗(u) = [S(u)] is an isomorphism. Since (e S(u))(i) =e u(i) by definition, it follows from Lemma 4.10 that u=v if S(u)e 'S(v). In fact, for anye i, we see thatu(i)−1v(i) =u(1s(i))−1v(1s(i)) = 1x1y for somexandyinob(G). Then 1xand 1y should be composable. This yields thatSe∗ is a monomorphism.
Let u be an element in aut(S(e G)). Then Theorem 4.5 implies that u is an autofunctor. We define a self-functoru0 onS(e G) by
u0(i) =u(i)u(1s(i))−1
for any i∈ob(S(eG)) =mor(G). Observe thatu(1s(i))−1 and u(i) are composable.
In fact, we haves(u(i)) =s(u(1s(i))) by [4, Claim 3.3].
We show thatu0 is an autofunctor; that is,u0 is bijective onob(S(e G)) =mor(G) and for any k ∈ mor(G), there exists l(k) ∈ mor(G) such that u0(Gk) ⊂ Gl(k). Suppose thatu0(i) =u0(i). Thent(u(1s(i))) =s(u0(i)) =s(u0(j)) =t(u(1s(i))). We see that the pair (u(1s(i)), u(1s(j))) is a morphism in S(e G) and hence (1s(i),1s(j)) is in S(e G). Observe that u has the inverse. Thus it follow thats(i) = s(j) and u(i) =u(j). We havei=j. This implies that u0 is bijective on ob(S(e G)) because mor(G) is finite.
Since uis a morphism of quasi-schemoids, it follows that for any k∈ mor(G), there exists l(k)0 ∈mor(G) such thatu(Gk)⊂ Gl(k)0. Suppose that (i, j) is in Gk. By definition, we havei−1j =k. Thens(i) =t(k) and s(k) =s(j). Moreover, it follows that (u0(i))−1u0(j) =u(1s(i))u(i)−1u(j)u(1s(j))−1=u(1t(k))l(k)0u(1s(k))−1. We can choose the last element asl(k)0mentioned above. Furthermore, we see that the autofunctoru0preserves the setGe◦={1x,|x∈ob(G)}, which is the set of base points ofS(e G); see [4, Section 3].
Let (j, i) and (l, k) be morphisms inGewhich are in the same elementGhfor some hinmor(G). Then we see thatj−1i=h=l−1kand hences(i) =s(k). Moreover, since u is a morphism in qASmd, it follows that there exists h0 ∈ mor(G) such that (u(j), u(i)) and (u(l), u(k)) are in the same elementGh0; that is, u(j)−1u(i) = u(l)−1u(k). Thus we have
u(j)−1u0(i) =u(j)−1u(i)u(1s(i))−1=u(l)−1u(k)u(1s(k))−1=u(l)−1u0(k).
Then Lemma 4.10 yields that u is homotopy equivalent to u0, which is a base points preserving automorphism, inqASmd. Let (qASmd)0be the category of quas- schemoids with base points. The result [4, Corollary 3.5] asserts that the functor Se : Gpd → (qASmd)0 is fully faithful. This enables us to conclude that Se∗ is
surjective. This completes the proof.
Acknowledgements. The author thanks Kentaro Matsuo who pointed out a mistake in the proof of Lemma 4.3 in a preliminary version of this manuscript.
5. Appendix
We refer the reader to [4, Section 2] for the definition of association schemoids and their categoryASmd. In this section, we consider a homotopy relation inASmd
with acylinderobtained by modifying the quasi-schemoid I= ([1], s) =K([1]) in Definition 3.1. Unfortunately, the result is trivial; see Assertion 5.1 below. Thus we need other cylinder to develop more interesting homotopy theory onASmd.
Lett: [1]→[1] be a contravariant functor defined byt(0) = 1 andt(1) = 0. Then Ie:= ([1], s, t) is an association schemoid. Observe that this is a unique association schemoid structure on the discrete schemoid I. Let F, G : (C, S, T)→ (D, S0, T0) be morphisms in ASmd. Then it is natural to define a homotopy relation F ∼G in ASmd by replacing the category qASmd with ASmd in Definition 3.1. More precisely, we writeF ∼Gif there exists a morphismH: (C, S, T)×Ie→(D, S0, T0) inASmdsuch thatH :F ⇒GorH :G⇒F; see Remark 3.2.
Assertion 5.1. LetF andGbe morphisms of association schemoids from (C, S, T) to (D, S0, T0). ThenF ∼Gif and only if F =G.
Proof. In order to prove the assertion, it sufficies to show that ifH : F ⇒ Gfor some H : (C, S, T)×Ie→(D, S0, T0) in ASmd, then F = G. Since the homotopy H is a morphism of association schmeoids, it follows that H ◦(T ×t) = T0H. The morphism H is a homotopy from F to G. Then F(f) = H(f,10) for any f ∈mor(C). This implies that
T0F(f) =T0H(f,10) = (H◦(T0×T))(f,10) =H(T(f),11) =GT(f) =T0G(f).
and henceF(f) =G(f) because (T0)2=T0 by definition. We have the result.
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