Homotopy Classiﬁcation of Categorical Torsorshera.ugr.es/doi/15028495.pdf · categorical groups,...

Applied Categorical Structures 9: 465–496, 2001.© 2001 Kluwer Academic Publishers. Printed in the Netherlands.

465

Homotopy Classification of Categorical Torsors

ANTONIO M. CEGARRA and ANTONIO R. GARZÓN�

Departamento de Álgebra, Univ. Granada, 18071 Granada, Spain.E-mail: [email protected]

(Received: 7 July 1998; accepted: 25 May 1999)

Abstract. The long-known results of Schreier on group extensions are here raised to a categor-ical level by giving a factor set theory for torsors under a categorical group (G,⊗) over a smallcategory B. We show a natural bijection between the set of equivalence classes of such torsorsand [B(B),B(G,⊗)], the set of homotopy classes of continuous maps between the correspondingclassifying spaces. These results are applied to algebraically interpret the set of homotopy classes ofmaps from a CW-complex X to a path-connected CW-complex Y with πi(Y ) = 0 for all i �= 1, 2.

Mathematics Subject Classifications (2000): 18D50, 18D30, 18G50, 55P57, 55U10.

Key words: categorical group, small category, torsor, classifying space, homotopy classes.

1. Introduction

A categorical group G = (G,⊗) is a monoidal groupoid in which each object hasa quasi-inverse with respect to the tensor product. If B is a small category, a torsorover B under a categorical group G is defined as a (Grothendieck) cofibrationp: E → B, such that for any object A ∈ B the fibre category EA is equivalent toG, via a given action of G on E . Thus, central group extensions are instances ofsuch torsors when groups are regarded as categories with only one object.

The main object of this work is to classify all equivalence classes of torsorsover a category B under a categorical group G, and to point out the topologicalsignificance of the result. This is carried out in the same way in which equiva-lence classes of central group extensions of a group G by an abelian group A areclassified by the cohomology group H 2(G,A), and then they are shown in one-to-one correspondence with homotopy classes of maps from an Eilenberg–Mac Lanespace of type K(G, 1) to an Eilenberg–Mac Lane space of type K(A, 2).

The cohomological classification of categorical torsors is done extending theSchreier analysis of extensions of groups, such as Breen in [2], Ulbrich in [34] orCarrasco and Cegarra in [9] did to solve similar classification problems. We explaina factor set theory for categorical torsors and conclude that, for each category Band categorical group G, torsors are classified by a cohomology set H

2(B,G)

which, for example, reduces to the usual cohomology of a small category [29] (and

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466 ANTONIO M. CEGARRA AND ANTONIO R.GARZON

therefore of a group) when abelian groups are taken as coefficients or, for example,to Dedecker’s non-abelian cohomology of groups [12], when a group is regardedas a small category and the coefficients are those categorical groups defined bycrossed modules [6]. We should point out that an alternative sheaf-theoretic proofof this classification theorem has been given by L. Breen in [1], where only strictcategorical groups, that is, crossed modules, are considered.

The topological classification of categorical torsors arises, on the one hand, dueto the fact that small categories are algebraic homotopy types of CW-complexes[28, 19], and, on the other hand, because categorical groups are algebraic homotopy2-types of path-connected CW-complexes [33, 6, 25, 10]. There is a nerve functorfrom the category of small categories to the category of simplicial sets; taking thegeometrical realization of the nerve Ner(B) of a category B, one produces theclassifying space B(B) and any CW-complex is homotopy equivalent to B(B) forsome small category B. There is a nerve functor from the category of categoricalgroups to the category of simplicial sets and this functor, composed with the geo-metrical realization functor, defines the classifying space B(G,⊗) of a categoricalgroup G. This is a path-connected CW-complex with trivial homotopy groups indimensions other than 1 and 2. Furthermore, any path-connected CW-complex X

with πj(X) = 0 for j > 2 is of the homotopy type of B(G,⊗) for a categoricalgroup G. Our main result here is that equivalence classes of torsors over a smallcategory B under a categorical group G are in one-to-one correspondence withhomotopy classes of continuous maps from a CW-complex with the homotopy typeof category B to a CW-complex with the homotopy type of categorical group G.

2. Categorical Group Actions

Monoidal categories and, in particular, categorical groups have been studied exten-sively in the literature and we refer to [24, 30, 21, 22] or [33] for the background.To fix the notation throughout the paper, let us recall that a monoidal categoryG = (G,⊗, a, I, l, r) consists of a category G, a functor ⊗: G × G → G, anobject I and natural isomorphisms

a = aX,Y,Z: X ⊗ (Y ⊗ Z)∼−→ (X ⊗ Y )⊗ Z,

l = lX: I ⊗X∼−→ X, r = rX: X ⊗ I

∼−→ X

(1)

such that for any objects X,Y,Z, T ∈ G the following diagrams commute

HOMOTOPY CLASSIFICATION OF CATEGORICAL TORSORS 467

X ⊗ (Y ⊗ (Z ⊗ T ))1⊗ a

a

X ⊗ ((Y ⊗ Z) ⊗ T )

a

(X ⊗ Y )⊗ (Z ⊗ T )

a

(X ⊗ (Y ⊗ Z))⊗ T

a⊗1

((X ⊗ Y )⊗ Z) ⊗ T

(2)X ⊗ (I ⊗ Y )

a

1⊗r

(X ⊗ I ) ⊗ Y

l⊗1

X ⊗ Y

In a monoidal category, an object X is 2-regular if the functors Y �→ X ⊗ Y

and Y �→ Y ⊗ X are equivalences. A categorical group is a monoidal category inwhich every arrow is invertible, that is, it is a monoidal groupoid, and every objectis 2-regular. In a monoidal groupoid, the necessary and sufficient condition thatevery object be regular is that for any object X, there exist another object X∗ andan isomorphism X ⊗ X∗ → I .

Suppose G and H are categorical groups. A homomorphism T = (T , µ):G → H consists of a functor T : G → H and a family of natural isomorphisms

µ = µX,Y : T (X ⊗ Y ) −→ T (X) ⊗ T (Y ), (3)

such that for all objects X,Y,Z in G the following diagram commutes

T ((X ⊗ Y) ⊗ Z)µ

T (X ⊗ Y) ⊗ T (Z)

µ⊗1

T (X ⊗ (Y ⊗ Z))

T (a)

µ

(T (X) ⊗ T (Y )) ⊗ T (Z)

T (X) ⊗ T (Y ⊗ Z)1⊗µ

T (X) ⊗ (T (Y ) ⊗ T (Z))

a

(4)

If T : G → H is a homomorphism, there exists an (unique) isomorphism

µ0: T (I )∼−→ I, (5)

such that the following two diagrams commute [30],

T (X ⊗ I )

µ

T (r)T (X)

T (X) ⊗ T (I )1⊗µ0

T (X) ⊗ I

r

T (I ⊗ X)

µ

T (l)T (X)

T (I ) ⊗ T (X)µ0⊗1

I ⊗ T (X)

l (6)


Given homomorphisms of categorical groups (T , µ), (T ′µ′): G → H, a ho-motopy (or morphism) from (T , µ) to (T ′, µ′) consists of a natural transformationε: T → T ′ such that, for any objects X,Y ∈ G, the following diagram commutes:

T (X ⊗ Y )µ

ε

T (X)⊗ T (Y )

ε⊗ε

T ′(X ⊗ Y )µ′

T ′(X) ⊗ T ′(Y ).

Thus we have a 2-category whose objects are the categorical groups, whose arrowsare the homomorphisms and whose 2-arrows or deformations are the homotopies.

EXAMPLE 2.1. The categorical group of self-equivalences in a category, Eq(C).Let C be a category. The objects of category Eq(C) are the equivalences F : C

→ C and the morphisms are the natural equivalences τ : F → G. The compositionin Eq(C) is given by the usual vertical composition of natural transformations:(σ τ)X = σXτX. It is clear that Eq(C) is a groupoid. The composition of the func-tors and the horizontal composition of the natural transformations define a tensorfunctor ⊗: Eq(C)× Eq(C) → Eq(C), that is, given τ : F → G and τ ′: F ′ → G′,then τ ′ ⊗ τ : F ′F → G′G is defined by (τ ′ ⊗ τ)X = τ ′

GXF′(τX) = G′(τX)τ ′

FX.Note that ⊗ is a functor as a consequence of the well-known Godement rules.

Thus, Eq(C) = (Eq(C),⊗, 1, 1C, 1, 1) is a categorical group in which theassociativity, left unit and right unit constraints are identity arrows.

EXAMPLE 2.2. A categorical group is deemed to be strict if the isomorphisms(1) are identities and the isomorphisms X ⊗ X∗ → I can be chosen to be anidentity. Any categorical group is equivalent to a strict categorical group. This fact,commonly assumed in the literature, can be obtained, for example, as consequenceof Proposition 1.5 and Theorem 2.6 in [10].

It is well known that strict categorical groups or, equivalently, groupoids in thecategory of groups, are the same as Whitehead crossed modules, [6]. Recall that acrossed module of groups is a system � = (H, π, ϕ, ρ), where ρ: H → π is agroup homomorphism and ϕ: π → Aut(H) is an action (so that π is an H -group)for which the following conditions are satisfied:

ρ(ah) = aρ(h)a−1; ρ(h)h′ = hh′h−1.

Given a crossed module �, the corresponding strict categorical group G(�)

can be described as follows: The objects are the elements of the group π ; an arrowh: a → b is an element h ∈ H with a = ρ(h)b. The composition is multiplicationin H . The tensor product is given by

(ah−→ b) ⊗ (c

h′−→ d) = (achbh′−→ bd).


EXAMPLE 2.3. Suppose T : G → H is an equivalence of groupoids. Then eachcategorical group structure on G transports along T to a categorical group structureon H. This in particular applies to the case when G is a strict categorical group andin this way one can obtain many algebraic examples of categorical groups.

EXAMPLE 2.4. The categorical group of loops of a pointed space, P2(X, ∗).Let us denote by P1(Y ) the fundamental groupoid of a topological space Y .If (X, ∗) is a pointed topological space with base point ∗ ∈ X, then P2(X, ∗) =

P1(%(X, ∗)), that is, the fundamental groupoid of the loop space %(X, ∗). Thus,the objects are the maps ω: I → X such that ω(0) = ∗ = ω(1), and the morphisms[f ]: ω → ω′ are homotopy classes rel end loops of homotopies f : ω → ω′ relend points. The composition of two morphisms in P2(X, ∗), [f ]: ω → ω′ and[g]: ω′ → ω′′ is defined by [g][f ] = [gf ], where gf : I × I → X is the map

(gf )(t, s) ={f (t, 2s), 0 ≤ s ≤ 1/2,

g(t, 2s − 1), 1/2 ≤ s ≤ 1.

Since the functor P1 preserves products, the map µ: %(X, ∗) × %(X, ∗) →%(X, ∗) defined by

µ(ω,ω′) ={ω(2t), 0 ≤ t ≤ 1/2,

ω′(2t − 1), 1/2 ≤ t ≤ 1(7)

induces a functor ⊗: P2(X, ∗) × P2(X, ∗) → P2(X, ∗) that is given on objects byω ⊗ ω′ = µ(ω,ω′) and, on morphisms, by [f ] ⊗ [g] = [f ⊗ g] where

(f ⊗ g)(t, s) ={f (2t, s), 0 ≤ t ≤ 1/2,

g(2t − 1, s), 1/2 ≤ t ≤ 1.

There is an associativity constraint a: (ω ⊗ ω′) ⊗ γ → ω ⊗ (ω′ ⊗ γ ) which isdefined as the homotopy class of the map A: I × I → X given by

A(u, v) =

ω( 4u

v+1), 0 ≤ u ≤ v+14 ,

ω′(4u− v − 1), v+14 ≤ u ≤ v+2

4 ,

γ ( 4u−2−v2−v

), v+24 ≤ u ≤ 1

(8)

and there are a unit object ∗, which is the constant map from I to ∗, and unitconstraints

l = [L]: ∗ ⊗ γ → γ, r = [R]: γ ⊗ ∗ → γ, (9)

where L,R: I × I → X are respectively defined by

L(u, v) ={

∗, 0 ≤ u ≤ 1−v2 ,

γ ( 2u+v−11+v

), 1−v2 ≤ u ≤ 1

(10)


and

R(u, v) ={γ ( 2u

v+1), 0 ≤ u ≤ v+12 ,

∗ v+12 ≤ u ≤ 1,

(11)

such that P2(X, ∗) = (P2(X),⊗, a, ∗, l, r) is a categorical group. Note that thegroup of connected components of P2(X, ∗) is π1(X, ∗) and the group of automor-phisms in the unit object is π2(X, ∗).

Note that in [10] we defined, for n ≥ 1, homotopy groupoids Pn(K, ∗) of apointed Kan complex (K, ∗). The first homotopy groupoid is the Poincaré groupoidof K, the second one is enriched with a categorical group structure, the third one isa braided categorical group and the higher ones are commutative Picard categories.If X is a pointed topological space and we consider S(X), the pointed total singularcomplex of X, the categorical groups P2(X, ∗) and P2(S(X), ∗) can be identified.

Let C be a category and G a categorical group. A G-action on C, see [20], con-sists of a functor G×C

ac−→ C, (X,A) �→ XA, together with natural isomorphisms

φ = φX,Y,A: (X⊗Y )A∼−→ X(YA), (12)

φ0 = φ0,A: IA∼−→ A, (13)

such that, for all objects X,Y,Z ∈ G and A ∈ C, the following diagrams commute

(X⊗(Y⊗Z))A

aA

φX((Y⊗Z)A)

Xφ′

((X⊗Y )⊗Z))A

φ

X(Y (ZA))

(X⊗Y )(ZA)

φ

(14)(X⊗I )A

φ

rA

X(IA)

Xφ0

XA

When there is a G-action on C, each object X ∈ G defines an equivalence

4(X): C → C, 4(X)(Af→ B) = Xf : XA → XB, and each morphism u: X → Y

in G defines a natural equivalence 4(u): 4(X) → 4(Y ), 4(u)A = uA: XA → YA.Thus, there is a homomorphism of categorical groups

4 = (4, φ): G → Eq(C), (15)


where φX,Y : 4(X ⊗ Y ) → 4(X)4(Y ) is given by (φX,Y )A = φX,Y,A. Conversely,given any category C and any homomorphism (4, φ) of G into Eq(C), we candefine a G-action on C by making XA = 4(X)(A) for each pair of objects X ∈ G

and A ∈ C, and uf : XA → YB as the diagonal of the commutative diagram

XA4(X)(f )

uf4(u)A

XB

4(u)B

YA 4(Y )(f)YB

for each pair of morphisms u: X → Y in G and f : A → B in C. Also, for anyobjects X,Y ∈ G and A ∈ C, we make φX,Y,A = (φX,Y )A: X⊗YA → X(YA) andφ0,A = (φ0)A: IA → A where φ0: 4(I) → 1C is the natural isomorphism (5)defined by the homomorphism (4, φ).

Therefore, to define a G-action on a category C is equivalent to giving a homo-morphism of G into the categorical group of self-equivalences in C.

EXAMPLE 2.5. Let G be a group and S any set and consider dis(G) and dis(S)the discrete categories they define. The multiplication inG determines a categoricalgroup dis(G) = (dis(G), ·, 1, e, 1, 1) and a dis(G)-action on dis(S) is the samething as a group action of G on S in the usual sense.

EXAMPLE 2.6. Let C be a category. Then, Eq(C) acts on C by the action cor-responding to the identity homomorphism 1Eq(C): Eq(C) → Eq(C), i.e. FA =F(A).

EXAMPLE 2.7. If C is any category and G any categorical group, we can definea G-action on C by making XA = A for each pair of objects X ∈ G and A ∈ C,and uf = f for any morphisms u in G and f in C; the natural isomorphisms φand φ0 are the identities. This G-action corresponds to the zero homomorphismG → Eq(C) and is called the trivial G-action.

EXAMPLE 2.8. Let G be any categorical group. We can let G act on itself byleft tensor multiplication, i.e., the G-action on G in which XY = X ⊗ Y , uv =u ⊗ v, and the natural isomorphisms φ and φ0 are given by the associativity andleft constraints. This is called the regular left action on G.

EXAMPLE 2.9. The P2(Y )-action on the track groupoid P1(YX).

If X and Y are spaces with base point ∗, then P1(YX) is the track groupoid

defined in [3], that is, the objects are the maps f : X → Y of spaces with base point


and the morphisms [4]: f → g are homotopy classes rel end maps of homotopies4: f → g rel base point.

Suppose thatX has non degenerated base point ∗. Then, for each map f : X→Y

and each loop ω ∈ %(Y ), we can choose a map 4f,ω: I × X → Y such that4f,ω(0, x) = f (x) and 4f,ω(t, ∗) = ω(t). We define ωf : X → Y by (ωf )(x) =4f,ω(1, x); thus 4f,ω is a free homotopy of f to ωf along ω.

The correspondence (ω, f ) �→ ωf extends to a functor P2(Y ) × P1(YX) →

P1(YX) that is defined on morphisms as follows: Let h: I × I → Y be a homo-

topy of ω to ω′ rel{0, 1} and let 4: I × X → Y be a homotopy of f to f ′ rel{∗},since the pair (I ×X, I × {∗} ∪ {0, 1} × X) = (I, {0, 1}) × (X, ∗) has the homo-topy extension property, the map from the subspace {0, 1} × I × X ∪ I × {0, 1}×X ∪ I × I × {∗} into Y defined by H(0, s, x) = 4(s, x), H(t, s, ∗) = h(t, s),H(t, 0, x) = 4f,ω(t, x) and H(t, 1, x) = 4f ′,ω′(t, x), has an extension H : I ×I ×X → Y . Let h7(s, x) = H(1, s, x); then h7: I ×X → Y is a homotopy of ωf toω′f ′ rel{∗} whose homotopy class depends only on the homotopy classes [h] and

[4] of h and 4 respectively. Then we define [h][4] = [h4]: ωf → ω′f ′.

To define the natural isomorphism φ: (ω⊗ω′)f → ω(ω′f ) for each pair of loops

ω,ω′ ∈ %(Y ) and each map f : X → Y , let us consider the map H : {0} × I × X

∪I × {0, 1} × X ∪ I × I × {∗} → Y defined by H(0, s, x) = f (x), H(t, 0, x) =4f,ω⊗ω′ (t, x),

H(t, 1, x) ={4f,ω′(2t, x), 0 ≤ t ≤ 1/2,4ω′f,ω(2t − 1, x), 1/2 ≤ t ≤ 1

and H(t, s, ∗) = ω(t). This map has an extension H : I × I ×X → Y . Let ϕ(s, x)= H(1, s, x); then ϕ: I × X → Y is a homotopy of (ω⊗ω′)f to ω(ω

′f ) rel{∗} and

we define φ = [ϕ]: (ω⊗ω′)f → ω(ω′f ).

Finally, for any map f : X → Y , from the homotopy extension property we caneasily deduce the existence of a map H : I × I × X → Y such that H(0, s, x) =f (x), H(t, 0, x) = 4f,∗(t, x), H(t, 1, x) = f (x) and H(t, s, ∗) = ∗; this is usedto define the natural isomorphism φ0: ∗f → f as the homotopy class of the mapϕ0(s, x) = H(1, s, x).

Let us note that the P2(Y )-action on P1(YX) induces, on the connected compo-

nents, the standard action of group π1(Y ) on the set [X,Y ] of homotopy classes ofmaps from X to Y .

EXAMPLE 2.10. The P2(X, ∗)-action on the groupoid P2(X,A, ∗).Given a subspace A ⊆ X and a base point ∗ ∈ A we define %(X,A, ∗) = {γ ∈

XI/γ (0) = ∗, γ (1) ∈ A} ⊆ XI with the induced topology and let P2(X,A, ∗) =P1(%(X,A, ∗)), the fundamental groupoid of %(X,A, ∗). Then the objects arethose paths in X with origin ∗ and ending in A, and the morphisms [4]: γ → γ ′are homotopy classes rel end paths of homotopies 4: I × I → X of γ to γ ′ suchthat 4(0, s) = ∗ and 4(1, s) ∈ A.

The map µ: %(X, ∗) × %(X,A, ∗) → %(X,A, ∗), defined as in (7), inducesa functor between the corresponding fundamental groupoids


⊗: P2(X, ∗) × P2(X,A, ∗) → P2(X,A, ∗),which is given on objects by (ω, γ ) �→ ω⊗ γ , the product path of ω and γ , and onmorphisms by [f ] ⊗ [γ ] = [f ⊗ γ ], where

(f ⊗ γ )(t, s) ={f (2t, s), 0 ≤ t ≤ 1/2,γ (2t − 1, s), 1/2 ≤ t ≤ 1.

The constraints φ: (ω⊗ω′)⊗ γ → ω⊗ (ω′ ⊗ γ ) and φ0: ∗γ → γ are definedas the homotopy classes of the maps A: I × I → X and L: I × I → X given in(8) and (10).

Let G be a categorical group and suppose that G acts on categories C and D . Afunctor G- equivariant T = (T , µ) = C → D consists of a functor T : C → Dand a family of natural isomorphisms

µ = µX,A: T (XA) −→ XT (A), (16)

such that for all objects X,Y ∈ G and A ∈ C the following diagrams commute:

T ((X⊗Y )A)µ

T (φ)

(X⊗Y )T (A)

φ

T (X(YA))

µ

X(Y (T (A)))

X(T (YA))

XT (µ)

(17)

T (IA)µ

T (φ0)

IT (A))

φ0

T (A)

3. Torsors under Categorical Groups

When a functor PE : E → B is given, category E is called a B-category and PE

the projection functor. A B-functor from a B-category E to a B-category F is afunctor F : E → F such that PF F = PE and a B-natural transformation (or B-homomorphism) of a B-functor F : E → F to a B-functor G: E → F is a naturaltransformation τ : F → G such that for any object ξ ∈ E , PF (τξ ) = 1PE (ξ) [18].

If E is a B-category, the categorical group of self-equivalences in E (see Ex-ample 2.1) contains as a categorical subgroup the one of B-self-equivalences,


Eq(E/B) ⊆ Eq(E), and we say that a categorical group G acts on the B-categoryE if we have a G-action on E such that the associated representation homomor-phism (15) factors through Eq(E/B). Then, a G-action on a B-category E consistsof a functor G × E

ac−→ E , (X, ξ) �→ Xξ , making the diagram

G × Eac

pr

E

PE

EPE

B

commutative, together with natural isomorphisms φ: (X⊗Y )ξ → X(Y ξ) and φ0: I ξ

→ ξ satisfying PE (φ) = 1PE (ξ) = PE (φ0) and such that diagrams (14) are commu-tative.

If G acts on B-categories E and F , a B-functor G-equivariant of E to F is afunctor G-equivariant T = (T , µ): E → F , which is a B-functor and such thatPF (µX,ξ ) = 1PE (ξ) for all objects X ∈ G and ξ ∈ E .

We now make the following:

DEFINITION 3.1. Let G be a categorical group. By a pseudo-torsor under G

over a category B (or a B-pseudo-torsor under G) we shall mean a B-categoryPE : E → B endowed with a simply-transitive G-action, in the sense that the in-

duced functor G × E(ac,pr)−−−−→ E ×B E is an equivalence of categories.

By a morphism between pseudo-torsors under G over B we mean a B-functorG-equivariant.

The next proposition gives a better description of the structure of a pseudo-torsor under a categorical group G showing, in particular, that every non-emptyfibre category is equivalent to G. We shall first recall some terminology relatedwith the concept of (Grothendieck) fibred categories [18, 17, 13].

Let E be a B-category. For any object A ∈ B, EA denotes the fibre categoryover A: its objects, called A-objects, are the elements of P−1

E (A); its morphisms,called A-morphisms, are the elements of P−1

E (1A). More generally, if f : A → B

is a morphism in B, a morphism in E , α: ξ → η, such that PE (α) = f is called anf -morphism; if ξ is an A-object and η is a B-object, Homf (ξ, η) denotes the setof all f -morphisms of ξ to η.

If f : A → B is a morphism in B, an f -morphism α: ξ → η is called cartesianif for any other f -morphism whose target is η, α′: ξ ′ → η, there exists a uniqueA-morphism β: ξ ′ → ξ such that αβ = α′, that is, α is cartesian if for anyA-object ξ ′ the map α∗: HomA(ξ

′, ξ ) → Homf (ξ′, η) is bijective. Dually, an

f -morphism α: ξ → η is termed co-cartesian if for any B-object η′ the mapα∗: HomB(η, η

′) → Homf (ξ, η′) is a bijection. If a morphism is cartesian and

co-cartesian it is called bi-cartesian.The B-category E is called a (co)fibred category (or PE : E → B a (co)fibration)

provided that for any morphism f : A → B in B and any B-object η (resp. A-


object ξ ) there exists a (co-)cartesian f -morphism with target η (source ξ ) and,moreover, the composition of (co-)cartesian morphisms is again (co-)cartesian.If the B-category E is fibred and cofibred it is called bifibred and the projectionfunctor PE is termed a bifibration.

PROPOSITION 3.2. Suppose that a categorical group G acts on a B-category E .The following statements are equivalent:

(i) E is a pseudo-torsor.(ii) (a) For any object A ∈ B and any two A-objects ξ, ξ ′, there exists an object

X ∈ G and an A-isomorphism Xξ∼−→ ξ ′.

(b) For any morphism f : A → B in B, any f -morphism α: ξ → η and anytwo objects X,Y ∈ G, the map HomG(X, Y ) → Homf (

Xξ, Y η), u �→ uα, isa bijection.

(iii) Every morphism in E is cartesian and for any object A of B and any A-objectξ , the functor ρξ : G → EA, X �→ Xξ , is an equivalence.

(iv) Every morphism in E is cartesian and for any object A of B, either EA isempty or there exists an A-object ξ such that ρξ : G → EA, X �→ Xξ , is anequivalence.

(i) Every morphism in E is co-cartesian and for any object A of B and anyA-object ξ the functor ρξ : G → EA, X �→ Xξ , is an equivalence.

(vi) Every morphism in E is co-cartesian and for any object A of B, either EA isempty or there exists an A-object ξ such that ρξ : G → EA, X �→ Xξ , is anequivalence.

Proof. The functor (ac, pr): G × E → E ×B E is full and faithful if forany two objects (X, ξ), (Y, η) ∈ G × E the map HomG×E ((X, ξ), (Y, η)) →HomE×BE((

Xξ, ξ), (Y η, η)), (u, α) �→ uα, is bijective. Suppose PE (ξ) = A andPE (η) = B; then the set HomE (ξ, η) is the disjoint union of the sets Homf (ξ, η)

with f ∈ HomB(A,B), and also HomG×E ((X, ξ), (Y, η)) = ⋃f (HomG(X, Y ) ×

Homf (ξ, η)) and HomE×BE ((Xξ, ξ), (Y η, η))= ⋃

f (Homf (Xξ, Y η)×Homf (ξ, η)).

Thus, (ac, pr) is full and faithful if and only if for any morphism in B, f : A → B,the map HomG(X, Y ) × Homf (ξ, η) → Homf (

Xξ, Y η) × Homf (ξ, η), (u, α) �→(uα, α), is a bijection. This is clearly equivalent to the condition (b) in (ii).

If the functor (ac, pr) is dense and ξ, ξ ′ are A-objects, the pair (ξ, ξ ′) be-longs to E ×B E and so there exists an object (X, η) of G × E and an isomor-phism in E ×B E , (α, β): (Xη, η)

∼−→ (ξ ′, ξ ). This means that α: Xη∼−→ ξ ′ and

β: η∼−→ ξ are isomorphisms in E with PE (α) = PE (β) and then the composite

Xξ(Xβ)−1−−−−→ Xη

α−→ ξ ′ is an A-isomorphism and so the property (a) in (ii) holds.Since, if one has an A-isomorphism α: Xξ → ξ ′, the pair (α, 1): (Xξ, ξ) → (ξ ′, ξ )is an isomorphism in E ×B E , we conclude that the density of the functor G×E →E ×B E is equivalent to the condition (a) in (ii).

We have proved that (i) and (ii) are equivalent and now we will show theirequivalence with (iii), the equivalence with (v) being completely analogous.


Let us suppose that E is a pseudo-torsor and ξ an A-object. For any two objectsX,Y ∈ G, the map HomG(X, Y ) → HomA(

Xξ, Y ξ), u �→ uξ , is a bijection due to(ii) (b) and so the functor G → EA, X �→ Xξ , is full and faithful; moreover, it isdense because of (ii) (a) and thus it is an equivalence.

To show that every morphism in E is cartesian, let α: ξ → η be an f -morphismwith f : A → B. Let us choose an A-isomorphism λ: ξ ′ ∼→ Xξ for some X ∈ G

and let φ0: I ξ∼→ ξ and φ0: I η

∼→ η the canonical isomorphisms. We have acommutative diagram

HomG(X, I )(1)

Homf (Xξ, I η)

λ∗φ0∗∼ Homf (ξ

′, η)

HomG(X, I )(2)

HomA(Xξ, I ξ)

(I α)∗

λ∗φ0∗∼ HomA(ξ

′, ξ )

α∗ (18)

where (1) is the map u �→ uα and (2) is the map u �→ uξ ; since E is a pseudo-torsor maps (1) and (2) are bijections and so α∗ is also a bijection and therefore αis cartesian.

Now we will prove (iii) ⇒ (ii). The condition (a) is clear; to prove (b), letf : A → B be a morphism in B, α: ξ → η an f -morphism and X,Y two objectsof G. Since Y is 2-regular there exists an object Z ∈ G and an isomorphism v: Y ⊗Z

∼−→ X. Then, we have a commutative diagram

HomG(X, Y )v∗r−1∗

∼

(−)α

HomG(Y ⊗ Z, Y ⊗ I )

(−)α

HomG(Z, I )

(−)α

Y⊗−∼

Homf (Zξ, I η)

Y (−)�

Homf (Xξ, Y η)

(vξ)∗(rη)−1∗∼ Homf (

(Y⊗Z)ξ, (Y⊗I )η)(φ∗)−1φ0∗

∼ Homf (Y (Zξ), Y (Iη))

and it is enough to show that the right vertical map (−)α: HomG(Z, I ) →Homf (

Zξ, I η) is a bijection. To see that, let us consider the commutative diagram

HomG(Z, I )(1)

HomA(Zξ, I ξ)

φ0∗∼

(I α)∗

HomA(Zξ, ξ)

α∗

HomG(Z, I )(2)

Homf (Zξ, I η)

φ0∗∼ HomA(

Zξ, η)

(19)

where (1) is the map u �→ uξ and (2) is the map u �→ uα. Since α is cartesian, α∗is a bijection and since the functor G → EA, X �→ Xξ , is an equivalence, map (1)is a bijection; thus (2) is a bijection.


Finally, let us show that (iv) implies (iii). Suppose ξ is an A-object such thatρξ : G → EA gives an equivalence, and let η be any other A-object. Then thereexists an A-isomorphism λ: Y ξ

∼−→ η for some object Y ∈ G and we have an

equivalence (X⊗Y )ξ∼−→ X(Y ξ)

Xλ−→ Xη between the functors ρξ (− ⊗ Y ) and ρη

G−⊗Y

ρη

G

ρξ

EA

Since − ⊗ Y is a self-equivalence in G we conclude that ρη is an equivalence.Analogously, (vi) ⇒ (v) and so the proof is completed. ✷As an immediate consequence the following holds:

COROLLARY 3.3. Let E be a pseudo-torsor under G over B. If f : A → B is amorphism in B, ξ is an A-object and η is a B-object such that Homf (ξ, η) �= ∅,then there exists a bijection

AutG(I )∼−→ Homf (ξ, η).

Proof. We have bijections AutG(I )∼−→ Homf (

I ξ, Iη)∼−→ Homf (ξ, η). ✷

We now wish to define the notion of torsor under a categorical group. First letus note the following corollary which is also an evident consequence of the aboveproposition:

COROLLARY 3.4. Let E be a pseudo-functor under a categorical group G suchthat all fibres are non-empty (i.e., the projection functor is surjective on objects).Then the following statements are equivalent:

(i) E is a bifibred category.(ii) E is a fibred B-category.

(iii) E is a cofibred B-category.(iv) For any morphism in B, f : A → B, and any B-object η, there exists an

f -morphism with target η.(v) For any morphism in B, f : A → B, and any A-object ξ there exists an

f -morphism with source ξ .(vi) The projection functor is surjective on morphisms, i.e., for any morphism f

in B there exists an f -morphism in E .

Proof. After Proposition 3.2, (ii) and (iv), and (iii) and (iv) are respectivelyequivalent. Since (ii) and (iii) together mean the same as (i), it is enough to show


that (iv), (v) and (vi) are equivalent. Suppose that (iv) holds and let f : A → B be amorphism in B and ξ an A-object. If we choose any f -morphism α: ξ ′ → η, thenξ and ξ ′ are A-objects and there exists an object X ∈ G and an A-isomorphismλ: ξ

∼−→ Xξ ′. The composition (Xα)λ gives an f -morphism with source ξ asrequired. Thus we have (vi) ⇒ (v). The proof of (vi) ⇒ (iv) is analogous. Sinceall fibres are non-empty it is obvious that (iv) or (v) imply (vi) and so the proof iscompleted. ✷DEFINITION 3.5. A B-torsor under a categorical group G is a B-pseudo-torsorunder G such that the projection functor PE : E → B is surjective on morphisms.

A morphism between B-torsors under G is a B-functor G-equivariant.

The composition of morphisms of torsors is defined in an obvious way andTors(B,G) will denote the category of B-torsors under G.

The next proposition establishes that the existence of a morphism E → E ′ isan equivalence relation between torsors. Then, Tors[B,G] will denote the set ofconnected components in the category of torsors.

PROPOSITION 3.6. Let T = (T , µ): E → D be a morphism in Tors(B,G).Then the underlying functor T : E → D is a B-equivalence and there exists aB-torsor morphism T ′: D → E .

Proof. To prove that T is a B-equivalence it is enough to prove, according to([18], Prop. 6.10), that for any object A ∈ B the induced functor between the fibrecategories TA: EA → DA is an equivalence. Then let ξ ∈ EA be any A-object in Eand let us consider the diagram of functors

Gρξ

ρT (ξ)

EA

TA

DA

where ρξ (X) = Xξ and ρT (ξ)(X) = XT (ξ). TheA-isomorphisms µX,ξ : T (Xξ)∼−→

XT (ξ) define a natural equivalence of TAρξ to ρT (ξ) and then Proposition 3.2, (iv)implies that TA is an equivalence. It remains to prove that there exists a B-torsormorphism (T ′, µ′): D → E . To do so, if ξ ′ ∈ D is an A-object, let us choose anA-object of E , T ′(ξ), and an A-isomorphism in D , λξ ′: T (T ′(ξ ′)) ∼−→ ξ ′; thenthe correspondence ξ �→ T ′(ξ ′) extends uniquely to a B-functor T ′: D → E suchthat λ: T T ′ → 1 is a B-equivalence. Now T ′ extends uniquely to a B-functorG-equivariant (T ′, µ′): D → E such that for all objects X ∈ G and ξ ′ ∈ D thefollowing diagram commutes:


T (XT ′(ξ ′))µ

T (µ′)

XT T ′(ξ ′)

Xλ

T T ′(Xξ ′) λ Xξ ′ ✷Next we give some examples which help to better understand the notion of

torsor under a categorical group.

EXAMPLE 3.7. Let G be any categorical group and B any category. The productcategory G×B is B-bifibred via the projection functor pr: G×B → B, (X,A) �→A, which is clearly surjective. Now we can let G act on the B-category G × B by

left tensor multiplication, G×G×B⊗×1−−−−→ B, i.e., X(Y,A) = (X⊗Y,A). Then,

G × B is a B-torsor under G that will be called the trivial (or split) B-torsor.Note that a B-torsor under G, E , is equivalent to the trivial one if, and only if,

the projection functor PE : E → B is split, i.e., if there exists a functor S: B → Esuch that PES = 1B . The proof of this fact is easy since the trivial B-torsor is splitby taking S(A) = (I, A) and S(f ) = (1I , f ) and then any B-torsor equivalent toit is also split. Conversely, if there exists S: B → E such that PES = 1B , then the

B-functor T : G × B → E defined by the composition G×B1×S−−−−→ G×E

ac−→E , i.e., T (XA) = XS(A), is G-equivariant via the isomorphism T (X(Y,A)) =X⊗Y S(A)

∼−→ X(YS(A)) = XT (Y,A). Thus, the B-torsor E is equivalent to thetrivial B-torsor.

EXAMPLE 3.8. Let us consider the action, described in Example 2.10, of the cat-egorical group of loops of a pointed space P2(X, ∗) on the groupoid P2(X,A, ∗)associated to a pointed subspace ∗ ∈ A ⊆ X.

There is a fibration P : %(X,A, ∗) → A given by P(γ ) = γ (1) that inducesa fibration between the corresponding fundamental groupoids P : P2(X,A, ∗) →P1(A). Thus, P2(X,A, ∗) is a bifibred P1(A)-category in which any morphism iscartesian (since any morphism is invertible) and it is plain to see that the action ofP2(X, ∗) on it is, in fact, an action on the P1(A)-category P2(X,A, ∗).

The object of this example is to demonstrate the following statement:

“P2(X,A, ∗) is a P1(A)-pseudo-torsor under P2(X, ∗) if π2(A, ∗1) = 0 forany point ∗1 ∈ A. Moreover, if A is contained in a path component of X, thenP2(X,A, ∗) is a P1(A)-torsor under P2(X, ∗).”

To prove that P2(X,A, ∗) is a pseudo-torsor it is enough to observe, accordingto Proposition 3.2, that for any path in X, δ: ∗ → ∗1, the functor ρδ = − ⊗δ: P2(X, ∗) → P2(X,A, ∗)∗1 , where P2(X,A, ∗)∗1 is the fibre category in ∗1, isan equivalence of groupoids. To see that, note first that composition with δ definesa fibre homotopy equivalence


T (XT ′(ξ ′))µ

T (µ′)

XT T ′(ξ ′)

Xλ

T T ′(Xξ ′) λ Xξ ′

which induces an equivalence δ⊗−: P2(X,A, ∗1)∗1 → P2(X,A, ∗)∗1 . In addition,the map ω �→ δ⊗ω⊗ δ−1 is a homotopy equivalence of %(X, ∗1) to %(X, ∗) that

induces an equivalence of groupoids P2(X, ∗1)δ⊗−⊗δ−1−−−−→ P2(X, ∗) (here δ−1(t) =

δ(1 − t) is the opposite path of δ). Then we have a diagram,

P2(X, ∗) −⊗δP2(X,A, ∗)∗1

P2(X, ∗1)

δ⊗−⊗δ−1

in P2(X,A, ∗1)∗1

δ⊗−

and it is enough to prove that the induced functor by the inclusion is an equivalencebetween the groupoids P2(X, ∗1) and P2(X,A, ∗1)∗1 . Now, both groupoids havethe same objects, which are the loops in X based on ∗1. If 4: I×I → X representsa morphism in P2(X,A, ∗1)∗1 , [4]: ω → ω′, then 4/1 × I is a loop in A ho-motopic to the constant ∗1 and, using the homotopy extension property of the pair(I×I, ∂I×I∪I×∂I ), we see that there exists 4′: I×I → X such that [4] = [4′]and 4′/1× I is the constant loop ∗1. Thus, 4′ represents a morphism in P2(X, ∗1),[4′]: ω → ω′, and therefore HomP2(X,∗1)(ω, ω

′) → HomP2(X,A,∗1)∗1(ω, ω′) is

surjective. To see that it is an injective map, suppose that HomP2(X,∗1)(ω, ω′) is

non-empty and that [f ]: ω → ω′ belongs to it. Then we have a commutativediagram

HomP2(X,∗1)(ω, ω′) in HomP2(X,A,∗1)∗1

(ω, ω′)

HomP2(X,∗1)(∗1, ∗1)

[f ]⊗− �

in HomP2(X,A,∗1)∗1(∗1, ∗1)

[f ]⊗−�

π2(X, ∗1)in

Ker(π2(X,A, ∗1)P−→ π1(A, ∗1))

and we deduce the injectivity from the exact sequence

0 = π2(A, ∗1) → π2(X, ∗1) → π2(X,A, ∗1) → π1(A, ∗1).

Then, P2(X, ∗1)in−→ P2(X,A, ∗1)∗1 is full and faithful and so it is an isomor-

phism of groupoids.


If, in addition, we suppose that for every point ∗1 ∈ A there exists a path in X,γ : ∗ → ∗1, then the projection functor P : P2(X,A, ∗) → P2(A) is surjective onobjects and therefore the pseudo-torsor is a torsor.

Note that this torsor is not equivalent to the trivial one since the projectionfunctor P2(X,A, ∗) → P1(A) restricted to the corresponding groups of automor-phisms in the constant loop ∗ is the morphism π2(X,A, ∗) → π1(A, ∗) which, ingeneral, is not a retraction and not even surjective.

EXAMPLE 3.9. IfG is a group, then G can be regarded as a groupoid with exactlyone object where the morphisms are the members of G and the composition law isthe group composition operation. If A is an abelian group then A can be regarded asa strict categorical group in which both composition and tensor product are definedby the group operation. Then, for any group G and any abelian group A, we havea bijection

Tors[G,A] ∼= Extcen(G,A)

between the set of equivalence classes of G-torsors under A and the set of equiva-

lence classes of central extensions of G by A: In fact, if 1 → A → Ep→ G → 1 is

a central extension of G by A, then E is a bifibred G-category with projection func-tor p: E → G and the categorical group A acts on this G-category by the functorA× E → E, (a, x) �→ ax. It is clear that the induced functor A× E → E ×G E

is an isomorphism and so E is a G-torsor under A. Conversely, let E be a G-torsor under A with PE : E → G the projection functor. Then E is a groupoid, anyobject of E is in the fibre over the unique object of G and this fibre category isequivalent to A, which also has only one object. Therefore, if we choose any object

e ∈ E , the inclusion AutE (e)in−→ E is a G-equivalence. Now, the restriction of

PE gives a surjective homomorphism PE : AutE(e) → G whose kernel is identifiedwith A by the map a �→ ae in such a way that we have a central group extension1 → A → AutE(e) → G → 1 that is equivalent, in the sense described above, tothe G-torsor E .

4. Schreier Theory for Categorical Torsors

If G and H are groups, the Schreier extension theorem, [31], gives a cohomologicalclassification of the extensions of H by G in terms of equivalence classes of factorsets, that is, mappings Z: G → Aut(H), t : G2 → H , satisfying:

C(tx,y)Zxy = ZxZy; Zx(ty,z)tx,yz = tx,ytxy,z;tx,1 = t1,x = 1; Z1 = idH ,

where, for any h ∈ H , C(h) is the inner automorphism defined by h.We are inspired by Schreier’s analysis of group extensions and also by more

recent works [2, 34] or [9] to make a corresponding analysis of categorical torsors.


This will allow us to conclude that a survey of all (equivalence classes) B-torsorsunder a prescribed categorical group G can be done in terms of 2-cocycles in thefollowing sense:

DEFINITION 4.1. Let G be a categorical group and B a category. A 2-cocycle onB with coefficients on G is a system, denoted (Z, t), consisting of the followingdata:

(a) For any morphism f : A → B of B an object Zf ∈ G is given.

(b) For any pair of composable morphisms in B, Af→ B

g→ C, a morphism in G,tg,f : Zf ⊗ Zg → Zgf is given.

These data must satisfy the following conditions:

(i) (Cocycle condition) For any three composable morphisms Af→B

g→Ch→D,

the diagram

(Zf ⊗ Zg) ⊗ Zh

tg,f ⊗1Zgf ⊗ Zh

th,gf

Zf ⊗ (Zg ⊗ Zh)

a

1⊗th,g

Zhgf

Zf ⊗ Zhg

thg,f

(20)

is commutative.(ii) (Normalization conditions.) For any morphism in B, f : A → B,

Z1A = I ; t1B,f = r: Zf ⊗ I → Zf ; tf,1A = l: I ⊗ Zf → Zf . (21)

We will denote Z2(B,G) the set of all 2-cocycles on B with coefficients on G.

Next we will show how each B-torsor under G has an associated “factor set”that is a 2-cocycle in the above sense and, as we will demonstrate later, turns out tobe an appropriate type of descent datum to rebuild the B-torsor. We will first note,in the following example, that Schreier’s factor sets are 2-cocycles.

EXAMPLE 4.2. Let H be a group. If we consider H as a category with exactlyone object, then it has the categorical group of self-equivalences Eq(H) associatedwith it as in Example 2.1.

The categorical group Eq(H) was studied by Duskin in [14], where it was calledthe groupoid in the category of groups of automorphisms of H , since it has thegroup Aut(H) as the objects and the holomorph group Hol(H) as the arrows. Moreprecisely, Eq(H) can be described as follows: The objects are the elements of thegroup Aut(H) of automorphisms of H ; an arrow h: ϕ → ψ is an element h ∈ H

with ϕ = C(h)ψ . The composition is the multiplication in H . The tensor productis given by

(ϕh→ ψ) ⊗ (ϕ′ h′→ ψ ′) = (ϕϕ′ hψ(h′)−−−−→ ψψ ′).


Now, suppose we have two groups H and G. Then, a 2-cocycle on (the category)G with coefficients on Eq(H), (Z, t) ∈ Z

2(G,Eq(H)), consists simply of twomaps Z: G → Aut(H) and t : G2 → H such that: (1) for any pair (x, y) ∈ G2,tx,y is a morphism in Eq(H) from ZxZy to Zxy , i.e. ZxZy = C(tx,y)Zxy; (2) forany (x, y, z) ∈ G3, (tx,y ⊗ 1Zz

)txy,z = (1Zx⊗ ty,z)tx,yz, or equivalently tx,ytxy,z =

Zx(ty,z)tx,yz; (3) Z1 = idH , t1,x = 1 = tx,1.Thus, the 2-cocycles of Z

2(G,Eq(H)) are exactly Schreier’s factor sets of groupextensions of G by H .

Below we assume that E is a B-torsor under a categorical group G with projec-tion functor PE : E → B.

For each object A ∈ B, choose an A-object FA ∈ E . Then, for each morphismof B, f : A → B, choose an A-object Zf ∈ G and an f -morphism ϒf : FA →Zf FB , with source FA and target Zf FB (this morphism exists since if we takeϒ : FA → η any f -morphism with source FA, then η and FB are B-objects andthere exists aB-isomorphism λ: η

∼−→ Zf FB ; the composite λϒ gives the requiredϒf ). When f = 1A is an identity morphism we take Z1A = I , the unit object, and

ϒ1A = φ−10 : FA

∼−→ IFA the canonical A-isomorphism.

Now, if Af→ B

g→ C is a pair of composable morphisms in B, and keeping inmind that ϒgf is co-cartesian, we observe that there exists a unique C-isomorphism(Zf ⊗Zg)FC

∼−→ (Zgf )FC making the following diagram commutative:

(Zf )FB

(Zf )ϒgZf ((Zg)FC)

φ−1

FA

ϒf

ϒgf

(Zf ⊗Zg)FC

(Zgf )FC

in addition, by the bijection HomG(Zf ⊗ Zg,Zgf ) → HomC((Zf ⊗Zg)FC,

(Zgf )FC),u �→ uFC , such a morphism is written as (tg,f )FC for a unique morphism in G,

tg,f : Zf ⊗ Zg → Zgf .

We claim that the system (Z, t) = ((Zf ), (tg,f )) is a 2-cocycle on B withcoefficients on G which we call a “Schreier system” or a “factor set” for the B-torsor.

Let us suppose Af→ B

g→ Ch→ D to be three composable morphisms in B.

Then we have

ϒ(hg)f = (thg,f )FDφ−1(Zf )ϒhgϒf

= (thg,f )FDφ−1Zf ((th,g)FD)

Zf (φ−1)Zf (Zg (ϒh))Zf ϒgϒf

(1)= (thg,f )FD(1Zf ⊗th,g )FDφ

−1Zf (φ−1)Zf (Zg (ϒh))Zf ϒgϒf ,


where (1) is a consequence of the naturality of φ, and analogously

ϒh(gf ) = (th,gf )FDφ−1(Zgf )ϒhϒgf

= (th,gf )FDφ−1Zgf ϒh

(tg,f )FCφ−1Zf ϒgϒf

(2)= (th,gf )FD(tg,f ⊗1Zh )FDφ

−1φ−1Zf (Zg (ϒh))Zf ϒgϒf

(3)= (th,gf )FD(tg,f ⊗1Zh )FD

a−1FDφ

−1Zf (φ−1)Zf (Zg (ϒh))Zf ϒgϒf ,

where (2) is a consequence both the functoriality of the action and the naturality ofφ, and (3) so is of (14).

Since ϒ(hg)f = ϒh(gf ) and any morphism in E is co-cartesian, then((thg,f )(1⊗th,g))FD = ((th,gf )(tg,f⊗1)a−1)FD , and therefore thg,f (1⊗ th,g)a = th,gf (tg,f ⊗1), as required for the cocycle condition.

Moreover, for any morphism f : A → B, since (Zf )φ0φ = rFB (14), thenϒ(1Bf ) = ϒf = rFBφ

−1(Zf )(φ−10 )ϒf , and hence t1B,f = r: Zf ⊗ I → Zf .

Analogously, tf,1A = l and so (Z, t) satisfies the normalization conditions.The Schreier system for the B-torsor E , (Z, t) ∈ Z

2(B,G), clearly dependson the choices of FA, Zf and ϒf . Let (Z′, t ′) be the new cocycle arising fromthe second choices F′

A, Z′f and ϒ ′

f . Then, for any object A ∈ B, FA and F′A are

two A-objects and there exists an object hA ∈ G and an A-isomorphism in B,uA: FA → (hA)F′

A; hence, for any morphism in B, f : A → B, we have a uniquemorphism in G,

ϕf : Zf ⊗ hB∼−→ hA ⊗ Z′

f

making the following diagram commutative:

FA

ϒf

uA

(Zf )FB

(Zf )uB Zf (hBF′B)

∼

(Zf ⊗hB)F′B

(φf )F′B

(hA)F′A

(hA)ϒ ′f

hA((Z′

f )F′B)

∼ (hA⊗Z′f )F′

B

It is straightforward (though tedious) to show that for any pair of composable

morphisms Af→ B

g→ C, the following diagram, in which we have omitted thecanonical associativity isomorphisms, is commutative:


Zgf ⊗ hCϕgf

hA ⊗ Z′gf

Zf ⊗ Zg ⊗ hC

tg,f ⊗1

1⊗ϕg

hA ⊗ Z′f ⊗ Z′

g

1⊗t ′g,f

Zf ⊗ hB ⊗ Z′g

ϕf ⊗1

(22)

Note also that for any object A ∈ B, the diagram

I ⊗ hAϕ1

l

hA ⊗ I

r

hA

(23)

is commutative.These observations suggest the following:

DEFINITION 4.3. Let B be a category and G a categorical group. Two 2-cocycles(Z, t) and (Z′, t ′) in Z

2(B,G) are called cohomologous if there exists a map

h: Obj(B) → Obj(G)

and, for each morphism f : A → B in B, there is a morphism in G

ϕf : Zf ⊗ hB∼−→ hA ⊗ Z′

f

satisfying the conditions of making diagrams (22) and (23) commutative.We say that the pair (h, ϕ) is a coboundary from (Z, t) to (Z′, t ′) and we write

(h, ϕ): (Z, t) → (Z′, t ′).

Being cohomologous is an equivalence relation between 2-cocycles. We willdenote by H

2(B,G) the quotient set of Z2(B,G) by the relation of being coho-

mologous.As we have seen before, the cohomology class of a Schreier system for a B-

torsor under G does not depend on the choices made and is called the Schreierinvariant of the torsor. Actually, it is an invariant of the equivalence class of thetorsor. To prove this, suppose that (T , µ): E → E ′ is a B-torsor morphism and let(Z, t) be the factor set of E after a choice of the f -morphism ϒf : FA → (Zf )FB .We can choose, for any object A ∈ B, the A-object of E ′, F′

A = T (FA), and forany morphism f : A → B, the f -morphism in E ′, ϒ ′

f , given by the composition

ϒ ′f : T (FA)

ϒf−→ T ((Zf )FB)µ−→ (Zf )T (FB).


Then, the resulting Schreier system for the B-torsor E ′ coincides with (Z, t)

and therefore E and E ′ have the same Schreier invariant.Next we will show the relationship between H

2(B,G) and other cohomologysets.

EXAMPLE 4.4. We would like to point out here that the cohomology set H2(G,

G(�)) of a group G, viewed as a category with only one object, with coefficientson the categorical group defined, as in Example 2.2, by a crossed module �, is thesame as Dededecker’s non-abelian (thin) cohomology set, H 2(G,�), of G withcoefficients in the crossed module � [12]. To show this, note that a 2-cocycle(Z, t) ∈ Z

2(G,G(�)) consists of two maps Z: G → π and t : G2 → H such that:(i) ZxZy = ρ(tx,y)Zxy ; (ii) (1Zx

⊗ ty,z)(tx,yz) = (tx,y ⊗ 1Zz)(txy,z) or equivalently,

(Zx)ty,ztx,yz = tx,ytxy,z; (iii) Z1 = 1, tx,1 = 1 = t1,x . Thus, (Z, t) represents exactlya Dedecker 2-cocycle.

EXAMPLE 4.5. Let G be a group regarded as a groupoid with exactly one objectand let A be an abelian group considered as a categorical group with only oneobject in which both composition and tensor product are defined by the groupoperation. Then our H

2(G,A) coincides with the usual Eilenberg–Mac Lane co-homology group of G with coefficients in the trivial G-module A or, equivalently,with the singular cohomology of the classifying space B(G) with coefficients in A.

EXAMPLE 4.6. If B is any small category and A is an abelian group consideredas a categorical group with exactly one object, then H

2(B, A) coincides with the2nd cohomology group of B with coefficients on the trivial B-module defined byRoos [29]. Thus, H

2(B, A) is the usual singular cohomology of the classifyingspace B(B) with coefficients in the abelian group A.

EXAMPLE 4.7. Let G and H be groups. Let us consider G as a category withonly one object and let dis(H) be the discrete category with only identities whoseobjects are the elements of H . The multiplication in H determines a categoricalgroup structure on dis(H) and Z

2(G, dis(H)) = HomGp(G,H), the set of allgroup homomorphisms of G into H , and H

2(G, dis(H)) is the set of conjugacyclasses of homomorphisms of G into H .

EXAMPLE 4.8. If a categorical group G is enriched with a symmetry c = (cA,B :A ⊗ B → B ⊗ A) [24], then G is called a Picard category or a group-likemonoidal category. If we regard a group G as a category with only one objectand G is a Picard category, then our H

2(G,G) coincides with the 2nd cohomologygroup defined by Takeuchi and Ulbrich [34], of the cosimplicial complex of Picardcategories (Cn(G,G))n≥0, where Cn is the category of all maps Gn → G, thatis, H

2(G,G) is the 2nd cohomology group of G with coefficients in the Picardcategory G considered with the trivial G-module structure.

Now we establish the main result of this section (cf. [1], Proposition 6.2).


THEOREM 4.9. Let B be a category and G a categorical group. Each element ofH

2(B,G) is the Schreier invariant of a, unique up to equivalence, B-torsor underG. Thus, there exists a bijection

Tors[B,G] ∼= H2(B,G).

Proof. Let (Z, t) ∈ Z2(B,G) be a 2-cocycle. Then, (Z, t) defines a pseudo-

functor B → Cat in the sense of Grothendieck [18], for which: all fibre cate-gories coincide with G; for each morphism f : A → B in B, the associated func-

tor is − ⊗ Zf : G → G; for each pair of composable morphisms Af→ B

g→ C,

the natural equivalence Cf,g: (− ⊗ Zg)(− ⊗ Zf )∼−→ (− ⊗ Zgf ) is given by the

composition

(− ⊗ Zf ) ⊗ Zg

Cf,g

a

− ⊗ Zgf

− ⊗ (Zf ⊗ Zg)

1⊗tg,f

Thus, (Z, t) [18], has canonically associated with it a cofibration E(Z,t)P−→ B

which can be described as follows: the objects of E(Z,t) are pairs (X,A) withX ∈ Obj(G) andA ∈ Obj(B); the arrows are pairs (u, f ): (X,A) → (Y, B)wheref : A → B is a morphism in B and u: X ⊗ Zf → Y is a morphism in G. The

composition of two morphisms (X,A)(u,f )−−−−→ (Y, B)

(v,g)−−−−→ (Z,C) is definedby (v, g)(u, f ) = (v ◦ u, gf ): (X,A) → (Z,C), where v ◦ u: X ⊗ Zgf → Z isthe unique morphism in G making the diagram below commutative

X ⊗ Zgfv◦u

Z

X ⊗ (Zf ⊗ Zg)

1⊗tg,f

a

Y ⊗ Zg

v

(X ⊗ Zf ) ⊗ Zg

u⊗1

This composition is associative and unitary thanks to the normalization and cocycleconditions of 4.1. The cofibration P: E(Z,t) → B is given py P((u, f )) = f . Notethat any morphism in E(Z,t) is cocartesian since G is a groupoid.

Moreover, the categorical group G acts on the B-category E(Z,t) by left tensormultiplication, that is, by putting Z(X,A) = (Z⊗X,A) for any objects Z,X ∈ G

and A ∈ B, and, for any morphisms v: Z → T in G and (u, f ): (X,A) → (Y, B)


in E(Z,t), by putting v(u, f ) = ((v⊗u)a−1, f ): (Z⊗X,A) → (T⊗Y,B) according

to the composition in G, (Z ⊗ X) ⊗ Zfa−1−→ Z ⊗ (X ⊗ Zf )

v⊗u−−−−→ Z ⊗ Y .

The natural isomorphisms φ: (X⊗Y )(Z ⊗ A)∼−→ X(Y (Z,A)) and φ0: I (Z,A)

∼−→ Z ⊗ A are given by the associativity and unit constraints of G.If A is an object of B, for any A-object (X,A) ∈ E(Z,t) we have an A-isomorp-

hism (1X⊗I , 1A): (X,A)∼−→ X(I,A) and it can inmediately be seen that for any

objects X,Y ∈ G, HomG(X, Y ) ∼= HomA(X(I,A), Y (I, B)). Thus, the functor

X �→ X(I,A) establishes an equivalence between G and the fibre category over A.So, by Proposition 3.2(v), E(Z,t) is a B-torsor under G.

Now, to describe a Schreier system for this B-torsor E(Z,t), we can choose theA-object FA = (I, A) for any object A ∈ B, and for each morphism f : A → B

the object Zf ∈ G and the f -morphism ϒf = (r−1l, f ): (I, A) → Zf (I, B) =(Zf ⊗I, B). It is straightforward to see that the resulting factor set for E(Z,t) relativeto that choice is simply the given 2-cocycle (Z, t).

Finally, let us suppose that E(Z,t) is the unique, up to equivalence, B-torsorwhose Schreier invariant is the cohomology class of (Z, t). Suppose that (Z′, t ′)is a factor set for a B-torsor E after the choices of an A-object FA for any objectA ∈ B and an f -morphism ϒf : FA → (Z′

f )FB for any morphism f : A → B inB. If (Z, t) and (Z′, t ′) are cohomologous by a coboundary (h, ϕ): (Z, t) →(Z′, t ′), then there exists a B-torsor morphism T : E(Z,t) → E defined by puttingT ((X,A)) = (X⊗hA)FA for each object (X,A) ∈ E(Z,t) and, for each morphism(u, f ): (X,A) → (Y, B) in E(Z,t), putting T ((u, f )): (X⊗hA)FA → (Y⊗hB)FB asthe unique f -morphism in E determined by the commutativity of the diagram

(X⊗hA)FA

T (u,f )

(X⊗hA)ϒf

(Y⊗hB)FB

(X⊗hA)((Z′

f )FB)(X⊗Zf ⊗hB)FB

(u⊗1)FB

(X⊗hA⊗Z′f )FB

∼ (1⊗ϕf )FB

Then, E(Z,t) and E represent the same element in Tors[B,G]. ✷As we saw in Example 3.7, Tors[B,G] is pointed by the class of the split

B-torsor G × B. The set H2(B,G) is also pointed by the class of the “trivial

cocycle”, i.e., the pair (Z, t) ∈ Z2(B,G) defined by Zf = I for any morphism

f : A → B in B, and tg,f = lI = rI : I × I → I for any pair of composable mor-

phisms Af→ B

g→ C. It is easy to see that the bijection Tors[B,G] ∼= H2(B,G)

is actually a bijection of pointed sets.


5. Homotopy Classification of Categorical Torsors

Let 7 be the category of ordered sets [n] = {0, 1, 2, ..., n}, n ≥ 0, and increas-ing mappings. Then the category of simplicial sets is the category of functorsK: 7op → Sets. In this section we use a few largely well-known definitions andfacts about simplicial sets and for the background we refer the interested reader to[26, 11, 16] or [19].

Below we regard each ordered set [n] as a category with exactly one arrowi → j if i ≤ j . Then a non decreasing map [n] → [m] is the same that a functor.

Let B be a small category. If Func([n],B) denotes the set of functors from[n] to B, n ≥ 0, then we have a simplicial set Func(−,B): 7op → Sets, which iscalled the (Grothendieck) nerve of the category and is denoted by Ner(B). Thus,Ner0(B) = Obj(B), the set of objects of B; Ner1(B) = Mor(B), the set of mor-phisms of B; for n ≥ 2, the n-simplices of Ner(B) are the composable sequences

of arrows of B of length n,A0f1→ A1 → · · · fn→ An and the i-th face of this simplex

is obtained by deleting the object Ai . For example, if [m] ∈ 7, Ner([m]) = 7[m]is the standard m-simplex.

For a small category B, its classifying space B(B) is the geometric realizationof the simplicial set Ner(B). If a group G is regarded as a category with exactly oneobject, then B(G) is a classifying space for the discrete group G in the traditionalsense; it is an Eilenberg–Mac Lane space of type K(G, 1) and so few homotopytypes thus occur. However, any CW-complex is homotopy equivalent to the classi-fying space of a small category, as Quillen showed [19]: the category of simplices7(K) associated to an arbitrary simplicial set K is that one whose objects are pairs(n, α) where n ≥ 0 and α ∈ Kn is an n-simplex in K; an arrow u: (n, α) → (m, β)

is an arrow u: [n] → [m] in 7 with the property α = u∗(β). Then there exists ahomotopy equivalence |K| � B(7(K)) between the geometric realization of Kand the classifying space of 7(K). If X is any CW-complex and we take S(X), thetotal singular complex of X, then X � |S(X)| � B(7S(X)).

When πi(X, ∗) = 0 for any i ≥ 2 and any base point ∗, then X has the homo-topy type of a groupoid. Indeed, X � B(P1(X)), where P1(X) is the fundamentalgroupoid of X.

Now, let G = (G,⊗, a, I, l, r) be a categorical group. We define the nerve ofG, denoted by Ner(G,⊗), as the simplicial set,

Ner(G,⊗) = Z2(−,G): 7op → Sets. (24)

If we recall the notion of 2-cocycle, Definition 4.1, then Ner0(G,⊗) = {I },Ner1(G,⊗) = Obj(G), the set of objects of G, and for n ≥ 2, an n-simplex ofNer(G,⊗) is a tuple

x = (xijk : Aij ⊗ Ajk → Aik)0≤i<j<k≤n


of morphisms in G (i.e. x consists of an object Aij for each 0 ≤ i < j ≤ n anda morphism xijk : Aij ⊗ Ajk → Aik for each 0 ≤ i < j < k ≤ n) such that, if0 ≤ i < j < k < l ≤ n, the diagram

(Aij ⊗Ajk) ⊗ Akl

xijk⊗1Aik ⊗ Akl

xikl

Aij ⊗ (Ajk ⊗ Akl)

a

1⊗xjkl

Ail

Aij ⊗ Ajl

xijk

(25)

is commutative.The faces of a 2-simplex x = (x: A01 ⊗A12 → A02) are d0(x) = A12, d1(x) =

A02 and d2(x) = A01, and the m-face of an n-simplex x = (xijk), n ≥ 3, isdm(x) = (xδm(i),δm(j),δm(k)), where δm: [n] → [n+1] is the injective non-decreasingmap that does not take the value m ∈ [n].

Let us note that if x = (xijk : Aij ⊗ Ajk → Aik) is an n-simplex of Ner(G,⊗),n ≥ 3, then x is completely determined by any three of its faces since a facedm(x) includes all the morphisms xijk such that m /∈ {i, j, k}. Then, if one knowsdm(x), dr(x) and ds(x), 0 ≤ m < r < s ≤ n, one also knows all xijk exceptxmrs ; but taking any i /∈ {m, r, s} (such an integer exists because n ≥ 3) and thecorresponding diagram (25) for i, m, r and s (in the corresponding order), we cansee that xmrs is also determined by the others.

An immediate consequence of the above observation is that the canonical sim-plicial map Ner(G,⊗) → Cosk3(Ner(G,⊗)) is an isomorphism, where Cosk3(−)

is the third Verdier’s coskeleton functor [35], and then the simplicial set Ner(G,⊗)

is isomorphic to the nerve of the categorical group G as defined in [10] (theredenoted by Ner2(G)). Thus the nerve functor is a homotopy functor on categoricalgroups (see [10], Prop. 3.1).

The classifying space of a categorical group G, B(G,⊗), is the geometricalrealization of the simplicial set Ner(G,⊗). The next proposition describes thehomotopy type of B(G,⊗) and was proved in [10].

PROPOSITION 5.1. Let G = (G,⊗, a, I, l, r) be a categorical group. Then:

(i) Ner(G,⊗) satisfies the Kan extension condition.(ii) The homotopy groups of B(G,⊗) are

πi(B(G,⊗)) ={ 0, i �= 2,

[G] the group of connected components of G, i = 1,AutG(I ) the group of automorphisms in I, i = 2.

(iii) There exists a canonical homotopy equivalence B(G) � %B(G,⊗). Thus,the classifying space of category G (forgetting the tensor structure) is a loopspace of the classifying B(G,⊗).


(iv) Any path-connected CW-complex with trivial homotopy groups in dimensionsother than 1 and 2 is homotopy equivalent to the classifying space of a cat-egorical group. Specifically, if X is such a space and ∗ ∈ X is a base point,then X � B(P2(X, ∗)) where P2(X, ∗) is the categorical group of loops de-scribed in Example 2.4. Furthemore, if X is homotopy equivalent to B(G,⊗)

for some G, then P2(X, ∗) and G are monoidal equivalent.

The above results show that categorical groups provide algebraic models forhomotopy 2-types of connected spaces. This fact is not new (see [33]), althoughthe better-known references consider the strict case, where Whitehead’s crossedmodules provide the algebraic models [25, 6].

The construction of the nerve of a categorical group gives simplicial sets thatare very easy to handle and whose simplices have nice geometrical description.This is the essential difference with the more general construction made by Jardinein [20]. Let us now describe some known constructions that are clearly related withthe above notion of nerve.

EXAMPLE 5.2. Let G be a group and consider the discrete groupoid dis(G) withonly identities. The multiplication in G determines a categorical group structureon dis(G) and Ner(dis(G),⊗) is isomorphic to the Eilenberg–Mac Lane complexK(G, 1), that is, it is a minimal complex withG as the unique non-trivial homotopygroup in dimension 1 [26].

If A is an abelian group regarded as a category with one object, then the mul-tiplication in A defines a structure of categorical group whose nerve is isomorphicto the Eilenberg–Mac Lane complex K(A, 2) [26].

If � is a crossed module and G(�) is the strict categorical group it defines (seeExample 4.4), then G(�) is a groupoid in the category of groups and Ner(G(�)) isa simplicial group. Therefore, we can apply Kan’s classifying complex functor W ,obtaining a simplicial set W(Ner(G(�))) that is isomorphic to our Ner(G(�),⊗).

Now we observe that 2-cocycles on a category B with coefficients on a cate-gorical group G can be interpreted as simplicial maps between the correspondingnerves.

THEOREM 5.3. Let B be a small category and G = (G,⊗, a, I, l, r) a categor-ical group. Then there is a natural bijection

(Ner(B),Ner(G,⊗)) ∼= Z2(B,G)

between the set of simplicial maps from Ner(B) to Ner(G,⊗) and the set of 2-cocycles on B with coefficients on G.

Moreover, two 2-cocycles are cohomologous if, and only if, their correspondingsimplicial maps are homotopic. Consequently, the above bijection induces anotherone,

[Ner(B),Ner(G,⊗)] ∼= H2(B,G),


between the set of homotopy classes of simplicial maps from Ner(B) to Ner(G,⊗)

and the 2-cohomology set of B with coefficients on G.Proof. If α: Ner(B) → Ner(G,⊗) is a simplicial map, then it is evident to

see, using the simplicial identities, that the pair (α1, α2) is actually a 2-cocycle inZ

2(B,G), which determines uniquely to α since, as mentioned above, for n ≥ 3every n-simplex of Ner(G,⊗) is determined by any three of its faces and then by its2-dimensional faces. Moreover, any 2-cocycle (Z, t) ∈ Z

2(B,G) arises in such away from a simplicial map α defined as follows: If x: [n] → B is an n-simplex ofNer(B), then α(x) = x∗(Z, t), where x∗: Z

2(B,G) → Z2([n],G) = Nern(G,⊗)

is the map induced by the functor x. Thus, α �→ (α1, α2) establishes the announcedbijection.

Now, let us consider the product category B × [1] and u0, u1: B → B × [1]the inclusion functors ui(A) = (A, i), i = 0, 1. Then, we claim that the sequence

Z2(B × [1],G)

u∗0

u∗1

Z2(B,G) H

2(B,G) (26)

is a coequalizer one: If (W, T ) ∈ Z2(B ×[1],G) is a 2-cocycle such that u∗

0(W, T )

= (Z, t) and u∗1(W, T ) = (Z′, t ′), we find a coboundary (h, ϕ): (Z, t) → (Z′, t ′)

by putting, for each object A ∈ B, hA = W(1A,ı), where ı: 0 → 1 is the unique non-identity morphism of category [1], and for each morphism f : A → B, ϕf : Zf ⊗hB → hA ⊗ Z′

f is the composition

Zf ⊗ hBT(f,1),(1,ı)

W(f,ı)

T −1(1,ı),(f,0)

hA ⊗ Z′f

(note that (f, ı) = (f, 1)(1A, ı) = (1B, ı)(f, 0) in B ×[1]). It is straightforward tosee that the cocycle conditions of (W, T ) are equivalent to the coboundary con-ditions of (h, ϕ) and the rest is left to the reader. Conversely, any coboundary(h, ϕ): (Z, t) → (Z′, t ′) defines a 2-cocycle (W, T ) on B × [1] with coeffi-cients on G such that u∗

0(W, T ) = (Z, t) and u∗1(W, T ) = (Z′, t ′) by putting,

for each morphism f : A → B in B, W(f,0) = Zf , W(f,1) = Z′f and W(f,ı) =

hA ⊗ Z′f , and for each pair of composable morphisms in B, A

f→ Bg→ C,

T(g,0),(f,0) = tg,f : Zf ⊗Zg → Zgf , T(g,1),(f,1) = t ′g,f : Z′f ⊗Z′

g → Z′gf , T(g,1),(f,ı) =

1 ⊗ t ′g,f : hA ⊗Z′f ⊗Z′

g → hA ⊗Z′gf and T(g,ı),(f,0): Zf ⊗ hB ⊗Z′

g → hA ⊗Z′gf

is the composition

Zf ⊗ hB ⊗ Z′g

ϕf ⊗1−−−−→ hA ⊗ Z′f ⊗ Z′

g

1⊗t ′g,f−−−−→ hA ⊗ Z′gf .

Since the functors (Ner(−),Ner(G,⊗)) and Z2(−,G) are naturally equivalent

and, for each category B, Ner(B × [1]) = Ner(B)× Ner([1]) = Ner(B)×7[1],by (26) we have the commutative diagram


Z2(B × [1],G)

�

Z2(B,G)

�

H2(B,G)

(Ner(B)× 7[1],Ner(G,⊗)) (Ner(B),Ner(G,⊗)) [Ner(B),Ner(G,⊗)]

and therefore the induced map H2(B,G) → [Ner(B),Ner(G,⊗)] is a bijection. ✷

The above theorem, together with Theorem 4.9, also establishes that there is anatural bijection between the collection of equivalence classes of B-torsors undera categorical group G and the collection of homotopy classes of simplicial mapsfrom Ner(B) to Ner(G,⊗), that is:

THEOREM 5.4. Let B be a small category and let G be a categorical group.There exists a natural bijection,

Tors[B,G] ∼= [Ner(B),Ner(G,⊗)].Nerves of categorical groups are Kan complexes and then, for any small cate-

gory B and any categorical group G, the geometrical realization functor induces abijection [11], [Ner(B),Ner(G,⊗)] ∼= [B(B), B(G,⊗)] between the set of homo-topy classes of simplicial maps from Ner(B) to Ner(G,⊗) and the set of homotopyclasses of continuous maps from the classifying space of B to the classifying spaceof the categorical group. Then, by Theorems 5.3 and 5.4, we have:

THEOREM 5.5. If a CW-complex X has the homotopy type of a category B anda CW-complex Y has the homotopy type of a categorical group G, then there arebijections:

Tors[B,G] ∼= [X,Y ] ∼= H2(B,G).

Let us recall (see Proposition 5.1) that a CW-complex Y has the homotopy typeof a categorical group if, and only if, Y is path-connected and πi(Y, ∗) = 0 for alli �= 1, 2. In this case, Y has the homotopy type of the categorical group of loopsP2(Y, ∗) or any other tensor equivalent to it, since tensor equivalent categoricalgroups have homotopy equivalent classifying spaces [10], in the same way thatequivalent categories also have homotopy equivalent classifying spaces [28].

Theorem 5.5 generalizes other known homotopy classification results, mainlythe following:

EXAMPLE 5.6. Let X, Y be Eilenberg–Mac Lane spaces, X of type K(G, 1) andY of type K(A, 2). Then X has the homotopy type of the category with only oneobject defined byG and Y has the homotopy type of the categorical group with onlyone object defined by A, in which both the composition and the tensor product aredefined by the group operation (see Example 5.2).


Since Tors[G,A] ∼= Extcen (G,A), the set of equivalence classes of central ex-tensions of G by A (Example 3.9) and H

2(G,A) ∼= H 2(G,A) the usual Eilenberg–Mac Lane cohomology of the group G with coefficients on A (Example 4.5), wesee that the bijections of Theorem 5.5 particularize to the Eilenberg–Mac Laneclassification and interpretation theorems:

Extcen (G,A) ∼= [X,Y ] ∼= H 2(G,A).

We conclude with more examples.

EXAMPLE 5.7. Let B be a small category and K(A, 2) an Eilenberg–Mac Lanespace. Then, H

2(B, A) = H 2(B(B), A), the singular cohomology of the classi-fying space B(B) with coefficients in A, and so a part of 5.5 is the well-knownbijection [B(B),K(A, 2)] ∼= H 2(B(B), A). However, the other part gives a newinterpretation of [B(B),K(A, 2)] in terms of the set Tors[B, A].EXAMPLE 5.8. Let X, Y be CW-complexes. Let us suppose that πi(X, ∗) =0 for all i ≥ 2 and any base point ∗ ∈ X0 (Xk denotes the k-skeleton of X),and that Y is path-connected such that πj(Y, ∗) = 0 for all j ≥ 3. Then, X ishomotopy equivalent to the classifying space of its fundamental groupoid P1(X)

and Y is homotopy equivalent to the classifying space of its categorical group ofloops P2(Y, ∗) for any ∗ ∈ Y 0. Thus, Theorem 5.5 gives bijections,

Tors[P1(X),P2(Y, ∗)] ∼= [X,Y ] ∼= H2(P1(X),P2(Y, ∗)).

In the above bijections, the fundamental groupoid P1(X) can be substituted byany other one equivalent to it. For example, we can consider the relative funda-mental groupoid [6], P1(X,X

0) which is the full subgroupoid of P1(X) whoseobjects are the points ∗ ∈ X0. The categorical group P2(Y, ∗) can also be sub-stituted by any other monoidal equivalent to it; for instance (see Example 4.4),we can take the strict categorical group W(Y, ∗) defined by Whitehead’s crossedmodule W = (π2(Y, Y

1, ∗), π1(Y1, ∗), ϕ, ∂), where ∂: π2(Y, Y

1, ∗) → π1(Y1, ∗)

is the boundary map and ϕ represents the standard action of π1(Y1, ∗) on

π2(Y, Y1, ∗). It is proved in [10] that P2(Y, ∗) is tensor equivalent to W(Y, ∗).

Thus, we also have bijections,

Tors[P1(X,X0),W(Y, ∗)] ∼= [X,Y ] ∼= H

2(P1(X,X0),W(Y, ∗)).

We shall note that, in the literature, there are other classification results for ho-motopy classes of maps into 2-types. For instance, Brown and Higgins in [5] showthat, if X is a CW-complex with skeletal filtration X and C is any crossed complex,there is a natural bijection of homotopy classes [X,BC] ∼= [πX, C] where πX isthe fundamental crossed complex of X and BC is the image of C under the classi-fying space functor B: Crossed complexes → CW -complexes. A similar result for3-types was given in [15]. The special case when C is a reduced crossed module


and with a different definition of BC was proved in [7]. These results seem tobe more adapted to computation since they use models for homotopy types whichsatisfy a general Van Kampen theorem. Next example shows that, in some cases,small categories can be easy to handle.

EXAMPLE 5.9. Let X be a CW-complex with the homotopy type of a polyhedron|K| associated to a simplicial complex K. If we regard K as the category definedby the ordered set of its simplices, then the classifying space B(K) is the polyhe-dron defined by the simplicial complex whose vertices are the simplices of K andwhose simplices are finite non-empty collections of simplices of K that are totallyordered. That is, B(K) = |Sd(K)| is the polyhedron defined by the barycentricsubdivision of K. Thus, X has the homotopy type of category K.

Then, if Y has the homotopy type of a categorical group G (i.e. Y is path-connected, πi(Y, ∗) = 0 for all i ≥ 3 and G is tensor equivalent to P2(Y, ∗)),Theorem 5.5 gives bijections,

Tors[K,G] ∼= [X,Y ] ∼= H2(K,G).

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