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1 .2 The ∞-category of ∞-groupoids

In document Contributions of the present thesis (página 37-40)

|Aut(x)|.

This is independent of the choice of the x in the same connected component since an arrow between two choices induces an isomorphism of vertex groups.

Homotopy equivalent groupoids have the same cardinality.

Example 1.1.6. If X is a finite set considered as a groupoid, then the groupoid cardinality coincides with the set cardinality. If G is a group considered as a one-object groupoid, then the groupoid cardinality is the inverse of the order of the group.

Remark 1.1.7. The groupoid cardinality is a standard construction in physics and combinatorics: one can sum over the different isomorphism classes of objects and for each object divide out by the order of its symmetry group.

The homotopy cardinality of a finite map of groupoids X−→ S isp

|X−→ Sp | := X

x∈π0S

|Xs|

|Aut(s)|δsQπ0S,

where Xs is the homotopy fibre, andQπ0S is the vector space spanned by iso-classes, denoted by the formal symbol δsfor s ∈ π0S. Remark that the cardinality of the basis object 1−−→ S in grpdpsq /Sis the basis vector δsinQπ0S.

1 .2 The ∞-category of ∞-groupoids

While most of the examples are formulated in the 2-category of groupoids, our theoretical results are in the setting of∞-categories, which is a natural generalisation, and is for most of our purposes as usable as the theory of category thanks to Joyal [35,36] and Lurie [46].

We review here the needed notions from the theory of∞-categories and we give a glimpse of homotopy linear algebra [24].

1.2.1 Infinity-groupoids, functor category, and diagrams

Our∞-categories are quasi-categories, these are simplicial sets satisfying the weak Kan condition: every inner horn admits a filler (not necessarily unique).

The theory of quasi-categories has been substantially developed by Joyal [35,

36] and Lurie [46]. There are other models of∞-categories, for example Segal categories, or complete Segal spaces, see [10]. A model-independent abstract formulation is being developed in the work in progress [50].

An ∞-groupoid is an ∞-category in which all morphisms are invertible.

They are precisely Kan complexes: simplicial sets in which every horn admits a filler (and not only the inner ones). We work in the category of ∞-groupoids, denotedS, following the notation of [25]. We sometimes use the word space instead of∞-groupoid. ∞-groupoids have an analogous role to sets in the 1-category theory.

Defining ∞-categories by describing the simplices in all dimensions, and verify filler conditions is more difficult than in the 1 or 2-category setting. Instead, we obtain new ∞-categories from already existing ones and constructions that guarantees we obtain ∞-categories. Between two objects X, Y of a∞-category C , there is a mapping space MapC(X, Y) which is an∞-groupoid. Between two ∞-categories, there is a functor ∞-category Fun(C , D), whose objects are the ∞-functors from C to D, morphisms are the corresponding homotopies, etc. A commutative diagram of shape I in an

∞-category is an object in the functor ∞-category Fun(I, C ). For example, a commutative triangle is an object in Fun(∆2,C ), a commutative square is an object in Fun(∆1× ∆1,C ).

1.2.2 Pullbacks, fibres

Given an∞-category C and a square σ: ∆1× ∆1 →C , denoted

X0 X

Y0 Y.

p0

q0 q

p

There is an isomorphism of simplicial sets ∆1× ∆1 ' ∆0? Λ22 (cone over Λ22) and it makes sense to ask whether or not σ is a limit diagram (a cone which is universal in the∞-categorical sense). If σ is a limit, we say σ is a pullback square, and write X0= X×YY0.

Lemma 1.2.1([46, Lemma 4.4.2.1]). Given a prism diagram of∞-groupoids

X X0 X00

Y Y0 Y00

y

in which the right-hand square is a pullback. Then the outer rectangle is a pullback if and only if the left-hand square is.

Remark 1.2.2. We talk about a prism, it is a ∆1× ∆2-diagram, so consisting of three squares and two triangles. We have not drawn the square whose horizontal sides are composites of the horizontal arrows. The triangles are not drawn either, they are the fillers that exist by the axioms of∞-categories.

Given a map of∞-groupoids p : X → S and an object s ∈ S, the fibre Xs

of p over s is the pullback

Xs X

1 S.

y p

psq

A map of∞-groupoids is a monomorphism when its fibres are either empty or contractible. If f : X → Y is a monomorphism, then there is a complement Z := Y\X such that Y ' X + Z; a monomorphism is essentially an equivalence from X onto some connected components of Y.

1.2.3 Homotopy linear algebra

Recall that the objects of the slice∞-category S/Iare maps of∞-groupoids with codomain I. For the terminal object ∗, we haveS/∗ 'S, as in the slice category in ordinary category theory. For every∞-groupoid I, we have the following fundamental equivalence (follows from [46, Theorem 2.2.1.2]):

S/I' Fun (I,S)

which takes X → I to the functor sending i to the fibre Xi. Pullback along a morphism f : J → I, defines an functor f:S/I → S/J. This functor is right adjoint to the functor f!:S/J→S/Igiven by post-composing with f.

A span is a pair of∞-groupoid maps with common domain I←− Mp −→ J.q It induces a functor between the slices by pullback and post-composition

S/I p

−→S/M q!

−→S/J.

A functor is linear if it is homotopy equivalent to a functor induced by a span. The following Beck-Chevalley rule holds for∞-groupoids: for any pullback square

J I

V U,

f

p y

q

g

the functors p!f, gq!:S/I → S/V are naturally homotopy equivalent (see [31] for the technical details regarding coherence of these equivalences). By the Beck-Chevalley rule, the composition of two linear functor is linear.

We denote by LIN the symmetric monoidal∞-category who objects are slice ∞-categories S/I and morphisms are linear functors, with the tensor product induced by the cartesian product:

S/I⊗S/J:=S/I×J, with neutral objectS ' S/1.

The∞-category S/Iplays the role of the vector space with basis I. The presheaf categorySI can be considered the linear dual of the slice category S/Isince

LIN(S/I,S) ' S/I'SI.

A span I ←− M −→ J defines both a linear functor S/I −→ S/J and the dual linear functorSJ→SI.

For an extended treatment of linear functors and homotopy linear algebra, we refer to [24].

1.2.4 Cardinality

An∞-groupoid X is locally finite if at each base point x the homotopy groups πi(X, x) are finite for i > 1 and are trivial for i sufficiently large. It is called finite if furthermore it has only finitely many components. We denote byF (following the notation of [26]) the∞-category of finite ∞-groupoids. A map is finite if each fibre is finite. A pullback of any homotopy finite map is again finite. A span I←− Mp −→ J and the corresponding linear functorq S/I−→ S/J

are finite if the map p is finite.

Proposition 1.2.3([24, proposition 4.3]). Let I, J, M be locally finite∞-groupoids and I←− Mp −→ J a finite span. Then the induced finite linear functorq S/I −→ S/J

restricts toF/I−→F/J.

The cardinality [7] of a finite∞-groupoid X is the alternating product of the cardinalities of the homotopy groups

|X| = X

x∈π0(X)

Y k=1

k(X, x)|(−1)k.

For a locally finite∞-groupoid S, there is a notion of cardinality |–|: F/SQπ0Ssending a basis element psq to the basis element δs=|psq|.

In document Contributions of the present thesis (página 37-40)