nLab
groupoid object in an (infinity,1)-category

Context

Group Theory

Internal (,1)(\infty,1)-Categories

(,1)(\infty,1)-Category theory

Contents

Idea

A group object in an ordinary category CC with pullbacks is an internal group. More generally, there is the notion of an internal groupoid in a category CC.

By the logic of vertical categorification, an internal \infty-group or internal ∞-groupoid may be defined as a group(oid) object internal to an (∞,1)-category CC with (∞,1)-pullbacks. As described there, in full generality this involves not only a weakening of the usual associativity and unit laws up to homotopy, but requires specification of coherence laws of these homotopies up to higher homotopy, and so on.

A group object in an (∞,1)-category generalizes and unifies two familiar concepts:

Of particular relevance are such group objects that define effective quotients

A groupoid object is then accordingly the many-object version of a group object.

But notice the following. Since this is defined internal to an (∞,1)-category, externally these look like genuine ∞-groupoid and ∞-group objects. For instance a group object in a (2,1)-category such as Grpd is, externally, a 2-group.

Also notice that if the ambient (,1)(\infty,1)-category is in fact an (∞,1)-topos, then every object in there may already be thought of as an “∞-groupoid with geometric structure” (see for instance the discussion at cohesive (∞,1)-topos, but this is true more generally). The relation between the internal groupoid objects then and the objects themselves is (an oid-ification) of that of looping and delooping. Notably for GG any internal group object (externally an ∞-group) the corresponging ordinary object is its delooping object BG\mathbf{B}G, and every pointed connected object in the (,1)(\infty,1)-topos arises this way from an internal group object.

A groupoid object

C 2C 1C 0 \cdots C_2 \stackrel{\to}{\stackrel{\to}{\to}} C_1 \stackrel{\to}{\to} C_0

being effective means that it is the Čech nerve

C 0× C 0//C 1C 0× C 0//C 1C 0C 0× C 0//C 1C 0C 0 \cdots C_0 \times_{C_0//C_1} C_0 \times_{C_0//C_1} C_0 \stackrel{\to}{\stackrel{\to}{\to}} C_0 \times_{C_0//C_1} C_0 \stackrel{\to}{\to} C_0

of its action groupoid C 0//C 1C_0//C_1 (the (∞,1)-colimit over its diagram)

C 2C 1C 0C 0//C 1:=colim iC i. \cdots C_2 \stackrel{\to}{\stackrel{\to}{\to}} C_1 \stackrel{\to}{\to} C_0 \to C_0//C_1 := colim_i C_i \,.

Accordingly, groupoid objects in an (,1)(\infty,1)-category play a central role in the theory of principal ∞-bundles.

Notice that one of the four characterizing properties of an (∞,1)-topos by the higher analog of the Giraud theorem is that every groupoid object is effective.

Definition (complete Segal-space style)

Groupoid object

The following definition follows in style the definition of a complete Segal space object.

Definition

For CC an (∞,1)-category, a groupoid object in CC is a simplicial object in an (∞,1)-category

A:Δ opC A : \Delta^{op} \to C

such that for all partitions SSS \cup S' of [n][n] that share precisely one vertex ss, we have that

A([n]) A(S) A(S) A({s}) \array{ A([n]) &\to & A(S) \\ \downarrow && \downarrow \\ A(S') &\to& A(\{s\}) }

is a (∞,1)-pullback diagram in CC. Here, by a partition SSS \cup S' of [n][n] that share precisely one vertex ss, we mean two subsets SS and SS' of {0,1,,n}\{0,1,\ldots,n\} whose union is {0,1,,n}\{0,1,\ldots,n\} and whose intersection is the singleton {s}\{s\}. The linear order on [n][n] then restricts to the linear order on SS and SS'.

The (,1)(\infty,1)-category of groupoid objects in CC is the full sub-(∞,1)-category

Grpd(C)Func(Δ op,C) Grpd(C) \hookrightarrow Func(\Delta^{op}, C)

of the (∞,1)-category of (∞,1)-functors on those objects that are groupoid objects.

This is HTT, prop. 6.1.2.6, item 4'' with HTT, def. 6.1.2.7.

Remark

If one requires the above condition only for those partitions that are order-preserving, then this yields the definition of a (pre-)category object in an (∞,1)-category.

Definition

A groupoid object A:Δ opCA : \Delta^{op} \to C is the Cech nerve of a morphism A 0BA_0 \to B if AA is the restriction of an augmented simplicial object A +:Δ a opCA^+ : \Delta^{op}_a \to C with A 0 +A 1 +A^+_0 \to A^+_{-1} as the morphism A 0BA_0 \to B, such that the sub-diagram

A 1 + A 0 + A 0 + A 1 + \array{ A^+_1 &\to& A^+_0 \\ \downarrow && \downarrow \\ A^+_0 &\to& A^+_{-1} }

of A +A^+ is a (∞,1)-pullback diagram in CC.

This is HTT, below prop. 6.1.2.11.

If AA is the Čech nerve of a morphism A 0A 1A_0 \to A_{-1}
then the groupoid object is deloopable in the groupoid sense.

Definition

A groupoid object A:Δ opCA : \Delta^{op} \to C is an effective quotient object if the (∞,1)-colimit diagram A +:Δ a opCA^+ : \Delta_a^{op} \to C exists, such that AA is the Cech nerve of A 0 +A 1 +A^+_0 \to A^+_{-1}, i.e. of A 0lim A A_0 \to \lim_\to A_\bullet.

Group object

A group object is a groupoid object U:Δ opCU : \Delta^{op} \to C for which U 0*U_0 \simeq * is a terminal object.

(HTT, def. 7.2.2.1).

It follows (HTT, prop. 7.2.2.4) that a group object is of the form

U=(G×GG*). U = \left( \cdots G \times G \stackrel{\to}{\stackrel{\to}{\to}} G \stackrel{\to}{\to} * \right) \,.

Relation to (,1)(\infty,1)-quotients

Remark

A groupoid object in an (,1)(\infty,1)-category

(A 2A 1A 0) \left( \cdots A_2 \stackrel{\to}{\stackrel{\to}{\to}} A_1 \stackrel{\to}{\to} A_0 \right)

is the (∞,1)-category analog of an internal equivalence relation on A 0A_0, which is just a pair of morphisms

RA 0. R \stackrel{\to}{\to} A_0 \,.

The colimit (coequalizer) of the latter diagram is the quotient of A 0A_0 by the relation RR.

Analogously, the (∞,1)-colimit

lim (Δ opAC) \lim_\to (\Delta^{op} \stackrel{A}{\to} C)

over the simplicial diagram A:Δ opCA : \Delta^{op} \to C is the corresponding (,1)(\infty,1)-quotient.

If we are given a model category presentation of the (∞,1)-category CC, then this (∞,1)-colimit is presented by a homotopy colimit over the corresponding simplicial diagraM a homotopy quotient .

Properties

Equivalent characterizations

We state in prop. 1 below a list of equivalent conditions that characterize a simplicial object in an (∞,1)-category as a groupoid object. This uses the following basic notions, which we review here for convenience.

Definition

For KK \in sSet a simplicial set, write Δ /K\Delta_{/K} for its category of simplices. For X 𝒞 Δ opX_\bullet \in \mathcal{C}^{\Delta^{op}} a simplicial object, write

X[K]:Δ /K opΔ opX𝒞 X[K] \colon \Delta^{op}_{/K} \to \Delta^{op} \stackrel{X}{\to} \mathcal{C}

for the precomposition of X X_\bullet with the canonical projection. Moreover, write

X(K)limX[K] X(K) \coloneqq \underset{\leftarrow}{\lim} X[K]

for the (∞,1)-limit over this composite (∞,1)-functor in 𝒞\mathcal{C} (if it exists). (Notice: square brackets for the composite functor, round brackets for its (,1)(\infty,1)-limit.)

Remark

For X 𝒞 Δ opX_\bullet \in \mathcal{C}^{\Delta^{op}} and KKK \to K' the following are equivalent

  1. the induced morphism of cone (,1)(\infty,1)-categoris 𝒞 X[K]𝒞 X[K]\mathcal{C}_{X[K]} \to \mathcal{C}_{X[K']} is an equivalence of (∞,1)-categories;

  2. the induced morphism of (∞,1)-limits X(K)X(K)X(K) \to X(K') is an equivalence.

(The first perspective is used in (Lurie), the second in (Lurie2).)

Proof

In one direction: the limit is the terminal object in the cone category, and so is preserved by equivalences of cone categories. (This direction appears as (Lurie, prop. 4.1.1.8)). Conversely, the limits is the object representing cones, and hence an equivalence of limits induces an equivalence of cone categories.

Proposition

Let 𝒞\mathcal{C} be an (,1)(\infty,1)-category incarnated explicitly as a quasi-category. Then a simplicial object in 𝒞\mathcal{C} is a groupoid object if the following equivalent conditions hold.

  1. If KKK \to K' is a morphism in sSet which is a weak homotopy equivalence and a bijection on vertices, then the induced morphism on slice-(∞,1)-categories

    𝒞 /X[K]𝒞 /X[K] \mathcal{C}_{/X[K]} \to \mathcal{C}_{/X[K']}

    is an equivalence of (∞,1)-categories (a weak equivalence in the model structure for quasi-categories).

  2. For every n2n \geq 2 and every 0in0 \leq i \leq n, the morphism 𝒞 /X[Δ n]𝒞 /X[Λ i n]\mathcal{C}_{/X[\Delta^n]} \to \mathcal{C}_{/X[\Lambda^n_i]} is an weak equivalence in the model structure for quasi-categories

  3. (…)

Using remark 3 this means equivalently that the simplicial object X X_\bullet is a groupoid precisely if the following

  1. X X_\bullet satisfies the ordinary Segal conditions and the morphism X(Δ 2)X(Λ 0 2)X(\Delta^2) \to X(\Lambda^2_0) is an equivalence.

  2. (…)

The first items appear as (Lurie, prop. 6.1.2.6). The second ones appear in the proofs of (Lurie2, prop. 1.1.8, lemma 1.2.25).

The (,1)(\infty,1)-category of groupoid objects

Proposition

The (,1)(\infty,1)-category of groupoid objects in CC is a reflective sub-(∞,1)-category

Grpd(C)Func(Δ op,C). Grpd(C) \stackrel{\overset{}{\leftarrow}}{\hookrightarrow} Func(\Delta^{op}, C) \,.

This is HTT, prop. 6.1.2.9. In nice cases the image of this reflective subcategory are the effective epimorphisms:

Proposition

If C=HC = \mathbf{H} in an (∞,1)-semitopos there is a natural equivalence of (∞,1)-categories

Grpd(H)(H I) eff Grpd(\mathbf{H}) \simeq (\mathbf{H}^I)_{eff}

between the (,1)(\infty,1)-category of groupoid objects in H\mathbf{H} and the full sub-(∞,1)-category of the arrow category of H\mathbf{H} (the (∞,1)-functor (∞,1)-category Func(Δ[1],H)Func(\Delta[1], \mathbf{H})) on the effective epimorphisms.

This appears below HTT, cor. 6.2.3.5.

Cech nerves

Write Δ a\Delta_a for the augmented simplex category (including the object [1][-1]).

Proposition

An augmented simplicial object A +:Δ a opCA^+ : \Delta_a^{op} \to C is the right Kan extension of its restriction to [1][-1] and [0][0]

{[1][0]} A + 0 C A + Δ a op \array{ \{[-1] \leftarrow [0]\} &\stackrel{A^+|_{\leq 0}}{\to}& C \\ \downarrow & \nearrow_{\mathrlap{A^+}} \\ \Delta_a^{op} }

precisley if A + 0A^+|_{\geq 0} is a groupoid object in CC and the diagram

A 1 A 0 A 0 A 1 \array{ A_1 &\to& A_0 \\ \downarrow && \downarrow \\ A_0 &\to& A_{-1} }

is a (∞,1)-pullback in CC.

AA is called the Cech nerve of A 0A 1A_0 \to A_{-1} if the equivalent conditions of this proposition are satisfied.

Effective quotients

Proposition

In C=C = ∞Grpd every groupoid object is an effective quotient, def. 3.

This is HTT, below remark 6.1.2.15 and HTT, cor. 6.1.3.20.

More generally, this is true for every (∞,1)-topos.

Proposition

In CC is an (∞,1)-topos, then every groupoid object in CC is an effective quotient, def. 3.

This is HTT, theorem 6.1.0.6 (4) iv).

Delooping

For 𝒳\mathcal{X} an (∞,1)-category with (∞,1)-pullbacks and for x:*Xx : * \to X a pointed object in 𝒳\mathcal{X}, its loop space object at xx is the (∞,1)-pullback

Ω xX:=* X* \Omega_x X := {*} \prod_{X} {*}

hence the object universally filling the diagram

Ω xX * x * x X. \array{ \Omega_x X &\to& * \\ \downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{x}} \\ * &\stackrel{x}{\to}& X } \,.

Since this is the beginning of the Cech nerve of *X* \to X, Ω xX\Omega_x X is naturally equipped with the structure of an \infty-group object in 𝒳\mathcal{X}.

Proposition

Let 𝒳\mathcal{X} be an (∞,1)-topos. Then the operation of forming loop space objects constitutes an equivalence of (∞,1)-categories

Ω:PointedConnected(𝒳)Grp(𝒳) \Omega : PointedConnected(\mathcal{X}) \stackrel{\simeq}{\to} Grp(\mathcal{X})

from the full sub-(∞,1)-category of the under-(∞,1)-category */𝒳*/\mathcal{X} of pointed objects on those that are also 0-connected (hence those that have an essentially unique point) with the (,1)(\infty,1)-category of group objects in 𝒳\mathcal{X}.

This is HTT, lemma 7.2.2.11 (1)

The inverse to Ω\Omega we write

B:Grp(𝒳)PointedConnected(𝒳). \mathbf{B} : Grp(\mathcal{X}) \to PointedConnected(\mathcal{X}) \,.

For GGrp(𝒳)G \in Grp(\mathcal{X}) we call BG\mathbf{B}G its delooping.

Examples

Group objects in an (,1)(\infty,1)-topos

When the ambient (∞,1)-category is an (∞,1)-topos then – by the \infty-Giraud axioms – all groupoid objects are effective, meaning that for

BG=lim U \mathbf{B}G = \lim_{\to} U_\bullet

the (∞,1)-colimit over the group object U U_\bullet we have that U U_\bullet is reproduced as the Cech nerve of *BG* \to \mathbf{B}G

(G×GG*)(*× BG*× BG**× BG**). \left( \cdots G \times G \stackrel{\to}{\stackrel{\to}{\to}} G \stackrel{\to}{\to} * \right) \simeq \left( \cdots {*}\times_{\mathbf{B}G}{*}\times_{\mathbf{B}G}{*} \stackrel{\to}{\stackrel{\to}{\to}} {*}\times_{\mathbf{B}G}{*} \stackrel{\to}{\to} * \right) \,.

The object BG\mathbf{B}G is the delooping object of the group object GG.

For more on this see also principal ∞-bundle.

Models for group objects in Grpd\infty Grpd

There is a model category structure that presents the (∞,1)-category of group objects in ∞Grpd: the ∞-groups.

  • The group objects GG themselves are modeled by a model structure on the category sGrpsGrp of simplicial groups.

  • Their delooping spaces BG\mathbf{B}G are modeled by a model structure on the category sSet 0sSet_0 of simplicial sets with a single vertex.

The operation of forming loop space objects constitutes a Quillen equivalence between these two model structures

Ω:sSet 0 QuillensGrp. \Omega : sSet_0 \stackrel{\simeq_{Quillen}}{\to} sGrp \,.

The Quillen equivalence itself is in section 6 there.

Proposition

There exists the transferred model structure on the category sGrpsGrp of simplicial groups along the forgetful functor

U:sGrpsSet Quillen U : sGrp \to sSet_{Quillen}

to the standard model structure on simplicial sets.

This means that a morphism in sGrpsGrp is a

  • weak equivalences

  • or fibration

precisely if it is so in sSet QuillensSet_{Quillen}.

This appears as (GoerssJardine, ch V, theorem. 2.3).

Proposition

There is a model structure on reduced simplicial sets sSet 0sSet_0 (simplicial sets with a single vertex) whose

  • weak equivalences

  • and cofibrations

are those in the standard model structure on simplicial sets.

This appears as (GoerssJardine, ch V, prop. 6.2).

Proposition

The simplicial loop space functor GG and the delooping functor W¯()\bar W(-) (discussed at simplicial group) constitute a Quillen equivalence

(GW¯):sGrW¯GsSet 0. (G \dashv \bar W) : sGr \stackrel{\overset{G}{\leftarrow}}{\underset{\bar W}{\to}} sSet_0 \,.

The (GW¯)(G \dashv \bar W)-unit and counit are weak equivalences:

XW¯GX X \stackrel{\simeq}{\to} \bar W G X
GW¯GG. G \bar W G \stackrel{\simeq}{\to} G \,.

This appears as (GoerssJardine, ch. V prop. 6.3).

References

Groupoid objects in (,1)(\infty,1)-categories are the topic of section 6.1.2 in

Model category presentations of groupoid objects in Grpd\infty Grpd by groupoidal complete Segal spaces are discussed in

  • Julia Bergner,

    Adding inverses to diagrams encoding algebraic structures, Homology, Homotopy and Applications 10 (2008), no. 2, 149–174. (arXiv:0610291)

    Adding inverses to diagrams II: Invertible homotopy theories are spaces, Homology, Homotopy and Applications, Vol. 10 (2008), No. 2, pp.175-193. (web, arXiv:0710.2254)

A standard textbook reference on the model categories presentation of \infty-groups in sSetsSet is chapter V of

Discussion from the point of view of category objects in an (∞,1)-category is in

Revised on October 22, 2013 20:21:40 by Colin Tan? (137.132.3.9)