nLab
Infinity-Grpd

Contents

Contents

Idea

Grpd\infty Grpd is the (∞,1)-category of ∞-groupoids, i.e. of (∞,0)-categories. This is the archetypical (∞,1)-topos, the home of classical homotopy theory.

Equivalently this means all of the following:

  1. Grpd\infty Grpd is the simplicial localization of the category Top k{}_k of (weakly Hausdorff) locally compact topological spaces at the weak homotopy equivalences. As such it is the ∞-category-enhancement of the classical homotopy category: τ 0(Grpd)\tau_0(\infty Grpd) \simeq Ho(Top), itself presented by the classical model structure on topological spaces: GrpdL wheTop k\infty Grpd \simeq L_{whe} Top_k.

  2. Grpd\infty Grpd is the simplicial localization of the category sSet of simplicial sets at the simplicial weak homotopy equivalences. As such it is the ∞-category-enhancement of the classical homotopy category: τ 0(Grpd)\tau_0(\infty Grpd) \simeq Ho(sSet), itself presented by the classical model structure on simplicial sets: GrpdL whesSet\infty Grpd \simeq L_{whe} sSet.

    Hence, as a Kan-complex enriched category (a fibrant object in the model structure on sSet-categories) Grpd\infty Grpd is the full sSet enriched-subcategory of sSet on the Kan complexes.

  3. Grpd\infty Grpd is the full sub-(∞,1)-category of (∞,1)Cat on those (∞,1)-categories that are ∞-groupoids.

Properties

As an (,1)(\infty,1)-topos

As an (∞,1)-topos Grpd\infty Grpd is the terminal (,1)(\infty,1)-topos: for every other (∞,1)-sheaf (∞,1)-topos H\mathbf{H} there is up to a contractible space of choices a unique geometric morphism (LConstΓ):HGrpd(LConst \dashv \Gamma) : \mathbf{H}\stackrel{\leftarrow}{\to} \infty Grpd – the global section geometric morphism. See there for more details.

Limits and colimits in Grpd\infty Grpd

Limits and colimits over a (∞,1)-functor with values in Grpd\infty Grpd may be reformulation in terms of the universal fibration of (infinity,1)-categories.

Let the (∞,1)-functor Z| GrpdGrpd opZ|_{Grpd} \to \infty Grpd^{op} be the universal ∞-groupoid fibration whose fiber over the object denoting some \infty-groupoid is that very \infty-groupoid.

Then let XX be any ∞-groupoid and

F:XGrpd F : X \to \infty Grpd

an (∞,1)-functor. Recall that the coCartesian fibration E FXE_F \to X classified by FF is the pullback of the universal fibration of (∞,1)-categories ZZ along F:

E F Z| Grpd X F Grpd \array{ E_F &\to& Z|_{Grpd} \\ \downarrow && \downarrow \\ X &\stackrel{F}{\to}& \infty Grpd }
Proposition

Let the assumptions be as above. Then:

  • The colimit of FF is equivalent to E FE_F:

    E FcolimF E_F \simeq colim F
  • The limit of FF is equivalent to the (∞,1)-groupoid of sections of E FXE_F \to X

    Γ X(E F)limF. \Gamma_X(E_F) \simeq lim F \,.
Proof

The statement for the colimit is corollary 3.3.4.6 in HTT. The statement for the limit is corollary 3.3.3.4.

Subcategories

The n-truncated objects of Grpd\infty Grpd are the n-groupoids (including (-1)-groupoids and the (-2)-groupoid).

category: category

Last revised on August 3, 2020 at 02:56:58. See the history of this page for a list of all contributions to it.