nLab Infinity-Grpd

Idea

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

2. $\infty Grpd$ 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: $\tau_0(\infty Grpd) \simeq$ Ho(sSet), itself presented by the classical model structure on simplicial sets: $\infty Grpd \simeq L_{whe} sSet$.

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

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

Properties

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

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

Limits and colimits in $\infty Grpd$

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

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

Then let $X$ be any ∞-groupoid and

$$F : X \to \infty Grpd$$

an (∞,1)-functor. Recall that the coCartesian fibration $E_F \to X$ classified by $F$ is the pullback of the universal fibration of (∞,1)-categories $Z$ along F:

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

Proposition

Let the assumptions be as above. Then:

• The colimit of $F$ is equivalent to $E_F$:

  $$E_F \simeq colim F$$

• The limit of $F$ is equivalent to the (∞,1)-groupoid of sections of $E_F \to X$

  $$\Gamma_X(E_F) \simeq lim F \,.$$

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 $\infty Grpd$ are the n-groupoids (including (-1)-groupoids and the (-2)-groupoid).

Related categories

• Set

• Grpd,

  $\infty Grpd$

  Top, Ho(Top)

• Cat, (∞,1)Cat

• (∞,n)Cat

category: category