nLab
Infinity-Grpd

$\mathrm{\infty }\mathrm{Grpd}$ is the (∞,1)-category of ∞-groupoids, i.e. of (∞,0)-categories.

It is the full subcategory of (∞,1)Cat on those (∞,1)-categories that are ∞-groupoids.

It is also the archetypical (∞,1)-topos.

Incarnations

As an $\mathrm{sSet}$-category

As an simplicially enriched category $\mathrm{\infty }\mathrm{Grpd}$ is the full SSet-enriched subcategory of SSet on Kan complexes.

As an enriched model category

$\mathrm{\infty }\mathrm{Grpd}$ is the (∞,1)-category that is presented by the Quillen model structure on simplicial sets.

As a Kan-complex enriched category this is the full sSet-subcategory on fibrant-cofibrant objects of the Quillen model structure on simplicial sets.

Under the homotopy hypothesis-theorem, this means that $\mathrm{\infty }\mathrm{Grpd}$ is also the full $(\mathrm{\infty },1)$-subcategory of Top on spaces of the homotopy type of a CW-complex.

Properties

As an $(\mathrm{\infty },1)$-topos

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

Limits and colimits in $\mathrm{\infty }\mathrm{Grpd}$

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

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

Then let $X$ be any ∞-groupoid and

$$F:X\to \mathrm{\infty }\mathrm{Grpd}$$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:

$$\begin{array}{ccc}{E}_F& \to & Z{\mid }_{\mathrm{Grpd}}\\ \downarrow & & \downarrow \\ X& \stackrel{F}{\to }& \mathrm{\infty }\mathrm{Grpd}\end{array}$$\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 $E_F$:

  $$E_F\simeq \mathrm{colim}F$$E_F \simeq colim F

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

  $${\Gamma }_X(E_F)\simeq \lim F.$$\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 $\mathrm{\infty }\mathrm{Grpd}$ are the n-groupoids. (including (-1)-groupoids and the (-2)-groupoid).

Related categories

• Set

• Grpd, $\mathrm{\infty }\mathrm{Grpd}$

• Cat, (∞,1)Cat

• (∞,n)Cat

category: category