# nLab Infinity-Grpd

### Context

#### categories of categories

$(n+1,r+1)$-categories of (n,r)-categories

$\infty Grpd$ 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.

# Contents

## Incarnations

### As an $sSet$-category

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

### As an enriched model category

$\infty Grpd$ 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 $\infty Grpd$ is also the full $(\infty,1)$-subcategory of Top on spaces of the homotopy type of a CW-complex.

## 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).

category: category

Revised on October 30, 2012 14:18:14 by Stephan Alexander Spahn (79.227.188.115)