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.
As an -category
As an simplicially enriched category is the full SSet-enriched subcategory of SSet on Kan complexes.
As an enriched model category
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 is also the full -subcategory of Top on spaces of the homotopy type of a CW-complex.
As an -topos
As an (∞,1)-topos is the terminal -topos: for every other (∞,1)-sheaf (∞,1)-topos there is up to a contractible space of choices a unique geometric morphism – the global section geometric morphism. See there for more details.
Limits and colimits in
Limits and colimits over a (∞,1)-functor with values in may be reformulation in terms of the universal fibration of (infinity,1)-categories.
Let the (∞,1)-functor be the universal ∞-groupoid fibration whose fiber over the object denoting some -groupoid is that very -groupoid.
Then let be any ∞-groupoid and
an (∞,1)-functor. Recall that the coCartesian fibration classified by is the pullback of the universal fibration of (∞,1)-categories along F:
Let the assumptions be as above. Then:
The colimit of is equivalent to :
The limit of is equivalent to the (∞,1)-groupoid of sections of
The statement for the colimit is corollary 184.108.40.206 in HTT. The statement for the limit is corollary 220.127.116.11.
The n-truncated objects of are the n-groupoids. (including (-1)-groupoids and the (-2)-groupoid).