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 $(\mathrm{\infty }\mathrm{,1})$-subcategory of Top on spaces of the homotopy type of a CW-complex.

Properties

• As an (∞,1)-topos $\mathrm{\infty }\mathrm{Grpd}$ is the terminal $(\mathrm{\infty }\mathrm{,1})$-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.

category: category