For an (∞,1)-category, a simplicial object in is an (∞,1)-functor
from the opposite category of the simplex category into .
The -category of simplicial objects in and morphisms between them is the (∞,1)-category of (∞,1)-functors
For instance (Lurie, def. 220.127.116.11).
A cosimplicial object in is a simplicial object in the opposite category .
Powering over simplicial sets
Assume that has all (∞,1)-limits. The following is a model for the powering of simplicial objects in by simplicial sets.
Let be an (∞,1)-category incarnated as a quasi-category, and let be a simplicial object. Then for any simplicial set, write
for the the composite (∞,1)-functor of with the projection from (the opposite category of) the category of simplices of , and write
for the (∞,1)-limit over it (if it exists).
This is discussed in (Lurie HTT 4.2.3, notation 18.104.22.168). See also around (Lurie 2, notation 1.1.7).
For the simplicial set consisting of two consecutive edges, we have for that
is the homotopy fiber product in
For and the following are equivalent
the induced morphism of cone -categoris is an equivalence of (∞,1)-categories;
the induced morphism of (∞,1)-limits is an equivalence.
(The first perspective is used in (Lurie), the second in (Lurie2).)
In one direction: the limit is the terminal object in the cone category, and so is preserved by equivalences of cone categories. (This direction appears as (Lurie, prop. 22.214.171.124)). Conversely, the limits is the object representing cones, and hence an equivalence of limits induces an equivalence of cone categories.
This is (Lurie, prop. 126.96.36.199).
If is an (∞,1)-topos then is a cohesive (∞,1)-topos over . For more see at cohesive (∞,1)-topos - Examples - Simplicial objects.
If is a locally cartesian closed (∞,1)-category whose internal language is homotopy type theory, then the internal language of is that homotopy type theory equipped with the axioms for a linear interval object. (…)
Geometric realization and filtering
The geometric realization of a simplicial object is, if it exists, the (∞,1)-colimit over the corresponding (∞,1)-functor .
Hence the geometric realization of a cosimplicial object – called its totalization – is the (∞,1)-limit over .
The geometric realization of the simplicial skeleta of
constitutes a filtering on the geometric realization of itself
If is a stable (∞,1)-category, then the the corresponding spectral sequence of a filtered stable homotopy type is the spectral sequence of a simplicial stable homotopy type.
The following statement is the infinity-Dold-Kan correspondence.
Let be a stable (∞,1)-category. Then the (∞,1)-categories of non-negatively graded sequences in is equivalent to the (∞,1)-category of simplicial objects in an (∞,1)-category in
Under this equivalence, a simplicial object is sent to the sequence of geometric realizations ((∞,1)-colimits) of its simplicial skeleta
This constitutes a filtering on the geometric realization of itself
(Higher Algebra, theorem 188.8.131.52)
Internal category objects
Simplicial objects in general (∞,1)-categories are discussed in
Related discussion is also in
Simplicial obects in stable (∞,1)-categories are discussed in