equivalences in/of $(\infty,1)$-categories
category object in an (∞,1)-category, groupoid object
For $\mathcal{C}$ an (∞,1)-category, a simplicial object in $\mathcal{C}$ is an (∞,1)-functor
from the opposite category of the simplex category into $\mathcal{C}$.
The $(\infty,1)$-category of simplicial objects in $\mathcal{C}$ and morphisms between them is the (∞,1)-category of (∞,1)-functors
For instance (Lurie, def. 6.1.2.2).
For $\mathcal{C}$ a 1-category a simplicial object in $\mathcal{C}$ is a simplicial object in the traditional sense of category theory.
A cosimplicial object in $\mathcal{C}$ is a simplicial object in the opposite category $\mathcal{C}^{op}$.
Assume that $\mathcal{C}$ has all (∞,1)-limits. The following is a model for the powering of simplicial objects in $\mathcal{C}$ by simplicial sets.
Let $\mathcal{C} \in QCat \hookrightarrow sSet$ be an (∞,1)-category incarnated as a quasi-category, and let $X \colon \Delta^{op} \to \mathcal{C}$ be a simplicial object. Then for $K \in sSet$ any simplicial set, write
for the the composite (∞,1)-functor of $X_\bullet$ with the projection from (the opposite category of) the category of simplices of $K$, and write
for the (∞,1)-limit over it (if it exists).
This is discussed in (Lurie HTT 4.2.3, notation 6.1.2.5). See also around (Lurie 2, notation 1.1.7).
The inclusion $\Delta_{/K}^{nd} \hookrightarrow \Delta_{K}$ of the full subcategory on non-degenerate simplicies is a homotopy cofinal functor (as discussed there). Therefore the $(\infty,1)$-limit in def. 3 may equivalently be taken over this category of non-degenerate simplices.
For $K = \Delta^1 \coprod_{\Delta^0} \Delta^1$ the simplicial set consisting of two consecutive edges, we have for $X_\bullet \in \mathcal{C}^{\Delta^\bullet}$ that
is the homotopy fiber product in
For $K = \Delta^n$ itself an $n$-simplex, for some $n \in \mathbb{N}$ the powering reduces to evaluation on that simplex:
This is because the category of non-degenerate simplices of an $n$-simplex has a terminal object (namely that $n$-simplex itself), and so its opposite category has an initial object and the $(\infty,1)$-limit over a diagram with initial object is given by evaluation at that initial object.
For $X_\bullet \in \mathcal{C}^{\Delta^{op}}$ and $K \to K'$ the following are equivalent
the induced morphism of cone $(\infty,1)$-categoris $\mathcal{C}_{X[K]} \to \mathcal{C}_{X[K']}$ is an equivalence of (∞,1)-categories;
the induced morphism of (∞,1)-limits $X(K) \to X(K')$ 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. 4.1.1.8)). Conversely, the limits is the object representing cones, and hence an equivalence of limits induces an equivalence of cone categories.
Let $X \colon \Delta^{op} \to \mathcal{C}$ be a simplicial object which is a groupoid object in an (∞,1)-category.
If $K \to K'$ is a morphism in sSet which is a weak homotopy equivalence and a bijection on vertices, then the induced morphism on slice-(∞,1)-categories
is an equivalence of (∞,1)-categories (a weak equivalence in the model structure for quasi-categories).
Equivalently, by remark 1, we have an equivalence
This is (Lurie, prop. 6.1.2.6).
If $\mathcal{C} = \mathbf{H}$ is an (∞,1)-topos then $\mathcal{C}^{\Delta^{op}}$ is a cohesive (∞,1)-topos over $\mathbf{H}$. For more see at cohesive (∞,1)-topos - Examples - Simplicial objects.
If $\mathcal{C}$ is a locally cartesian closed (∞,1)-category whose internal language is homotopy type theory, then the internal language of $\mathcal{C}^{\Delta^{op}}$ is that homotopy type theory equipped with the axioms for a linear interval object. (…)
The geometric realization ${\vert X_\bullet \vert}$ of a simplicial object $X_\bullet$ is, if it exists, the (∞,1)-colimit over the corresponding (∞,1)-functor $X_\bullet \;\colon\; \Delta^{op} \to \mathcal{C}$.
Hence the geometric realization of a cosimplicial object $\Delta^{op} \to \mathcal{C}^{op}$ – called its totalization – is the (∞,1)-limit over $\Delta \to \mathcal{C}$.
The geometric realization of the simplicial skeleta of $X_\bullet$
constitutes a filtering on the geometric realization of $X_\bullet$ itself
If $\mathcal{C}$ 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 $\mathcal{C}$ be a stable (∞,1)-category. Then the (∞,1)-categories of non-negatively graded sequences in $C$ is equivalent to the (∞,1)-category of simplicial objects in an (∞,1)-category in $\mathcal{C}$
Under this equivalence, a simplicial object $X_\bullet$ is sent to the sequence of geometric realizations ((∞,1)-colimits) of its simplicial skeleta
This constitutes a filtering on the geometric realization of $X_\bullet$ itself
(Higher Algebra, theorem 1.2.4.1)
A pre-category object in an (∞,1)-category $\mathcal{C}$ is a simplicial object which satisfies the Segal conditions;
a category object in an (∞,1)-category is a pre-category object which also satisfies the univalence axiom;
a groupoid object in an (∞,1)-category is a category object all of whose morphisms are equivalences under composition.
Simplicial objects in general (∞,1)-categories are discussed in
Related discussion is also in
Simplicial obects in stable (∞,1)-categories are discussed in