nLab simplicial object in an (infinity,1)-category

Definition

Definition

For $\mathcal{C}$ an (∞,1)-category, a simplicial object in $\mathcal{C}$ is an (∞,1)-functor

$$X \colon \Delta^{op} \to \mathcal{C}$$

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

$$\mathcal{C}^{\Delta^{op}} = Func_\infty(\Delta^{op}, \mathcal{C}) \,.$$

For instance (Lurie, def. 6.1.2.2).

Remark

For $\mathcal{C}$ a 1-category a simplicial object in $\mathcal{C}$ is a simplicial object in the traditional sense of category theory.

Definition

A cosimplicial object in $\mathcal{C}$ is a simplicial object in the opposite category $\mathcal{C}^{op}$.

Properties

Powering over simplicial sets

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.

Definition

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

$$X[K] \;\colon\; \Delta_{/K}^{op} \longrightarrow \Delta^{op} \stackrel{X}{\longrightarrow} \mathcal{C}$$

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

$$X(K) \;\colon\; \underset{\leftarrow}{\lim} \left( \Delta_{/K}^{op} \longrightarrow \Delta^{op} \stackrel{X}{\longrightarrow} \mathcal{C} \right)$$

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).

Remark

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. may equivalently be taken over this category of non-degenerate simplices.

Example

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

$$X(K) \simeq X_1 \underset{X_0}{\times} X_1$$

is the homotopy fiber product in

$$\array{ && X(K) \\ & \swarrow && \searrow \\ X_1 &&&& X_1 \\ & {}_{\mathllap{\partial_1}}\searrow && \swarrow_{\mathrlap{\partial_0}} \\ && X_0 } \,.$$

Example

For $K = \Delta^n$ itself an $n$-simplex, for some $n \in \mathbb{N}$ the powering reduces to evaluation on that simplex:

$$X(\Delta^n) \simeq X_n \,.$$

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.

Remark

For $X_\bullet \in \mathcal{C}^{\Delta^{op}}$ and $K \to K'$ the following are equivalent

1. the induced morphism of cone $(\infty,1)$-categoris $\mathcal{C}_{X[K]} \to \mathcal{C}_{X[K']}$ is an equivalence of (∞,1)-categories;

2. 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).)

Proof

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.

Proposition

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

$$\mathcal{C}_{/X[K]} \to \mathcal{C}_{/X[K']}$$

is an equivalence of (∞,1)-categories (a weak equivalence in the model structure for quasi-categories).

Equivalently, by remark , we have an equivalence

$$X(K) \to X(K') \,.$$

This is (Lurie, prop. 6.1.2.6).

Cohesion

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.

Internal language

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. (…)

Geometric realization and filtering

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 $X_\bullet \;\colon\; \Delta^{op} \to \mathcal{C}$.

$${\vert X_\bullet \vert} \coloneqq \underset{\longrightarrow}{\lim}_n X_n \,.$$

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 $X_\bullet$

$${\vert sk_0 X_\bullet \vert} \to {\vert sk_1 X_\bullet \vert} \to {\vert sk_2 X_\bullet \vert} \to \cdots \,.$$

constitutes a filtering on the geometric realization of $X_\bullet$ itself

$${\vert X_\bullet \vert} \simeq \underset{\longrightarrow}{\lim}_n {\vert sk_n X_\bullet \vert} \,.$$

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.

$\infty$-Dold-Kan correspondence

The following statement is the infinity-Dold-Kan correspondence.

Proposition

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 $\mathcal{C}$

$$Fun(N(\mathbb{Z}_{\geq 0}), C) \simeq Fun(N(\Delta)^{op}, C) \,.$$

Under this equivalence, a simplicial object $X_\bullet$ is sent to the sequence of geometric realizations ((∞,1)-colimits) of its simplicial skeleta

$${\vert sk_0 X_\bullet \vert} \to {\vert sk_1 X_\bullet \vert} \to {\vert sk_2 X_\bullet \vert} \to \cdots \,.$$

This constitutes a filtering on the geometric realization of $X_\bullet$ itself

$${\vert X_\bullet \vert} \simeq \underset{\longrightarrow}{\lim}_n {\vert sk_n X_\bullet \vert} \,.$$

(Higher Algebra, theorem 1.2.4.1)

Examples

Internal category objects

• 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.

Related concepts

• simplicial object

  • simplicial set

• semi-simplicial object

  • semisimplicial set

• simplicial homotopy theory

• homotopy Kan fibration

References

Simplicial objects in general (∞,1)-categories are discussed in

• Jacob Lurie, section 6.1.2 of Higher Topos Theory

Related discussion is also in

• Jacob Lurie, (∞,2)-Categories and the Goodwillie Calculus (arXiv:0905.0462)

Simplicial obects in stable (∞,1)-categories are discussed in

• Jacob Lurie, section 1.2.4 of Higher Algebra