nLab simplicial infinity-groupoid

Definition

Definition

A simplicial $\infty$-groupoid or simplicial anima is an (∞,1)-functor

$$X \colon \Delta^{op} \to \infty\mathrm{Grpd}$$

from the opposite category of the simplex category into the (∞,1)-category ∞Grpd of ∞-groupoids.

Examples

• A $(\infty,1)$-precategory $\mathcal{C}$ is a simplicial infinity-groupoid which satisfies the Segal conditions;

• A $(\infty,1)$-category is a $(\infty,1)$-precategory which also satisfies the univalence axiom;

• An $\infty$-groupoid or discrete $(\infty,1)$-category is a $(\infty,1)$-category all of whose morphisms are equivalences under composition.

 (∞,1)-category of simplicial ∞-groupoids

The $(\infty,1)$-category of simplicial $\infty$-groupoids and morphisms between them is the (∞,1)-category of (∞,1)-functors

$$\infty\mathrm{Grpd}^{\Delta^{op}} = Func_\infty(\Delta^{op}, \infty\mathrm{Grpd}) \,.$$

$\infty\mathrm{Grpd}^{\Delta^{op}}$ is the classifying $(\infty,1)$-topos for linear intervals.

There is an inclusion

$$\infty\mathrm{Grpd} \hookrightarrow (\infty,1)\mathrm{Cat} \hookrightarrow \infty\mathrm{Grpd}^{\Delta^{op}}$$

of $\infty$-groupoids and of $(\infty,1)$-categories inside $\infty\mathrm{Grpd}^{\Delta^{op}}$. Furthermore, the Sierpinski $(\infty,1)$-topos embeds into $\infty\mathrm{Grpd}^{\Delta^{op}}$

$$\infty\mathrm{Grpd}^{\Delta^1} \hookrightarrow \infty\mathrm{Grpd}^{\Delta^{op}}$$

There is also an automorphic $(\infty,1)$-functor

$$(-)^\op : \infty\mathrm{Grpd}^{\Delta^\op} \to \infty\mathrm{Grpd}^{\Delta^\op}$$

on $\infty\mathrm{Grpd}^{\Delta^{op}}$ which takes a simplicial $\infty$-groupoid to its opposite simplicial $\infty$-groupoid. When restricted to the $(\infty,1)$-subcategory $(\infty,1)\mathrm{Cat} \hookrightarrow \infty\mathrm{Grpd}^{\Delta^\op}$, the $(-)^\op$ $(\infty,1)$-functor takes $(\infty,1)$-categories to its opposite $(\infty,1)$-category.

The simplicial interval $\Delta^1 \in \infty\mathrm{Grpd}^{\Delta^{op}}$ (regarded under (∞,1)-Yoneda embedding) is a tiny object: there is an amazing right adjoint

$$(-)^{1/\Delta^1} : \infty\mathrm{Grpd}^{\Delta^\op} \to \infty\mathrm{Grpd}^{\Delta^\op}$$

on $\infty\mathrm{Grpd}^{\Delta^{op}}$.

Cohesion

Since $\infty\mathrm{Grpd}$ is an (∞,1)-topos, $\infty\mathrm{Grpd}^{\Delta^{op}}$ is a cohesive (∞,1)-topos over $\infty\mathrm{Grpd}$:

$$\infty\mathrm{Grpd}^{\Delta^{op}} \stackrel{\Pi_I}{\stackrel{\longrightarrow}{\stackrel{\overset{Disc_I}{\longleftarrow}}{\stackrel{\overset{\Gamma_I}{\longrightarrow}}{\underset{coDisc_I}{\longleftarrow}}}}} \infty\mathrm{Grpd} \,.$$

Here

• $\Pi_I$ sends a simplicial $\infty$-groupoid to the homotopy colimit over its components, hence to its “geometric realization” as seen in $\infty\mathrm{Grpd}$.

• $\Gamma_I$ evaluates on the 0-simplex;

• $Disc_I$ sends a simplicial $\infty$-groupoid to the simplicial object which is simplicially constant on $A$.

Hence cohesion of $\infty\mathrm{Grpd}^{\Delta^{op}}$ relative to $\infty\mathrm{Grpd}$ expresses the existence of a discrete and directed notion of path.

The simplicial interval $\Delta^1 \in \infty\mathrm{Grpd}^{\Delta^{op}}$ (regarded under (∞,1)-Yoneda embedding) exhibits the cohesion of $\infty\mathrm{Grpd}^{\Delta^{op}}$ over $\infty\mathrm{Grpd}$, in that the relative shape modality $\Pi_I$ is equivalent to the localization at $\Delta^1$

$$\Pi_I \simeq L_{\Delta^1} \,.$$

 Internal logic

The $(\infty,1)$-category of simplicial $\infty$-groupoids is the internal logic is given by simplicial type theory, a cohesive modal homotopy type theory equipped with the axioms for a linear interval and an axiom of cohesion for the linear interval.

 Models

Simplicial $\infty$-groupoids can be modeled by bisimplicial sets.

Related concepts

• infinity-groupoid

• cubical infinity-groupoid

• (infinity,1)-functor

• simplex category

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

• simplicial set

• simplicial object

• simplicial type theory

• bisimplicial set

• opposite simplicial infinity-groupoid