Simplicial diagram

In mathematics, especially algebraic topology, a simplicial diagram is a diagram indexed by the simplex category (= the category consisting of all $[n]=\{0,1,\cdots n\}$ and the order-preserving functions).

Formally, a simplicial diagram in a category or an ∞-category C is a contraviant functor from the simplex category to C. Thus, it is the same thing as a simplicial object but is typically thought of as a sequence of objects in C that is depicted using multiple arrows

$\cdots \underset{\rightrightarrows }{\rightrightarrows }{U}_2\underrightarrow{\rightrightarrows }{U}_1\rightrightarrows U_0$

where $U_n$ is the image of $[n]$ from $\mathrm{\Delta }$ in C.

A typical example is the Čech nerve of a map $U\to X$; i.e., $U_0=U,U_1=U{\times }_{X}U,\dots$. If F is a presheaf with values in an ∞-category and $U_{\bullet }$ a Čech nerve, then $F(U_{\bullet })$ is a cosimplicial diagram and saying $F$ is a sheaf exactly means that $F(X)$ is the limit of $F(U_{\bullet })$ for each $U\to X$ in a Grothendieck topology. See also: simplicial presheaf.

If $U_{\bullet }$ is a simplicial diagram, then the colimit

$[U_{\bullet }]:=\underset{[n]\in \mathrm{\Delta }}{\underrightarrow{\lim }}{U}_n$

is called the geometric realization of $U_{\bullet }$. For example, if $U_n=X\times G^n$ is an action groupoid, then the geometric realization in Grpd is the quotient groupoid $[X/G]$ which contains more information than the set-theoretic quotient $X/G$. A quotient stack is an instance of this construction (perhaps up to stackification).

The limit of a cosimplicial diagram is called the totalization of it.

Augmented simplicial diagram

Sometimes one uses an augmented version of a simplicial diagram. Formally, an augmented simplicial diagram is a contravariant functor from the augmented simplex category ${\mathrm{\Delta }}_{\text{<mrow data-mjx-texclass="ORD">aug</mrow>}}$ where the objects are $[n]=\{0,1,\dots ,n\},n\ge -1$ and the morphisms order-preserving functions.

References

• Khan, Adeel A. (2023), Lectures on Algebraic Stacks, https://www.preschema.com/papers/stacksncts.pdf 

Further reading

• https://ncatlab.org/nlab/show/simplicial+diagram
• https://ncatlab.org/nlab/show/totalization
• Rosona Eldred, Tot primer (2008) [1]