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

$\cdots \,{\underset {\rightrightarrows }{\rightrightarrows }}\,U_{2}\,{\underset {\rightarrow }{\rightrightarrows }}\,U_{1}\,\rightrightarrows \,U_{0}$

where $U_{n}$ is the image of $[n]$ from $\Delta$ 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

$[U_{\bullet }]:=\varinjlim _{[n]\in \Delta }U_{n}$

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 is the quotient groupoid which contains more information than the set-theoretic quotient $X/G=[X\times G\rightrightarrows X]$. A quotient stack is an instance of this construction (perhaps up to stackification).

References

• Khan, Adeel A. (2023), Lectures on Algebraic Stacks (PDF)

Further reading

• https://ncatlab.org/nlab/show/simplicial+diagram