Simplicial diagram

In mathematics, especially algebraic topology, a simplicial diagram is a diagram indexed by the simplex category (= the category consisting of all ${\displaystyle [n]=\{0,1,\cdots n\}}$ 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

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

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

A typical example is the Čech nerve of a map ${\displaystyle U\to X}$; i.e., ${\displaystyle U_{0}=U,U_{1}=U\times _{X}U,\dots }$.^[1] If F is a presheaf with values in an ∞-category and ${\displaystyle U_{\bullet }}$ a Čech nerve, then ${\displaystyle F(U_{\bullet })}$ is a cosimplicial diagram and saying ${\displaystyle F}$ is a sheaf exactly means that ${\displaystyle F(X)}$ is the limit of ${\displaystyle F(U_{\bullet })}$ for each ${\displaystyle U\to X}$ in a Grothendieck topology. See also: simplicial presheaf.

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

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

is called the geometric realization of ${\displaystyle U_{\bullet }}$.^[2] For example, if ${\displaystyle U_{n}=X\times G^{n}}$ is an action groupoid, then the geometric realization is the quotient groupoid ${\displaystyle [X/G]}$ which contains more information than the set-theoretic quotient ${\displaystyle X/G}$.^[3] 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.^[4]

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 ${\displaystyle \Delta _{\textrm {aug}}}$ where the objects are ${\displaystyle [n]=\{0,1,\dots ,n\},\,n\geq -1}$ and the morphisms order-preserving functions.

Notes

1. Khan 2023, Definition 2.1.6.
2. Khan 2023, Notation 4.1.6.
3. Khan 2023, Ch. 4, before § 4.1.
4. Khan 2023, Definition 3.2.3.

References

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

Further reading

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