Simplicial Diagram

In mathematics, especially algebraic topology, a simplicial diagram is a diagram indexed by the simplex category (= the category consisting of all = \ 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
:$\cdots \, \underset\rightrightarrows \, U_2 \, \underset\rightrightarrows \, U_1 \, \rightrightarrows\, U_0$
where $U_n$ is the image of /math> 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 $U_$ a Čech nerve, then $F(U_)$ is a cosimplicial diagram and saying $F$ is a sheaf exactly means that $F(X)$ is the limit of $F(U_)$ for each $U \to X$ in a Grothendieck topology. See also: simplicial presheaf.

If $U_$ is a simplicial diagram, then the colimit
: _:= \varinjlim_ U_n
is called the geometric realization of $U_$. For example, if $U_n = X \times G^n$ is an action groupoid, then the geometric realization in Grpd is the quotient groupoid /G/math> 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 $\Delta_$ where the objects are = \, \, n \ge -1 and the morphisms order-preserving functions.

Notes

References

*

Further reading

* https://ncatlab.org/nlab/show/simplicial+diagram
* https://ncatlab.org/nlab/show/totalization
* Rosona Eldred, Tot primer (2008


Algebraic topology
Functors

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