Simplicial diagram
In mathematics, especially algebraic topology, a simplicial diagram is a diagram indexed by the simplex category.
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
where is the image of from in C.
A typical example is the Čech nerve of a map ; i.e.,. If F is a presheaf with values in an ∞-category and a Čech nerve, then is a cosimplicial diagram and saying is a sheaf exactly means that is the limit of for each in a Grothendieck topology. See also: simplicial presheaf.
If is a simplicial diagram, then the colimit
is called the geometric realization of. For example, if is an action groupoid, then the geometric realization in Grpd is the quotient groupoid which contains more information than the set-theoretic quotient. A quotient stack is an instance of this construction.
The limit of a cosimplicial diagram is called the totalization of it.