Simplicial diagram

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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

where is the image of from in C.

A typical example is the Čech nerve of a map ; i.e., .[1] 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 .[2] 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 .[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 where the objects are and the morphisms order-preserving functions.

Notes

References

Further reading

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