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 [ n ] = { 0 , 1 , ⋯ n } {\displaystyle [n]=\{0,1,\cdots n\}} {\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

⋯ ⇉ ⇉ U 2 ⇉ → U 1 ⇉ U 0 {\displaystyle \cdots \,{\underset {\rightrightarrows }{\rightrightarrows }}\,U_{2}\,{\underset {\rightarrow }{\rightrightarrows }}\,U_{1}\,\rightrightarrows \,U_{0}} {\displaystyle \cdots \,{\underset {\rightrightarrows }{\rightrightarrows }}\,U_{2}\,{\underset {\rightarrow }{\rightrightarrows }}\,U_{1}\,\rightrightarrows \,U_{0}}

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

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

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

[ U ∙ ] := lim → [ n ] ∈ Δ ⁡ U n {\displaystyle [U_{\bullet }]:=\varinjlim _{[n]\in \Delta }U_{n}} {\displaystyle [U_{\bullet }]:=\varinjlim _{[n]\in \Delta }U_{n}}

is called the geometric realization of U ∙ {\displaystyle U_{\bullet }} {\displaystyle U_{\bullet }}.[2] For example, if U n = X × G n {\displaystyle U_{n}=X\times G^{n}} {\displaystyle U_{n}=X\times G^{n}} is an action groupoid, then the geometric realization in Grpd is the quotient groupoid [ X / G ] {\displaystyle [X/G]} {\displaystyle [X/G]} which contains more information than the set-theoretic quotient X / G {\displaystyle X/G} {\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 Δ aug {\displaystyle \Delta _{\textrm {aug}}} {\displaystyle \Delta _{\textrm {aug}}} where the objects are [ n ] = { 0 , 1 , … , n } , n ≥ − 1 {\displaystyle [n]=\{0,1,\dots ,n\},\,n\geq -1} {\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

Further reading