nLab nice simplicial topological space




topology (point-set topology, point-free topology)

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topological homotopy theory

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A nice simplicial topological space is a simplicial topological space that satisfies certain extra properties that make it well behaved in homotopy theory, notably so that its geometric realization of simplicial spaces is its homotopy colimit.



Let X:Δ opTopX : \Delta^{op} \to Top be a simplicial topological space.

Such XX is called

  • good if all the degeneracy maps X n1X nX_{n-1} \hookrightarrow X_n are all closed cofibrations;

  • proper if the inclusion sX nX ns X_n \hookrightarrow X_n of the degenerate simplices is a closed cofibration, where sX n= is i(X n1)s X_n = \bigcup_i s_i(X_{n-1}).

In other words this says: X X_\bullet is proper if it is cofibrant in the Reedy model structure [Δ op,Top Strom] Reedy[\Delta^{op}, Top_{Strom}]_{Reedy} on simplicial objects with respect to the Strøm model structure on Top.

The notion of good simplicial topological space goes back to (Segal 1973), that of proper simplicial topological space to (May).




A good simplicial topological space is proper.

A proof appears as Lewis, corollary 2.4 (b). A generalization of this result is in RobertsStevenson.


For X X_\bullet any simplicial topological space, then |SingX |{|Sing X_\bullet|} is good, hence proper, and the natural morphism

|SingX |X {|Sing X_\bullet|} \to X_\bullet

is degreewise a weak homotopy equivalence.

This follows by results in (Lewis).


Since for XTopX \in Top the map |SingX|X|Sing X| \to X is a cofibrant resolution in the standard Quillen model structure on topological spaces, we have that

|SingX |X |Sing X_\bullet| \to X_\bullet

is a degreewise weak homotopy equivalence. In particular each space |SingX n||Sing X_n| is a CW-complex, hence in particular a locally equi-connected space. By (Lewis, p. 153) inclusions of retracts of locally equi-connected spaces are closed cofibrations, and since degeneracy maps are retracts, this means that the degeneracy maps in |SingX ||Sing X_\bullet| are closed cofibrations.

Models for the homotopy colimit


Let X X_\bullet be a simplicial topological space. Then there is a natural weak homotopy equivalence

X hocolim nΔX n \Vert X_\bullet\Vert \simeq hocolim_{n \in \Delta} X_n

from its fat geometric realization of simplicial topological spaces to the homotopy colimit over the simplicial diagram X:Δ opTopX : \Delta^{op} \to Top.

If moreover X X_\bullet is proper, then the natural morphism X|X| {\Vert X\Vert} \to {|X|} is a weak homotopy equivalence, and hence also the ordinary geometric realization is a model for the homotopy colimit.


That the geometric realization of simplicial topological spaces of a proper simplicial space is is homotopy colimit follows from the above fact that proper spaces are Reedy cofibrant, and using the general statement discussed at homotopy colimit about description of homotopy colimits by coends.


In the case X X_\bullet that is a good simplicial topological space, a direct (i.e., not using the fact that goodness implies properness) proof that X|X| \Vert X\Vert \to |X| is a weak homotopy equivalence has been sketched by Graeme Segal and then refined by Tammo tom Dieck.


The definition of proper simplicial space goes back to

  • Peter May, The Geometry of Iterated Loop Spaces , Lecture Notes in Mathematics, 1972, Volume 271(1972), 100-112 (pdf)

May originally said strictly proper for what now is just called proper .

The definition of good simplicial space goes back to

The implication goodpropergood \Rightarrow proper seems to be a folk theorem. Its origin is maybe in

  • L. Gaunce Lewis, Jr., When is the natural map XΩΣXX \to \Omega \Sigma X a cofibration? , Trans. Amer. Math. Soc. 273 (1982), 147–155.

A generalization of the statement that good implies proper to other topological concrete categories and a discussion of the geometric realization of WGW¯GW G \to \bar W G for GG a simplicial topological group is in

Comments on the relation between properness and cofibrancy in the Reedy model structure on [Δ op,Set][\Delta^{op}, Set] are made in

Last revised on October 1, 2021 at 10:46:14. See the history of this page for a list of all contributions to it.