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Stream: theory: algebraic topology

Topic: geometric realization of simplicial topological spaces

Cloudifold (Jun 02 2022 at 14:51):

Lemma 4.12 in https://ncatlab.org/nlab/show/geometric+realization+of+simplicial+topological+spaces
says the Geometric realization between proper simplicial spaces sends object-wise weak homotopy equivalence $f_q : X_q \to Y_q$ to weak equivalence $|f| : |X| \to |Y|$.

I think there is another proof if additionally X and Y are "strictly proper simplicial space" (in <<The Geometry of Iterated Loop Spaces>>). Using the fact that Geometric realization preserve homotopy fiber, Theorem 11.12 in <<The Geometry of Iterated Loop Spaces>> and the proposition 3.1 in https://ncatlab.org/nlab/show/n-connected+continuous+function.

It seems that the difficulty is to show that the object-wise homotopy fiber $F_{f_q}$ is strictly proper,
and I don't know how to deal with basepoint in $|Y|$

Cloudifold (Jun 03 2022 at 16:40):

I also wonder how to prove "Pushouts along Hurewicz cofibrations preserve weak homotopy equivalences" in the proof of Lemma 4.12 in https://ncatlab.org/nlab/show/geometric+realization+of+simplicial+topological+spaces

Cloudifold (Jun 03 2022 at 16:57):

Cloudifold said:

I also wonder how to prove "Pushouts along Hurewicz cofibrations preserve weak homotopy equivalences" in the proof of Lemma 4.12 in https://ncatlab.org/nlab/show/geometric+realization+of+simplicial+topological+spaces

I found proof in Proposition 1.1 https://people.math.binghamton.edu/malkiewich/hocolim_bar.pdf