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Results tagged with simplicial-stuff
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user 32022
For questions about simplicial sets, simplicial (co)algebras and simplicial objects in other categories; geometric realization, Dold-Kan correspondence, simplicial resolutions etc.
3
votes
Geometric realization of simplicial spaces and finite limits
To avoid leaving this question open:
Assuming we work in the category of compactly generated spaces, geometric realization commutes with pullbacks.(It's crucial that we use the compactly generated pr …
- 2,974
1
vote
Generalize $\pi_0(B\mathcal{C})\cong\{\text{objects}\}/\{\text{morphisms}\}$ to categories i...
If everything takes place in the category of compactly generated spaces, it holds $$\pi_0(BC)=\pi_0(obC)/\tilde{},$$ where two path components in the object space get identified, if there are objects …
- 2,974
11
votes
4
answers
1k
views
Topological Grothendieck Construction
Let $C$ be a small category and $F\colon C^{op}\rightarrow Set$ a functor. The Grothendieck construction is the category $F\wr C$ with objects being pairs $(c,x)$ where $c$ is a object of $C$ and $x\i …
- 2,974
1
vote
The reflexive free-category comonad-resolution is a cofibrant replacement of the discrete s...
Here's a direct way of seeing it, without using the Quillen equivalence to simplicial sets equipped with the Joyal model structure:
The cofibrant objects in the Bergner model structure are the 'simpl …
- 2,974