All Questions
Tagged with simplicial-categories at.algebraic-topology
10 questions
6
votes
1
answer
540
views
"Universal" triangulated category
Let $\mathcal{C}$ be some category. One way to map this category into a triangulated category is to take the category of simplicial objects $s\mathcal{C}$ (which is an $\infty$-category), take its ...
- 4,386
6
votes
1
answer
613
views
Stabilization of a generic pointed model category
Let $\mathcal C$ be a pointed model category. It is well-known that its homotopy category $\mathrm{Ho}(\mathcal C)$ is naturally a $\mathrm{Ho}(\underline{\mathrm{sSet}}_*)$-category, where $\mathrm{...
- 2,399
6
votes
0
answers
408
views
The k-ification of the compact-open topology for weak Hausdorff compactly generated spaces
Let CGWH be the category of weak Hausdorff compactly generated spaces; see e.g.
N.P. Strickland. THE CATEGORY OF CGWH SPACES: Preprint available from
https://neil-strickland.staff.shef.ac.uk/courses/...
5
votes
1
answer
340
views
About elegant Reedy categories
I discovered today the notion of elegant Reedy category introduced in the paper Reedy categories and the $\Theta$-construction of Julia E. Bergner and Charles Rezk. An interesting property of such ...
- 1,609
4
votes
1
answer
191
views
Homotopy coherent space maps induces homotopy coherent chain complex morphisms
It is an elementary fact that a map $f:X \to Y$ between spaces induces a chain complex morphism $f_* : C_*(X) \to C_*(Y) $. This allows one to transfer 1-category theoretic arguments from spaces to ...
- 2,152
3
votes
2
answers
161
views
The exactness of the associated chain complex of a simplicial free abelian group over a finite set and the normalization theorem
Update:
Now I know why my method fails. But I still wanna know how to work out the original question, that is to show the exactness of the chain complex $C_*(X)$ except for two positions $n=0,N-1$.
...
- 441
3
votes
1
answer
293
views
Does $\mathbf{Top}$ admit a simplicial structure
A simplicial category $\mathcal{C}$ is an $\mathbf{sSet}$-enriched category which is bitensored, in that the functors $\mathbf{Map}(-,X)$ and $\mathbf{Map}(X,-)$ admit left adjoints for every $X\in \...
- 333
2
votes
0
answers
53
views
Eilenberg–Zilber-type theorem for Map([n],A), where the degeneracy maps for [n] are forgotten
The following statement should be immediately implied by Eilenberg–Zilber theorem if the sequences $(i_0,\ldots,i_k)$ below are only monotone. But I need the strict monotone version which I believe to ...
- 461
2
votes
0
answers
114
views
Does twisted arrows commute with the simplicial nerve construction?
Let $\mathcal{C}$ be a simplicial category, and let $N(\mathcal{C})$ be its simplicial nerve. We can form the category of twisted arrows as a simplicial category $TwArr(\mathcal{C})$
Now Lurie's ...
- 21
1
vote
0
answers
122
views
Cohomology of bisimplicial set is the cohomology of the total simplicial set?
Given a bisimplicial manifold (or set, topological space) there is the bar construction, which assigns to it a simplicial manifold, called the total simplicial manifold.
Now stepping back for a moment,...
- 1,198