6
votes
Beck-Chevalley condition on pushouts
No, the Beck-Chevalley condition does not hold for all pushout squares in a regular category, and not even if the category is exact, or a pretopos, or even a topos. In fact, here is a counterexample ...
- 66.7k
6
votes
Accepted
Can one bypass the geometric realization in the definition of algebraic $K$-theory?
I believe there is no good notion of homotopy groups for an arbitrary simplicial set S.
It depends on what “good” means.
Kan's original definition works for arbitrary pointed simplicial sets: $$\def\...
- 37.8k
4
votes
Accepted
Waldhausen S-construction for exact categories
Let me add to Tim's answer that this also holds when suitably defining all of those terms in the higher categorical context, as does Barwick in https://arxiv.org/pdf/1212.5232. Remark also that there ...
- 189
4
votes
Waldhausen S-construction for exact categories
Yes, if $\mathcal C$ is a Quillen-exact category, then the Waldhausen category $S_n(\mathcal C)$ is also a Quillen-exact category. The admissible monos are the Waldhausen cofibrations, and the ...
- 63.9k
1
vote
Accepted
On image of map $\text{Ext}^1_R(X,F)\to \text{Ext}^1_R(X,G)$ induced by $R$-linear map of free modules $F\to G$ with entries in the maximal ideal
This is true for any $\mathrm{Ext}$-degree (and, in fact, without many hypotheses except that $\mathfrak m$ is finitely generated and $F$ is free).
Let $(x_1,\dots,x_n)$ be generators of the maximal ...
- 52.6k
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