Questions tagged [exact-categories]
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11
questions
3
votes
0answers
105 views
$K$-theory with respect to two different choices of quasi-isomorphisms
This question is related to another question asked here. Let's assume we have an exact category $C$ that consists of specific vector bundles on a variety. Furthermore assume $C$ is idempotent complete ...
6
votes
1answer
311 views
Beck-Chevalley condition on pushouts
Let $C$ be a regular category with pushouts and $S(X)$ is the lattice of subjects of $X$. For every arrow $f\colon X\to A$, pulling back along $f$ gives a map $f^*\colon S(A)\to S(X)$ which has a left ...
4
votes
1answer
159 views
Are product / coproduct projections / inclusions 'semistrict'?
Let $\mathbf{C}$ be a category with zero object, kernels, and cokernels. Then, a morphism $f\colon A\rightarrow B$ in $\mathbf{C}$ is semistrict iff the canonical map $\operatorname{Coker}(\ker (f))\...
5
votes
1answer
528 views
Extension-closed subcategories of triangulated categories as “almost exact” categories
Did anybody study those subcategories of triangulated categories that are closed with respect to "extensions" (in the sense of distinguished triangles; in particular, any such $B$ is additive)? If we ...
5
votes
1answer
316 views
Does the stable category of a nice exact category embed in (the underlying category of) a derivator?
In Derivators, Pointed Derivators, and Stable Derivators, Moritz Groth gives as an example of a non-invertible morphism with trivial cone an inclusion $f:X\to I$. Here $X$ is an object of injective ...
1
vote
0answers
49 views
Is the product of two categories with cofibrations still a category with cofibrations?
Given two $k$-linear categories $\mathcal C$ and $\mathcal D$ which are "categories with cofibrations" (in the Waldhausen sense),
is the product category $\mathcal C\times \mathcal D$ still a category ...
1
vote
0answers
125 views
Could one recover the relative K-theory from the quotient derived category?
Let $A\to B$ be a full embedding of exact categories that induces an embedding $D^b(A)\to D^b(B)$. My question is: what can one say about the relation of the homotopy cofibre $K(A)\to K(B)$ (the ...
7
votes
0answers
246 views
Split exact categories arising naturally
If you're interested in the $K$-theory of rings, a useful feature of the exact category of finitely generated projective (or free) modules is that it is split exact, i.e. every short exact sequence is ...
3
votes
1answer
327 views
Given an exact category, viewed as a site, do there exist non-additive sheaves?
Suppose given an exact category $\mathcal{C}$. The following question arises while proving the Gabriel-Quillen-Laumon embedding theorem following Laumon [1].
Laumon constructs an abelian category $\...
5
votes
0answers
89 views
Ex/reg toposes without generic monomorphisms
A generic monomorphism is a monomorphism of which every monomorphisms in the same category is a pullback; it is like a subobject classifier without uniqueness. I am looking for a locally Cartesian ...
8
votes
1answer
825 views
Quasi-isomorphisms in exact categories
I am trying to understand quasi-isomorphisms in an exact category as defined via the mapping cylinder. I would like to know whether these form a category of weak equivalences in the sense of ...
