# Questions tagged [exact-categories]

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### Idempotent completeness

We say a category $\mathcal{N}$ is exact if it is additive and is endowed with an exact structure. In brief, it is an additive category with a predetermined class of short exact sequences in its ...

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### 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 $S$. However, when $S$ is fibrant - meaning that $S\to *$ is a fibration - there is a definition. The singular ...

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### 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

$\DeclareMathOperator\Ext{Ext}$Let $(R,\mathfrak m)$ be a Noetherian local ring. Let $F,G$ be finitely generated free $R$-modules and $f:F\to G$ be an $R$-linear map such that $f(F)\subseteq \mathfrak ...

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### $K$-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 ...

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### 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 ...

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### 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))\...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $\...

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### 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 ...

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### 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 ...