This article is about exact categories in the sense of Quillen. For exact categories in the sense of Barr, see regular category. For exact categories in the sense of Buchsbaum, see abelian category.
In mathematics, specifically in category theory, an exact category is a category equipped with short exact sequences. The concept is due to Daniel Quillen and is designed to encapsulate the properties of short exact sequences in abelian categories without requiring that morphisms actually possess kernels and cokernels, which is necessary for the usual definition of such a sequence.
In mathematics, specifically in category theory, an exactcategory is a category equipped with short exact sequences. The concept is due to Daniel Quillen...
exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category)...
of exactcategories, forming a very special case of regular categories. As mentioned above, the category of all abelian groups is an abelian category. The...
In mathematics, exactness may refer to: ExactcategoryExact functor Landweber exact functor theorem Exact sequence Exactness of measurements Accuracy...
cope with functors that fail to be exact, but in ways that can still be controlled. Let P and Q be abelian categories, and let F: P→Q be a covariant additive...
Grothendieck group of an exactcategory A {\displaystyle {\mathcal {A}}} . Simply put, an exactcategory is an additive category together with a class of...
are the derived category of an abelian category, as well as the stable homotopy category. The exact triangles generalize the short exact sequences in an...
In category theory, a branch of mathematics, the exact completion constructs a Barr-exactcategory from any finitely complete category. It is used to...
pairs, satisfying certain exactness conditions. In that way, regular categories recapture many properties of abelian categories, like the existence of images...
definitions, it is possible to define the derived category of an exactcategory (Keller 1996). The derived category is a natural framework to define and study...
In category theory, a branch of mathematics, a Frobenius category is an exactcategory with enough projectives and enough injectives, where the classes...
sense in any category with kernels and cokernels. The most common type of exact sequence is the short exact sequence. This is an exact sequence of the...
functor Hom(P, -): R-Mod → Ab is an exact functor, where R-Mod is the category of left R-modules and Ab is the category of abelian groups. When the ring...
A Category 5 Atlantic hurricane is a tropical cyclone that reaches Category 5 intensity on the Saffir–Simpson hurricane wind scale, within the Atlantic...
abelian category 0 → A → a B → b C → 0 {\displaystyle 0\to A\mathrel {\stackrel {a}{\to }} B\mathrel {\stackrel {b}{\to }} C\to 0} is called split exact if...
AB5 categories are the AB3 categories in which filtered colimits of exact sequences are exact. Grothendieck categories are the AB5 categories with a...
{A}})} of bounded chain complexes on an exactcategory A {\displaystyle \scriptstyle {\mathcal {A}}} . The category S n C {\displaystyle \scriptstyle S_{n}{\mathcal...
Grothendieck category are exact (i.e. the coproduct of a family of short exact sequences is again a short exact sequence), but products need not be exact. A functor...
similar to quotient categories, the difference being that with Serre quotients all involved categories are abelian and all functors are exact. Serre quotients...
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