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| {{otheruses4|exact categories in the sense of Quillen|exact categories in the sense of Barr|regular category|exact categories in the sense of Buchsbaum|abelian category}}
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| In [[mathematics]], an '''exact category''' is a concept of [[category theory]] due to [[Daniel Quillen]] which is designed to encapsulate the properties of [[short exact sequence]]s in [[abelian category|abelian categories]] without requiring that morphisms actually possess [[kernel (category theory)|kernels and cokernels]], which is necessary for the usual definition of such a sequence.
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| ==Definition==
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| An exact category '''E''' is an [[additive category]] possessing a [[class (set theory)|class]] ''E'' of "short exact sequences": triples of objects connected by arrows
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| : <math>M' \to M \to M''\ </math>
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| satisfying the following axioms inspired by the properties of [[short exact sequence]]s in an [[abelian category]]:
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| * ''E'' is closed under isomorphisms and contains the canonical ("split exact") sequences:
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| ::<math> M' \rightarrow M' \oplus M''\rightarrow M'';</math>
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| * Suppose <math>M \to M''</math> occurs as the second arrow of a sequence in ''E'' (it is an '''admissible epimorphism''') and <math>N \to M''</math> is any arrow in '''E'''. Then their [[pullback (category theory)|pullback]] exists and its projection to <math>N</math> is also an admissible epimorphism. [[Dual (category theory)|Dually]], if <math>M' \to M</math> occurs as the first arrow of a sequence in ''E'' (it is an '''admissible monomorphism''') and <math>M' \to N</math> is any arrow, then their [[pushout (category theory)|pushout]] exists and its coprojection from <math>N</math> is also an admissible monomorphism. (We say that the admissible epimorphisms are "stable under pullback", resp. the admissible monomorphisms are "stable under pushout".);
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| * Admissible monomorphisms are [[kernel (category theory)|kernel]]s of their corresponding admissible epimorphisms, and dually. The composition of two admissible monomorphisms is admissible (likewise admissible epimorphisms);
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| * Suppose <math>M \to M''</math> is a map in '''E''' which admits a kernel in '''E''', and suppose <math>N \to M</math> is any map such that the composition <math>N \to M \to M''</math> is an admissible epimorphism. Then so is <math>M \to M''.</math> Dually, if <math>M' \to M</math> admits a cokernel and <math>M \to N</math> is such that <math>M' \to M \to N</math> is an admissible monomorphism, then so is <math>M' \to M.</math>
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| Admissible monomorphisms are generally denoted <math>\rightarrowtail</math> and admissible epimorphisms are denoted <math>\twoheadrightarrow.</math> These axioms are not minimal; in fact, the last one has been shown by {{Harvard citations|txt=yes|last=Keller|first=Bernhard|year=1990}} to be redundant.
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| One can speak of an '''exact functor''' between exact categories exactly as in the case of [[exact functor]]s of abelian categories: an exact functor <math>F</math> from an exact category '''D''' to another one '''E''' is an [[additive functor]] such that if
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| :<math>M' \rightarrowtail M \twoheadrightarrow M''</math>
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| is exact in '''D''', then
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| :<math>F(M') \rightarrowtail F(M) \twoheadrightarrow F(M'')</math>
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| is exact in '''E'''. If '''D''' is a subcategory of '''E''', it is an '''exact subcategory''' if the inclusion functor is fully faithful and exact.
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| ==Motivation==
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| Exact categories come from abelian categories in the following way. Suppose '''A''' is abelian and let '''E''' be any [[strictly full subcategory|strictly full]] additive subcategory which is closed under taking [[extension (algebra)|extension]]s in the sense that given an exact sequence
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| :<math>0 \to M' \to M \to M'' \to 0\ </math>
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| in '''A''', then if <math>M', M''</math> are in '''E''', so is <math>M</math>. We can take the class ''E'' to be simply the sequences in '''E''' which are exact in '''A'''; that is,
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| :<math>M' \to M \to M''\ </math>
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| is in ''E'' iff
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| :<math>0 \to M' \to M \to M'' \to 0\ </math>
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| is exact in '''A'''. Then '''E''' is an exact category in the above sense. We verify the axioms:
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| * '''E''' is closed under isomorphisms and contains the split exact sequences: these are true by definition, since in an abelian category, any sequence isomorphic to an exact one is also exact, and since the split sequences are always exact in '''A'''.
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| * Admissible epimorphisms (respectively, admissible monomorphisms) are stable under pullbacks (resp. pushouts): given an exact sequence of objects in '''E''',
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| ::<math>0 \to M' \xrightarrow{f} M \to M'' \to 0,\ </math>
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| :and a map <math>N \to M''</math> with <math>N</math> in '''E''', one verifies that the following sequence is also exact; since '''E''' is stable under extensions, this means that <math>M \times_{M''} N</math> is in '''E''':
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| ::<math>0 \to M' \xrightarrow{(f,0)} M \times_{M''} N \to N \to 0.\ </math>
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| * Every admissible monomorphism is the kernel of its corresponding admissible epimorphism, and vice-versa: this is true as morphisms in '''A''', and '''E''' is a full subcategory.
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| * If <math>M \to M''</math> admits a kernel in '''E''' and if <math>N \to M</math> is such that <math>N \to M \to M''</math> is an admissible epimorphism, then so is <math>M \to M''</math>: See {{Harvard citations|txt=yes|last=Quillen|year=1972}}.
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| Conversely, if '''E''' is any exact category, we can take '''A''' to be the category of [[exact functor|left-exact functor]]s from '''E''' into the category of [[abelian group]]s, which is itself abelian and in which '''E''' is a natural subcategory (via the [[Yoneda lemma|Yoneda embedding]], since Hom is left exact), stable under extensions, and in which a sequence is in ''E'' if and only if it is exact in '''A'''.
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| ==Examples==
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| * Any abelian category is exact in the obvious way, according to the construction of [[#Motivation]].
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| * A less trivial example is the category '''Ab'''<sub>tf</sub> of [[torsion-free abelian group]]s, which is a strictly full subcategory of the (abelian) category '''Ab''' of all abelian groups. It is closed under extensions: if
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| ::<math>0 \to A \to B \to C \to 0\ </math>
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| :is a short exact sequence of abelian groups in which <math>A, C</math> are torsion-free, then <math>B</math> is seen to be torsion-free by the following argument: if <math>b</math> is a torsion element, then its image in <math>C</math> is zero, since <math>C</math> is torsion-free. Thus <math>b</math> lies in the kernel of the map to <math>C</math>, which is <math>A</math>, but that is also torsion-free, so <math>b = 0</math>. By the construction of [[#Motivation]], '''Ab'''<sub>tf</sub> is an exact category; some examples of exact sequences in it are:
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| ::<math>0 \to \mathbb{Z} \xrightarrow{\left(\begin{smallmatrix} 1 \\ 2 \end{smallmatrix}\right)} \mathbb{Z}^2 \xrightarrow{(-2, 1)} \mathbb{Z} \to 0,</math>
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| ::<math>0 \to \mathbb{Q} \to \mathbb{R} \to \mathbb{R}/\mathbb{Q} \to 0,</math>
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| ::<math>0 \to d\Omega^0(S^1) \to \Omega^1_c(S^1) \to H^1_{\text{dR}}(S^1) \to 0,</math>
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| :where the last example is inspired by [[de Rham cohomology]] (<math>\Omega^1_c(S^1)</math> and <math>d\Omega^0(S^1)</math> are the [[closed and exact differential forms]] on the [[circle group]]); in particular, it is known that the cohomology group is isomorphic to the real numbers. This category is not abelian.
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| * The following example is in some sense complementary to the above. Let '''Ab'''<sub>t</sub> be the category of abelian groups ''with'' torsion (and also the zero group). This is additive and a strictly full subcategory of '''Ab''' again. It is even easier to see that it is stable under extensions: if
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| ::<math>0 \to A \to B \to C \to 0\ </math>
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| :is an exact sequence in which <math>A, C</math> have torsion, then <math>B</math> naturally has all the torsion elements of <math>A</math>. Thus it is an exact category; some examples of its exact sequences are
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| ::<math>0 \to \mathbb{Z}/2\mathbb{Z} \to \mathbb{Z}/4\mathbb{Z} \to \mathbb{Z}/2\mathbb{Z} \to 0,</math>
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| ::<math>0 \to \mathbb{Z}/2\mathbb{Z} \xrightarrow{(1,0,0)} (\mathbb{Z}/2\mathbb{Z})^2 \oplus \mathbb{Z} \to (\mathbb{Z}/2\mathbb{Z}) \oplus \mathbb{Z} \to 0,</math>
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| ::<math>0 \to (\mathbb{Z}/2\mathbb{Z}) \oplus \mathbb{Z} \to (\mathbb{Z}/2\mathbb{Z})^2 \oplus \mathbb{Z} \xrightarrow{(0,1,0)} \mathbb{Z}/2\mathbb{Z} \to 0,</math>
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| :where in the second example, the <math>(1,0,0)</math> means inclusion as the first summand, and in the last example, the <math>(0,1,0)</math> means projection onto the second summand. One interesting feature of this category is that it illustrates that the notion of cohomology does not make sense in general exact categories: for consider the "complex"
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| ::<math>\mathbb{Z}/2\mathbb{Z} \xrightarrow{(1,0,0)} (\mathbb{Z}/2\mathbb{Z})^2 \oplus \mathbb{Z} \xrightarrow{(0,1,0)} \mathbb{Z}/2\mathbb{Z}</math>
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| :which is obtained by pasting the marked arrows in the last two examples above. The second arrow is an admissible epimorphism, and its kernel is (from the last example), <math>(\mathbb{Z}/2\mathbb{Z}) \oplus \mathbb{Z}</math>. Since the two arrows compose to zero, the first arrow [[mathematical jargon#factor through|factors through]] this kernel, and in fact the factorization is the inclusion as the first summand. Thus the quotient, if it were to exist, would have to be <math>\mathbb{Z}</math>, which is not actually in '''Ab'''<sub>t</sub>. That is, the cohomology of this complex is undefined.
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| ==References==
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| * {{cite journal
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| | last = Keller
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| | first = Bernhard
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| | title = Chain complexes and stable categories
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| | year = 1990
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| | journal = [[Manuscripta Mathematica]]
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| | volume = 67
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| | pages = 379–417
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| | quote = Appendix A. Exact Categories
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| | doi = 10.1007/BF02568439
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| | ref = harv
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| }}
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| * {{Cite document
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| | last = Quillen
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| | first = Daniel
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| | authorlink = Daniel Quillen
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| | chapter = Higher algebraic K-theory: I
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| | title = Higher K-Theories
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| | year = 1972
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| | series = Lecture Notes in Mathematics
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| | publisher = Springer
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| | volume = 341
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| | doi = 10.1007/BFb0067053
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| | pages = 85–147
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| | ref = harv
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| | postscript = <!--None-->
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| | isbn = 978-3-540-06434-3
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| }}
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| [[Category:Additive categories]]
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| [[Category:Homological algebra]]
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