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allows one to express every object as a quotient of a direct sum of copies of the generator. Finding a cogenerator allows one to express every object as a subobject of a direct product of copies of the cogenerator. One is often interested in projective generators (even finitely generated projective
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For example, the integers are a generator of the category of abelian groups (since every abelian group is a quotient of a
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generators, called progenerators) and minimal injective cogenerators. Both examples above have these extra properties.
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is 0. Even more is true: the * operation takes a homomorphism
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are objects which envelope other objects as an approximation.
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such that for every nonzero object H there exists a non
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All of this can also be done for continuous modules
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515:is an injective cogenerator in a category of
137:, a branch of mathematics, the concept of an
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16:For other things called "cogenerators", see
57:. Unsourced material may be challenged and
216:Assuming one has a category like that of
121:Learn how and when to remove this message
397:being a cogenerator says precisely that
310:abelian group. Given any abelian group
250:; and one can form direct products of
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55:adding citations to reliable sources
195:such that for every nonzero object
284:). This is the origin of the term
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362:is a left module over the ring
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18:Cogenerator (disambiguation)
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509:Tietze extension theorem
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70:"Injective cogenerator"
446:* is 0 if and only if
401:* is 0 if and only if
212:The abelian group case
139:injective cogenerator
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503:In general topology
450:is 0. It is thus a
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254:until the morphism
228:until the morphism
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143:Pontryagin duality
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166:zero object
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458:from left
248:surjective
111:March 2016
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308:divisible
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