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Results tagged with abelian-categories
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user 126667
8
votes
Accepted
Does the product (by an object) in an abelian category ever have a right adjoint?
In an additive category the functor F(–) = – × A = – ⊕ A cannot have a right adjoint unless A = 0. If F had a right adjoint then it would preserve coproducts and in particular A = F(0) = F(0 ⊕ 0) = F …
- 25.2k
15
votes
Accepted
How cavalier can I be when demanding a category have direct sums?
The answer to the first question is yes. If A and B have a direct sum A ⊕ B in C, then there are inclusions iA : A → A ⊕ B, iB : B → A ⊕ B and projections pA : A ⊕ B → A, pB : A ⊕ B → B such that pAi …
- 25.2k
48
votes
Accepted
Can a topos ever be an abelian category?
No. In fact no nontrivial cartesian closed category can have a zero object 0 (one which is both initial and final), as then for any X, 0 = 0 × X = X. (The first equality uses the fact that – × X com …
- 25.2k
18
votes
Is there an infinity × infinity lemma for abelian categories?
For an $n \times n$ square, the "lemma" follows by applying the spectral sequence of a double complex in two different ways: if we first take homology along columns, then the $E^2$ page is $0$, while …
- 25.2k
6
votes
Kernels and cokernels of multicomplex homomorphisms
If I'm not mistaken, multicomplexes in $\mathcal{A}$ are the same as unbounded cochain complexes in another category constructed from $\mathcal{A}$ as follows:
its objects are $\mathbb{Z}$-graded se …
- 25.2k