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Results tagged with homological-algebra
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user 11640
(Co)chain complexes, abelian Categories, (pre)sheaves, (co)homology in various (possibly highly generalized) settings, spectra, derived functors, resolutions, spectral sequences, homotopy categories. Chain complexes in an abelian category form the heart of homological algebra.
7
votes
Free Objects in Functor Categories
To amplify Qiaochu's answer (and answer S. Carnahan's question), I'd like to add that the forgetful functor $[\mathcal{C}, \textbf{Ab}]_{\textbf{Ab}} \to [\operatorname{ob} \mathcal{C}, \textbf{Set}]$ …
- 15.9k
3
votes
Accepted
Is this square a push-out square?
The right square is a pushout square as soon as the first downward arrow is an epimorphism, and we do not even need $f$ and $g$ to be monomorphisms. This is true in any abelian category.
Since diagr …
- 15.9k
5
votes
Accepted
Why do we need the axiom MS3 for localizing categories?
The axiom LMS3 you ask about directly corresponds to the axiom that any parallel pair of arrows in a filtered category can be coequalised.
The idea is to consider the full subcategory of $\mathcal{C}_ …
- 15.9k
14
votes
Accepted
Projectives and Injectives in Functor Categories
For each object $c$ in $\mathcal{C}$, let $c^* : [\mathcal{C}, \mathcal{A}] \to \mathcal{A}$ be evaluation at $c$. It is an exact functor, so if a left adjoint $c_! : \mathcal{A} \to [\mathcal{C}, \ma …
- 15.9k
6
votes
Accepted
Is any abelian category a subcategory of $\mathrm{Ab}^I$?
Observe that $\bigoplus_I : \textbf{Ab}^I \to \textbf{Ab}$ is a conservative exact functor: it is right exact by general nonsense, it preserves monomorphisms (because e.g. $\bigoplus_{i \in I} A_i$ is …
- 15.9k
6
votes
Accepted
Reference for constructing tensor products of finitely presented functors
This "tensor product" is also known as the weighted colimit in enriched category theory. The short answer is that all the isomorphisms you are interested in always exist, provided the objects you are …
- 15.9k
9
votes
Is every additive, left exact functor isomorphic to a hom functor?
Here is a (stupid in some sense) counterexample.
Suppose $A$ is not trivial and $\kappa$ is a cardinal greater than the cardinality of $\textrm{Hom}_A (X, M)$ for all f.g. $A$-modules $X$ and $M$.
The …
- 15.9k
5
votes
Accepted
Do pretopoi have cohomology and homotopy groups?
There's a long story that can be told here but I will try to be brief.
In one sense, the answer is yes – you can certainly define cohomology and homotopy groups and so on for pretoposes and have them …
- 15.9k
5
votes
On the difference between a projective chain complex and a level-wise projective chain complex
Here is a general nonsense fact: if $F \dashv U : \mathcal{A} \to \mathcal{B}$ is an adjunction and $U$ preserves epimorphisms, then $F$ preserves projective objects. Epimorphisms in $\textrm{Ch}(R)$ …
- 15.9k