All Questions
Tagged with cohomology ct.category-theory
29
questions
3
votes
0
answers
86
views
Possible relation between causal-net condensation and algebraic K theory
Causal-net condensation is a natural construction which takes a symmetric monoidal category or permutative category $\mathcal{S}$ as input date and produces a functor $\mathcal{L}_\mathcal{S}: \mathbf{...
- 153
5
votes
1
answer
420
views
Does coproduct preserve cohomology in differential graded algebra category
Consider two cochain DGA (differential graded algebras) named $A$ and $B$. By "coproduct" of two DGA I mean the category theory coproduct, not the coalgebra coproduct. It is defined in "...
- 159
9
votes
0
answers
520
views
In Mann's six-functor formalism, do diagrams with the forget-supports map commute?
One of the main goals in formalizing six-functor formalisms is to obtain some sort of "coherence theorem", affirming that "every diagram that should commute, commutes". In these ...
- 1,139
9
votes
0
answers
285
views
How acyclic models led to idea of model categories
The Wikipedia article about Acyclic models notices that the way that they were used in the proof of the Eilenberg–Zilber theorem laid the foundation stone to the idea of the model category.
Could ...
- 5,780
5
votes
1
answer
273
views
Axioms of derivators
I would like to enter the world of derivators. We can find a little history here and there about the limitations of triangulated categories and the motivation to enhance them, but also to compute ...
- 884
0
votes
3
answers
388
views
Are two different definitions for Čech cohomology equivalent?
In Spanier's book Algebraic Topology (Chapter 6 section 7) he defines Čech cohomology in terms of the nerves of open coverings.
I wish to know if this is equivalent, for a topological space A closed ...
13
votes
2
answers
784
views
Do pretopoi have cohomology and homotopy groups?
Grothendieck topoi have cohomology: the abelian category of abelian group objects in a topos has enough injectives, hence one can consider the right derived functors of the global sections functor ...
- 133
2
votes
1
answer
576
views
The nerve of the Ising category
A semi-simplicial complex (see Moore's 1958 paper: Semi-simplicial complexes and Postnikov systems, available at http://www-math.mit.edu/~hrm/kansem/moore-semi-simplicial-complexes.pdf, cf MathSciNet) ...
- 10.4k
5
votes
0
answers
161
views
Is there a "Kunneth isomorphism" for internal hom of chain complexes?
If $X^\bullet$ and $Y^\bullet$ are chain complexes over a field, we know from Kunneth theorem that
$$H^*((X\otimes Y)^\bullet)\cong H^*(X^\bullet)\otimes H^*(Y^\bullet) $$
I want to know if there is a ...
- 51
5
votes
0
answers
156
views
Extension groups in quotient categories
Let $\mathcal{A}$ be an abelian category and let $\mathcal{B}$ be a Serre subcategory of $\mathcal{A}$. We can form the quotient category $\mathcal{A}/\mathcal{B}$, and the canonical functor $Q:\...
- 1,479
3
votes
1
answer
314
views
Is every middle exact functor a derived functor?
Assume for the sake of simplicity we are working with categories of modules over some ring. Call a functor $F$ middle exact if for an exact sequence $ 0 \to A \to B \to C \to 0 $, we have that $FA \to ...
- 943
3
votes
1
answer
736
views
Are cohomology functors sheaves?
Question is the following:
Is the functor $H^n_{dR}:\text{Man}\rightarrow \text{Set}$ a sheaf with respect to open cover topology on $\text{Man}$?
More generally, are cohomology functors sheaves in ...
- 5,983
5
votes
1
answer
362
views
When is $\mathcal{D}(\mathcal{F}):\mathcal{D}(\mathcal{A})\to \mathcal{D}(\mathcal{B})$ fully faithful?
Let $\mathcal{A}$ and $\mathcal{B}$ be two abelian categories and let $\mathcal{F}:\mathcal{A}\to \mathcal{B}$ be an additive functor. Assume that $\mathcal{F}$ is exact and let $\mathcal{D}(\mathcal{...
- 1,479
6
votes
1
answer
466
views
Category of spaces/sheaves
Consider the following category $\mathcal C$:
An object of $\mathcal C$ is a pair $(X,\mathcal F)$ where $X$ is a space and $\mathcal F$ is a sheaf on $X$.
A morphism $(X,\mathcal F)\to(Y,\mathcal G)$...
- 18.5k
20
votes
3
answers
1k
views
Unifying "cohomology groups classify extensions" theorems
It is common for the first or second degree of various cohomologies to classify extensions of various sorts. Here are some examples of what I mean:
1) Derived functor of hom, $\text{Ext}^1_R(M, N)$. ...
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