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May
10
comment Textbooks on higher category theory
The question is far from being settled. That is why there are no textbooks.
May
10
comment Pullbacks in the Ind-completion
@AdeelKhan It's not a set-theoretic obstacle. $\mathbf{Ind}(\mathcal{C})$ is reflective in $\mathbf{Psh}(\mathcal{C})$ if and only if it is cocomplete, but that doesn't always happen.
May
10
comment Pullbacks in the Ind-completion
@MeesdeVries It is true that the canonical fully faithful embeding $\mathbf{Ind}(\mathcal{C}) \to \mathbf{Psh}(\mathcal{C})$ preserves filtered colimits and all limits. But that doesn't tell you which limits exist or not.
May
10
comment Pullbacks in the Ind-completion
@AdeelKhan No. First you would need to know that $\mathbf{Ind}(\mathcal{C})$ is reflective and secondly you would need to know that the reflector preserves finite limits. The first doesn't always happen and I don't know whether the second ever happens.
May
7
comment Definition of (left) resolution
First, build a morphism between $P$ and $A[0]$ using $\epsilon$. Then show it is a quasi-isomorphism if and only if $P_1 \to P_0 \to A \to 0$ is exact.
May
7
comment Definition of (left) resolution
Well, you haven't used the quasi-isomorphism condition...
May
7
comment An example of a coproduct of sheaves in the category of presheaves that is not a sheaf
Well, consider something silly like decomposing the empty set into two copies of itself vs three copies of itself...
May
7
comment An example of a coproduct of sheaves in the category of presheaves that is not a sheaf
Don't you have to quotient out by a suitable equivalence relation?
May
7
comment Question on the definition of a locally presentable category
No, it does not.
May
6
comment De Rham-Weil theorem
There are lots of different cohomology groups lying around. For instance, yes – an acyclic complex has vanishing cohomology groups. But taking global sections of an acyclic complex doesn't yield an acyclic complex.
May
6
comment Continuous functors
Sure, take $\mathcal{D} = \emptyset$. Then the condition is vacuous.
May
6
comment Is the diagonal map $\mathbb{C} \to \prod_{i=1}^\infty \mathbb{C}$ an etale map of rings?
No. Your confusion comes from the fact that Spec does not take infinite products to disjoint unions.
May
6
comment Regular epimorphisms in the category of simple, undirected graphs
This question is slightly complicated by the fact that $\mathbf{Grph}$ does not have all coequalisers.
May
6
comment Reference request: (categorical) commutative algebra text
You could start with a textbook on abelian categories. But at some point you're going to have to actually do some calculations with modules and rings...
May
6
comment “Stable model categories are categories of modules” - Clarification about a few things
On (7): you can't compose a left Quillen equivalence with a right Quillen equivalence, so sometimes you only get a "chain" or "zigzag" of Quillen equivalences.
May
5
comment what is the precise definition of a morphism defined over $k$?
The "local" picture of an automorphism is not another automorphism, however.
May
5
comment what is the precise definition of a morphism defined over $k$?
Well, instead of thinking about automorphisms, first generalise to morphisms between arbitrary varieties over $k$. Then specialise to affine varieties and use the correspondence with $k$-algebras.
May
5
comment Is the diagonal map $\mathbb{C} \to \prod_{i=1}^\infty \mathbb{C}$ an etale map of rings?
@yogeshmore No, it is not. A morphism of affine schemes is étale if and only if the corresponding ring homomorphism is étale.
May
4
comment Covariant and Contravariant Functor of Fixed Set Question - Category of Sets
The question is imprecise. After all, the formulae $X \mapsto X^A$ and $X \mapsto A^X$ do not specify what happens to morphisms. The point is to define actual functors whose object parts are as specified – so there's really nothing much to it at all.
May
2
comment Classifying space infinite totally ordered set contractible
@NajibIdrissi $\omega_1$ has a bottom element, so its nerve is contractible (in the strong sense). More generally, any category with an initial object has contractible nerve.