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Aug
11
comment Choosing projective replacement to be functorial
1. Classical derived functors are functors – between the derived categories. 2. Look up Cartan–Eilenberg resolutions. 3. A functorial projective replacement should be enough. 4. There is only one possible meaning.
Aug
11
comment Can you integrate on a scheme?
Integrate what?
Aug
11
comment Derived functors definition
That is the assumption, yes. Please read carefully.
Aug
10
comment Sifted colimits of models of a Lawvere theory.
I'm not so certain that filtered colimits + reflexive coequalisers suffice for all sifted colimits, but at any rate the conclusion about finitary monads on $\mathbf{Set}$ is correct. It's less clear to me what happens for monads of higher accessibility rank, though.
Aug
10
comment Examples of preadditive categories
The total category of modules is not preadditive.
Aug
9
comment Definition of exact sequence of functors.
That is correct. In fact, the category of functors (with values in a given abelian category) is itself an abelian category, so this does not need a separate definition.
Aug
7
comment Show that $A \lor B ⊢ B \lor A$
There's more than one way of stating the elimination rule as well. For instance, we could get rid of $\to$ and instead have the premises $\Gamma, p \vdash r$ and $\Gamma, q \vdash r$.
Aug
6
comment This is just the Eilenberg-Moore category, right?
I guess you mean to ask about the category of functors that preserve small products. The answer is yes, the proof is in the comments at the second linked page.
Aug
6
comment Show that $A \lor B ⊢ B \lor A$
I did not downvote. And, as you say, the OP has not specified the rules. It is entirely possible that a constructive proof is required.
Aug
6
comment Show that $A \lor B ⊢ B \lor A$
This is terrible – unnecessary proof by contradiction!
Aug
6
comment Relationship between simplicial complex and abstract simplicial complex
The first definition is not completely precise. The second definition is a way of making it precise.
Aug
6
comment Do graphs form groups under addition?
You may be looking for the concept of a monoidal category.
Aug
6
comment Transitive closure of a relation in categorical logic
You can define it in $\mathbf{Rel}$ as a poset-enriched category.
Aug
4
comment open sets in affine space are not affine varieties - easy proof
"Generalised principal open set" is a funny name for this. As it turns out, every Zariski open set is of this form.
Aug
4
comment Can I define a site as a category endowed with a pretopology instead of a topology?
It depends on how you define "induced Grothendieck topology". One way is more or less trivial but doesn't tell you what the covering sieves are; the other way tells you what the covering sieves are but requires more checking.
Aug
4
comment Can I define a site as a category endowed with a pretopology instead of a topology?
Yes. That is the whole point of the definition of "induced Grothendieck topology".
Aug
3
comment Does an equivalence of $G$-sets and $H$-sets imply an isomorphism of $G$ and $H$?
Indeed. In the case of linear representations, the fibre functor is indispensable.
Aug
1
comment Uncountable Kronecker Delta?
You can certainly define $\delta$ for arbitrary index sets. Continuity has nothing to do with it.
Aug
1
comment How to compute (co)limits of enriched categories?
(1) is not the cartesian product. It is the tensor product. The cartesian product has hom-objects given by cartesian products, of course.
Aug
1
comment Are there any free or fascist boolean algebras?
The category of boolean algebras is a full reflective subcategory of the category of Heyting algebras, by a very standard argument.