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Oct
11
comment (Are there) subtleties in the definition of 'biproduct'
Well, consider an infinite set $X$. Then for cardinality reasons, $X + X \cong X \times X$, yet we do not say that it is a biproduct.
Oct
10
comment Under which assumptions counit of the adjunction $f^* f_* \to 1$ is epimorphic?
The counit of an adjunction is a componentwise epimorphism if and only if the right adjoint is faithful. However, being an epimorphism is not the same as being surjective...
Oct
10
comment Hartshorne Chapter II exercise 5.7 on Invertible sheaves
That's a good approach. You might want to think about what happens when you tensor with the residue field. What does that imply about the number of generators you need?
Oct
10
comment does a commutative diagram implies pull-back?
No. The dimensions don't even have to match.
Oct
10
comment Show that C is a split exact chain complex if and only if the identity map on C is null homotopic.
Start by unfolding the definitions.
Oct
8
comment Seeking help to understand a simple Kripke model
You can prove in intuitionistic logic that LEM is equivalent to DNE. So LEM fails if and only if DNE fails.
Oct
8
comment How to show $\mathbb{A}_k^2 - \{ (x,y) \}$ is not an affine scheme
Well, it is a matter of fact that the inclusion induces an isomorphism of coordinate rings. So if both the domain and the codomain were affine then it would have to be an isomorphism, but it isn't even bijective on points.
Oct
8
comment Colimits in the 2-category of partial functions (which is locally posetal)
More precisely: if bi(co)limits or pseudo(co)limits exist, then they are ordinary (co)limits in particular. This is not true for a general bicategory.
Oct
8
comment Colimits in the 2-category of partial functions (which is locally posetal)
Bi(co)limits and pseudo(co)limits in locally posetal categories reduce to ordinary (co)limits (assuming existence). They beocme more interesting in locally preordered categories.
Oct
8
comment Does Extremal Mono imply Split Mono in (Epi, Regular Mono)-factorization categories?
Every regular monomorphism is in particular extremal. So you are claiming that every regular mono splits, which is of course false.
Oct
7
comment Single $\Delta$-complex structure on $S^3$
Well, you can take $\Delta^3$ and collapse the boundary to a point. But that's not a $\Delta$-complex.
Oct
7
comment Axioms of Abelian Category
You should spell out what (2) and (3) are. It is annoying to have to load a 764-page PDF to find two sentences.
Oct
6
comment Two different definitions of sheaf of $K$-modules and tensor products.
If your space is locally connected then there is a straightforward argument showing that the two coincide, following the outline you suggest. But it should be true in general.
Oct
6
comment Injective resolution of complexes equivalent to regular definition
Unless your definition of "complex" excludes negative degrees, this is an extra condition in general.
Oct
5
comment De Rham Cohomology is Sheaf Cohomology
Yes. You don't mean to say it's trivial, do you?
Oct
5
comment De Rham Cohomology is Sheaf Cohomology
Your argument is correct. However, it only shows that you have an exact sequence – still a long way from calculating sheaf cohomology.
Oct
5
comment Is there accepted terminology for algebraic structures whose every subalgebra is free?
One might say "hereditarily free".
Oct
4
comment Category theory? Logic? Anyone experienced this like me?
Having a transitive model is not as convenient. When category theorists adopt a universe axiom it is always for convenience...
Oct
3
comment Elementary motivations for free resolutions
One might ask the same questions about higher Ext or Tor groups...
Oct
3
comment Weighted colimits,hom-functor,Usage of Yoneda lemma
No, I refuse. That step is very basic and you must be able to work out at least that much for yourself if you want to get anywhere with category theory.