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 Yearling
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Apr
8
comment Is there a characterization of coverings in subcanonical pretopologies?
Nice! Do you have a reference?
Apr
2
comment Filtered colimits are exact in abelian categories
The exactness of filtered colimits is exactly the axiom (AB5) (given by A. Grothendieck in his famous Tōhoku paper). There are a couple of interesting remarks about this axiom: an equivalent form of (AB5), which is worked out magnificently here, and the incompatibility with axiom (AB5*), which gives you many examples of abelian categories which do not satisfy (AB5), $\mathbf{Ab}^{\text op}$ in particular.
Mar
16
comment A quasi-isomorphism between the total complex of a Cartan-Eilenberg resolution and the complex per se.
The axiom (AB4) ensures the commutativity of homology with coproducts. A more general result, from which this exercise follows as corollary, is proven in Lemma 12, Section III.7.13 of Methods of Homological Algebra by S. Gelfand and Y. Manin.
Mar
10
comment A quasi-isomorphism between the total complex of a Cartan-Eilenberg resolution and the complex per se.
Why don't you try to use a spectral sequence for this double complex? The canonical filtration for rows should degenerate at sheet 2, giving you exactly the isomorphisms between the homologies you're looking for.
Mar
10
comment Are split exact sequences exact in the opposite direction?
If you have the first short exact sequence and $\ell f = 1_A$, then you can choose $r$ in such a way $gr= 1_C$ and the second sequence is short exact. This is Proposition 2.68 "Splitting maps" of Abelian Categories by P. Freyd.
Feb
25
comment Associativity of the tensor product of dendroidal sets
A complete and nice answer by G. Heuts can be found here.
Feb
25
revised Associativity of the tensor product of dendroidal sets
Miscalculations corrected
Feb
20
comment Associativity of the tensor product of dendroidal sets
I copied the same thread on MO, as well.
Feb
20
revised Associativity of the tensor product of dendroidal sets
Added more background
Feb
20
revised Associativity of the tensor product of dendroidal sets
typo
Feb
20
comment Associativity of the tensor product of dendroidal sets
Right! I improved the questions based on this remark.
Feb
20
revised Associativity of the tensor product of dendroidal sets
Improved the questions thanks to a comment.
Feb
19
asked Associativity of the tensor product of dendroidal sets
Feb
19
comment If $\{f_1,…,f_n\}$ generate $R$ then does $\{f_1^N,…,f_n^N\}$
Suppose that $R$ is a commutative ring with unit. For any $n$-tuple of positive integers $k_1, \dots, k_n$, try to show that the ideal generated by the elements $f_1^{k_1}, \dots, f_n^{k_n}$ cannot be contained in any prime ideal. So they generate the ideal $(1) = R$.
Jan
16
awarded  Yearling
Dec
11
awarded  Caucus
Oct
24
comment Colored operads as finitely essentially algebraic theory.
Ok, I think I got it (I misunderstood the finitary condition). Thank you again!
Oct
24
answered Colored operads as finitely essentially algebraic theory.
Oct
24
comment Colored operads as finitely essentially algebraic theory.
Thank for the patience, Zhen Lin. I admit that it is straightforward and this is a way in which I would have made it. Probably there is something I misinterprete, but the definition on nlab requires the signature only to have functional symbols! Further, I thought the signature also have to be a finite set, in oder to be finitely presentable. That is why I have some trouble.
Oct
24
comment Colored operads as finitely essentially algebraic theory.
I have only read the definitions given in the nlab page, hoping that it was accurate enough. (I could not find essentially algebraic theory in the Elephant, although I am quite sure it treats them).