784 reputation
718
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location Barcelona
age 28
visits member for 3 years, 1 month
seen Aug 6 at 19:12
I'm a PhD student at the Universitat Autònoma de Barcelona.

Jul
22
comment Ring epimorphisms, and epimorphism in the category of small preadditive cats
@JeremyRickard, Thanks for your comments! Since there is a bounty ending tomorrow, if you want to post your partial answer with some details, I will accept it before tomorrow. Then it will be possible to edit. Also, I think that to have this partial answer could help others to give a complete answer.
Jul
21
comment Ring epimorphisms, and epimorphism in the category of small preadditive cats
So you are asking whether $\Phi$ in my question is surjective on objects, provided $\phi$ is an epimorphism. I do not know if this is true but I do not see any easy counterexample right now.
Jul
14
asked Ring epimorphisms, and epimorphism in the category of small preadditive cats
Jul
10
answered Spliting of short exact exact sequences of partially ordered groups
Jul
8
comment Spliting of short exact exact sequences of partially ordered groups
Also, are you considering just Abelian groups, or arbitrary group?
Jul
8
comment Spliting of short exact exact sequences of partially ordered groups
It depends what you actually want. In Abelian categories, the fact of having a $\gamma$ which is right inverse to $\beta$, tells you that $G=H\oplus G/H$. On the other hand, in general categories that condition just tells you that $G/H$ is a retract of $G$, but not a summand. Of course, in any case, you want $\gamma$ to be a morphism in your category, so in this case you want it to be an order homomorphism but, the mere existence of such $\gamma$ may not imply the splitting.
Jul
4
awarded  Yearling
Mar
16
awarded  Enlightened
Mar
16
awarded  Nice Answer
Mar
16
answered Regarding the definition of S-flows over a category (given a monoid S)
Mar
12
comment Lattice of subobjects of a particular coproduct
oh, you are right, if you want to post it as an answer I'd be happy to accept it.
Mar
12
comment Lattice of subobjects of a particular coproduct
Sorry, I wanted to say by an automorphism of the ring. I mean, let $\phi:R\to R$ be an automorphism. Using $\phi$ you can see $R$ as an $R-R$-bimodule (differently than in the standard way) and you can tensor by that bimodule.
Mar
12
asked Lattice of subobjects of a particular coproduct
Jan
29
awarded  Notable Question
Dec
13
comment Exactness of an additive left Kan extension
Many thanks for your edit! The Noetherian case (when you have Kernels) was clear to me but I did not know how to proceed in the general case...
Dec
13
accepted Exactness of an additive left Kan extension
Dec
12
comment Exactness of an additive left Kan extension
So do you claim that also the left adjoint between the categories of contravariant functors on $fp(R)$ and $fp(S)$ is exact provided $\phi_!$ is exact? I'm trying to prove this but I do not see how to prove that the slice category is directed... could you add some details for this other case? (If you prefer I can ask "officially" a new question)
Nov
25
comment Exactness of an additive left Kan extension
@FilippoAlbertoEdoardo I'm not defining $F_!$ as a restriction, I define $F$ as a restriction. $F_!$ is the additive left Kan extension to $F$, in particular, while $F$ is a functor $fp(R)\to fp(S)$, $F_!$ goes from the category of additive functors on $fp(R)$ to Abelian groups, to the same category of presheaves over $fp(S)$.
Nov
24
comment Exactness of an additive left Kan extension
You are right, $\mathbb Z\to \mathbb Q$ is flat (an Abelian group is flat if and only if it is torsion free). So $f^*$ cannot be restricted to a functor $fp(\mathbb Q)\to fp(\mathbb Z)$, but I am not asking that. I am just restricting $f_!$ and using its exactness (in your example it is clear that the image of a finitely presented module is a finite dimensional rational vector space).
Nov
24
comment Exactness of an additive left Kan extension
OK sorry! Now I see the problem, there was a missprint (now corrected). $f_!$ had to be $\phi_!$