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Jun
30 |
answered | Does projective imply flat? |
Feb
15 |
comment |
In a closed monoidal abelian category, are the compact projectives a monoidal subcategory?
I have not thought about the uniqueness of this monoidal structure. I guess picking an associator boils down to choosing an automorphism of $M \otimes M$ (the three-fold tensor product of $(0,\mathbb{Z})$), subject to some sort of cocycle condition on $M\otimes M \otimes M$ (the four-fold tensor product of $(0,\mathbb{Z})$). The remaining conditions should follow by writing general presheaves as colimits of representables. The trick with just choosing a skeleton of $\mathrm{Ab}$ is sort of a cop out, but it shows existence of a counterexample with the least amount of effort. |
Feb
15 |
answered | In a closed monoidal abelian category, are the compact projectives a monoidal subcategory? |
Dec
13 |
revised |
Exactness of an additive left Kan extension
Added details for the dual result. |
Nov
25 |
answered | Exactness of an additive left Kan extension |
Nov
10 |
awarded | Yearling |
Sep
24 |
awarded | Autobiographer |
Jul
23 |
awarded | Nice Answer |
May
12 |
awarded | Enlightened |
May
12 |
awarded | Nice Answer |
Apr
19 |
comment |
Universal property of module categories over monads
Is it clear that $\tilde{G}$ preserves coproducts? If the categories or the monad are not additive, then coproducts in the category of modules (algebras) are usually quite different from coproducts in the base category (e.g. groups vs. sets). If $T$ is additive and finitary (preserves filtered colimits), this is certainly not a problem. |
Feb
1 |
revised |
Semiadditivity and dualizability of 2
added 7 characters in body |
Feb
1 |
revised |
Semiadditivity and dualizability of 2
updated expired link |
Jan
25 |
comment |
Pushouts in the category of adjunctions
On the other hand, if you consider the 2-category of say lfp categories and adjunctions whose right adjoint does preserve filtered colimits, then you do get the desired pushouts. This 2-category is equivalent to finitely cocomplete categories with functors preserving finite colimits, and this is 2-monadic over the category of small categories. Since the 2-monad in question has rank, this 2-category has all bicolimits. I think it would be interesting to see if there are counterexamples if the categories involved do not have (many) colimits. |
Jan
25 |
comment |
Pushouts in the category of adjunctions
I don't think this works quite as intended. First of all, you often talk about limits when you really mean colimits (both in the definition of $X_i$ and $X_{\infty}$). But right adjoints need not preserve colimits (not even filtered colimits in general), so your argument that $G_1 X_{\infty}=G_2 Y_{\infty}$ does not work. |
Jan
25 |
answered | Does trace handle composition in a traced symmetric monoidal category? |
Jan
25 |
awarded | ct.category-theory |
Jan
24 |
answered | Push-outs of fully faithful (enriched) functors |
Jan
7 |
comment |
Is a composite of (co)monadic adjunctions (co)monadic?
Sorry, the functor in question sends a graph to the disjoint union of its set of arrows and its set of vertices, not just its set of arrows. The two nontrivial elements $s,t$ of the monoid act trivially on the set of objects, and send arrows to their source and target respectively. The usual relations that source of the target is target etc. have to be imposed. |
Jan
7 |
awarded | Commentator |