bio | website | math.uwo.ca/~dschaepp |
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visits | member for | 5 years, 5 months |
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I am a postdoc at the University of Western Ontario.
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 |
Jan 7 |
comment |
Is a composite of (co)monadic adjunctions (co)monadic?
Well, it is also monadic over $\mathbf{Set}\times \mathbf{Set}$, but the functor which sends a graph to its set of arrows is also monadic. In fact, there is a three element monoid in $\mathbf{Set}$ whose category of actions is equivalent to graphs (see exercise (GRMN) on page 107 of "TTT"). |