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I am a postdoc at the University of Western Ontario.


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