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bio website mimuw.edu.pl/~mrp
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seen Apr 6 at 14:52

I eat vegetarians.

If you repeat ``p or not p'' often enough, it becomes the truth.


Apr
5
revised An isomorphism of categories
Some minor corrections
Apr
3
comment A question on the Grothendieck construction
Your quick candidate is almost the right one --- since you take a colimit in Cat, you cannot obtain a non-trivial 2-category. The right way is to postcompose $I^{op}\to Cat$ with the embedding $Cat \to 2Cat$ and take the lax colimit. This construction is sometimes called a "2-Grothendieck construction" (notice, that in case of 2-categories, there are four different Grothendieck constructions).
Apr
3
answered An isomorphism of categories
Mar
31
comment An isomorphism of categories
@ZhenLin, thanks for the explanation :-)
Mar
31
comment An isomorphism of categories
Second, how is $j$ defined? Does it send everything from $\mathrm{FinSet}$ to the terminal $1$ (what does $*$ mean?)? If so, then $j$ is terminal in $[\mathrm{FinSet}, C]$ and $j\circ f$ is terminal in $[I^{op}, C]$. Thus $[I^{op}, C]/{(j\circ f)} = [I^{op}, C]$ --- I do not understand how, in the general case, could $[I^{op}, C]$ be isomorphic to $[F^{op}, C]$. For example, if $I = 1$ then $F$ is just an arbitrary finite set.
Mar
31
comment An isomorphism of categories
@ZhenLin, MaMing, I have no idea what you are talking about. First, there are typing errors in the question: a) an adjoint to $[F^{op},C] \rightarrow [I^{op},C]$ has type $[I^{op},C] \rightarrow [F^{op},C]$, b) you cannot apply $j$ to $f$, so perhaps you mean $j\circ f$, right?, c) it is not clear which category of elements you have in mind (i.e. you cannot compose $I^{op} \rightarrow C$ with $F \rightarrow I$; do you compose with the dual of the projection?). (cont)
Mar
20
comment The most unexpected and/or the least natural category theory theorem?
Todd, some of my students are sometimes better than myself (so I wonder if it is not high time to find another job :-) But, what I was really trying to say is: if we think that something is unnatural (in category theory), then we have to rethink it and study harder. There are a lot of things in category theory that seems to be unnatural/unexpected to me, and the only reason why I refrain from posting about them is to not get comments like the first one under your post :-)
Mar
20
comment The most unexpected and/or the least natural category theory theorem?
@PeterArndt, no --- you are given positive connectives and quantifications, so it suffices to define finite disjunctions in the internal logic. But this is easy when you can quantify over all propositions.
Mar
20
comment The most unexpected and/or the least natural category theory theorem?
I can't comment on your "hands-on way" because the link is broken (is there a problem with ncatlab?). However, I used to give these constructions (i.e. coproducts and coequalisers in the internal logic of a topos) as a take-home exam exercise for undergraduate students. So I guess these constructions can't be really difficult :-)
Mar
12
comment Adjoining adjoints in a 2-category
@DimitriChikhladze, you may always strictify a weak 2-category to a strict 2-category, however, I think for $\mathit{Mod}(\mathbb{C})$ there should be a (simple) canonical choice of an equivalent strict 2-category. I will come to this comment later, when time allows...
Mar
12
revised Are left adjoints a left adjoint?
Should have written 2-poset instead of (2,1)-category
Mar
12
revised Adjoining adjoints in a 2-category
Not sure, why I wrote (2,1)-category instead of 2-poset
Mar
12
revised Adjoining adjoints in a 2-category
additional explanation
Mar
12
answered Adjoining adjoints in a 2-category
Mar
12
answered Are left adjoints a left adjoint?
Mar
7
awarded  Yearling
Feb
13
awarded  Nice Answer
Jan
2
comment It looks so coKleisli, but it's not. What is it?
David, what do you mean by "discrete monoidal subcategory"?
Jan
1
comment Formalizations of category theory in proof assistants
@AndrejBauer, I think OP made the question CW to indicate that this is a question with candies --- to answer it, you should edit the question by removing one of the listed links, and put the link as an answer :-)
Dec
31
comment Can We Decide Whether Small Computer Programs Halt?
BTW, Joel's answer is nice :-)