bio | website | mimuw.edu.pl/~mrp |
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location | Poland | |
age | ||
visits | member for | 3 years, 1 month |
seen | Apr 6 at 14:52 | |
stats | profile views | 1,999 |
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 :-) |