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bio website mimuw.edu.pl/~mrp
location Poland
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visits member for 3 years, 9 months
seen Dec 21 at 23:47

I eat vegetarians.

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


Dec
21
comment Cantor's theorem for presheaves?
@QiaochuYuan, you're right --- it's not obvious (i.e. what I have written is wrong), because there can be non-trivial compositions in $\mathbb{C}$. Yes, it is obvious if $\mathbb{C}$ is small (we can take the coproduct over the image of $F$, to get an object that by the assumption of essentially surjectivness would have a monomorphism from any object, what would lead to a contradiction). On the other hand, I don't think that there are foundational issues here --- what if we asked the same question about $\mathbf{FinSet}$ instead of $\mathbf{Set}$?
Dec
21
comment Cantor's theorem for presheaves?
This means that there are at least $2^{|\mathbb{C}|}$ non-isomorphic objects in $\mathbf{Set}^{\mathbb{C}^{op}}$, so by classical Cantor's argument there can be no essentially surjective functor $\mathbb{C} \rightarrow \mathbf{Set}^{\mathbb{C}^{op}}$.
Dec
21
comment Cantor's theorem for presheaves?
Todd, isn't this obvious? Let us consider sets $1$ and $2$. There are functions $! \colon 2 \rightarrow 1$ and $j \colon 1 \rightarrow 2$. Therefore any function from objects $|\mathbb{C}^{op}| \rightarrow \{1, 2\}$ uniquely extends to a functor $\mathbb{C}^{op} \rightarrow \mathbf{Set}$ lying over $!$ and $j$. Indeed, for the uniqueness, if $F, G \colon \mathbb{C}^{op} \rightarrow \mathbf{Set}$ are two such functors with $F \approx G$, then for every $A \in \mathbb{C}$ we have $F(A) = G(A)$. (cont...)
Dec
15
comment Existence of internal toposes/inner models in a topos
Thanks, David, I'll take a look into the paper. But, anyway, I think it was Jean Benabou who first introduced the notion of an internal topos (BTW: I meant JEAN Benabou, of course. I can't believe I've written John. Shame on me...)
Dec
15
revised Adjoint of simplicial left Kan extension
in the mentioned context the subscript 0 actually means "underlying set"
Dec
15
comment Existence of internal toposes/inner models in a topos
David, what do you mean by "arithmetic universe" defined by Joyal? And what is your definition of an "internal topos"? Are your definitions of arithmetic universe and internal topos the same as given by John Benabou (and later described in "Problemes dans les topos")?
Dec
14
answered Adjoint of simplicial left Kan extension
Dec
12
comment The (un)reasonable (non-)ubiquity of the Grothendieck construction
I am still confused. Are you interested in enriched functors $X \rightarrow \mathbb{V}$ or classical functors $C \rightarrow \mathbf{Cat}(\mathbb{V})$?
Dec
12
comment Adjoint of simplicial left Kan extension
It is generally true that if $\mathcal{E}$ is cotensored and $R$ preserves cotensors, then the underlying adjunction $L \dashv R$ lifts to enriched adjunction (by classical Yoneda lemma). And your $R$ clearly preserves any limit that exists in $\mathcal{E}$.
Dec
12
comment The (un)reasonable (non-)ubiquity of the Grothendieck construction
I do not understand how Definition 3.1 from the paper answers your question --- this definition deals with a completely trivial situation of a (lax) functor from a (classical) category to a (classical) 2-category.
Dec
4
comment Map of adjunctions
@PiotrAchinger, thanks --- I haven't thought about the problem this way. Dimitri, good point, thanks.
Dec
3
revised Map of adjunctions
Corrected typos
Dec
3
revised Map of adjunctions
Added motivation
Dec
3
asked Map of adjunctions
Nov
12
awarded  Nice Answer
Oct
22
revised Completion of a category
Added missing "op"
Oct
22
comment Completion of a category
@GlenMWilson, I've finally written the whole answer --- hope you'll find it interesting.
Oct
22
revised Completion of a category
The second part of the disscussion
Oct
22
awarded  Necromancer
Oct
21
answered Completion of a category