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bio website mimuw.edu.pl/~mrp
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I eat vegetarians.

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


Oct
14
comment Relations In Category Theory
@ToddTrimble, but, actually, you do not have to assume that $C$ has products to define associative compositions (it suffices to assume that $C$ has pullbacks and stable images).
Oct
14
comment Relations In Category Theory
In fact, the above is the standard definition of an internal relation: an internal relation is a span of morphisms that are jointly mono. In case the category has binary products such relations can be represented as single monomorphisms into cartesian products. The latter definition is a bit less convenient to work with, but it is much easier to generalize it to non-canonical notions of subobjects, therefore some authors use it as their primary definition.
Sep
24
awarded  Autobiographer
Sep
1
comment Internal categories in an endofunctor category
I think the problem with such monoidal structures is that they generally do not preserve equalisers.
Aug
31
comment Categories with binary relations as objects
@Lehs, I am always happy when someone downvotes my answers --- it means that what I have written is not completely trivial :-) However, I think that downvoting an answer from someone who is trying to help you and make your student-level question a bit more interesting is a really strange way to say "thank you". Anyway, I have just made my final edit --- you may still be unhappy with it, but on no condition I will provide you a student-level (i.e. a low-level) answer to the question --- MO is not the right place to provide such answers.
Aug
31
revised Categories with binary relations as objects
Generalization of the proof
Aug
30
comment Categories with binary relations as objects
@Lehs, the counter-example to what? Because I have proven my claim, there are no counter-examples to it :-)
Aug
30
comment Reference request: Book of topology from “Topos” point of view
@StevenGubkin, I think a more appropriate question is: "have you read it backward?" (there are a lot of good books about "topos from topology point of view"; so I guess, it suffices to pick any and read it backward :-)
Aug
30
revised Categories with binary relations as objects
More detailed answer
Aug
30
comment Categories with binary relations as objects
Another question is whether one condition implies the other, but this is a simple student exercise and as such is not suitable for MO :-)
Aug
30
comment Categories with binary relations as objects
Your construction is exactly the above construction for $\mathbb{C} = \mathbf{Rel}$ --- you have just "relabeled" $N$ with $M_2$, and $M$ with $M_1^\mathit{co}$. In other words, because the category of relations is self-dual, you could mistakenly write relation $M_1$ in the wrong direction. I think, this is the crucial observation --- that you drew a wrong diagram. I am writing about this, because I made a very similar "mistake" once --- I drew a distributor in "a wrong direction", and it took me a few days until I realized, that the property I had been looking for was completely obvious.
Aug
30
comment Categories with binary relations as objects
@Lehs, I am not sure if I understand you --- I meant that given any 2-category $\mathbb{C}$, you may build a category of "weakly commutative twisted diagrams", whose objects are morphisms $R \colon X \rightarrow Y$, $R' \colon X' \rightarrow Y'$ in $\mathbb{C}$ and whose morphisms from $R$ to $R'$ are triples $\langle M \colon X' \rightarrow X, N \colon Y \rightarrow Y', \tau \colon N \circ R \circ M \rightarrow N \rangle$, where $M, N$ are morphisms in $\mathbb{C}$ and $\tau$ is a 2-morphism in $\mathbb{C}$. (cont)
Aug
30
answered Categories with binary relations as objects
Aug
24
revised Continuous relations?
Fixed English
Aug
23
comment Time in Girard's Geometry of Interaction
Sometimes, some people say some things, not because the things are true, deep, or in any sense smart, but to make other people think. Such people, are usually called "controversial" (sometimes even "excentric") by the community. For sure, Y.J. Girard is one of the greatest logicians of our time; yet he is a bit "controversial" :-)
Aug
23
answered Continuous relations?
Aug
19
comment Explicit description of the oplax limit of a functor to Cat?
@JasonGross, this question was answered in more detail before (see Mike's answer and my explanation): mathoverflow.net/questions/164343/…
Aug
18
comment In the category of sets epimorphisms are surjective - Constructive Proof?
@AndreasCaicedo, I do not understand why you have removed tag "set-theory". As for me, the question is exactly about set theory (moreover, it is formulated in such a way, that it is about the standard set theory).
Aug
18
comment In the category of sets epimorphisms are surjective - Constructive Proof?
@ZhenLin, aws: yes, you're, of course, right --- it seems that I'm getting old and seeing things which are not :-) So, to conclude the series of comments: "surjection iff epi" holds in every pretopos.
Aug
18
comment In the category of sets epimorphisms are surjective - Constructive Proof?
(...) but I don't see any reason why every $\Pi \Sigma$-pretopos, or any other notion of a predicative universe should be balanced (of course, CZF is balanced, so technically your claim is correct, but I'd like to see the exact reason behind this fact).