bio  website  ncatlab.org/nlab/show/… 

location  Adelaide, Australia  
age  31  
visits  member for  5 years, 1 month 
seen  23 mins ago  
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Pure mathematician interested in category theory and foundations.
49m

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Cantor's theorem for presheaves?
Why can you assume that C has $\kappa$many isomorphism classes? The discrete category with objects sets with cardinality smaller than or equipollent with some cardinal $\lambda \gt \kappa$ is locally small, but doesn't fit your proof... (the result is true, but for a different reason) 
4h

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Cantor's theorem for presheaves?
You could also ask not for essentially surjectivity to fail, but some other 2categorical analogue of for stronger notions of epimorphism, so there may be a vanilla epimorphism, but not, for instance, a descent morphism. 
7h

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Cantor's theorem for presheaves?
@Michal that would be like taking the "countable universe" $\mathbb{N}$, as a special case of some inaccessible $\lambda$. But restricting to finite categories $\mathbb{C}$, while I think to special a case to give a general proof, might be interesting... 
Dec 18 
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Existence of internal toposes/inner models in a topos
@MichalR.Przybylek  I've gotten my hands on some lecture notes of Joyal from 1974. His definition of arithmetic universe at that point is a pretopos such that there are free internal categories on internal graphs. Maietti's definition (pretopos with parameterised list objects) implies Joyal's definition. 
Dec 16 
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Existence of internal toposes/inner models in a topos
@Michal fair enough, I know the history of these ideas is convoluted... 
Dec 15 
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Existence of internal toposes/inner models in a topos
Ah, that's what I was worried about, incompleteness phenomena. Thanks also for explaining what Joyal actually proved, in my limited reading around on the issue I'd not seen it written down. 
Dec 15 
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Existence of internal toposes/inner models in a topos
@Michal Joyal's definition has never been published or even circulated, so I can only point to work of Maietti (published in Theory and Appl. of Categories) and Paul (see link in his answer below). An internal topos is a model of the finite limit theory of toposes (I take the axioms to be pullbacks, Cartesian closed, subobject classifier, NNO, all given by specified operations). This is probably the same as what is in Problemes dans les topos, I don't have it in front of me. 
Dec 15 
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Existence of internal toposes/inner models in a topos
Thanks, Paul. I learned of the arithmetic universe stuff from your answer here on MO, then following references. I guess I wanted reassurance on the internal free topos, but I'm really mystified by what should be analogues of inner models. The best I can think of is to have a topos fibred over a base topos that is the externalisation of an internal topos, and arising from some syntactic construction. 
Dec 15 
revised 
Existence of internal toposes/inner models in a topos
added 259 characters in body; edited tags; edited title 
Dec 15 
asked  Existence of internal toposes/inner models in a topos 
Dec 12 
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Model bicategories
Or perhaps by writing @Aaron MazelGee ... 
Dec 12 
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What is operator tmf?
If vertex algebras are "the" answer, then mathoverflow.net/questions/124637 should be relevant. 
Dec 4 
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What do loop groups and von Neumann algebras have to do with elliptic cohomology?
I mean that the sentence doesn't seem to be grammatical, but maybe that's me. 
Dec 4 
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What do loop groups and von Neumann algebras have to do with elliptic cohomology?
What did you mean by this sentence: " One picture of tmf whose accuracy I can't comment on is that it should look at least a bit like K(ku)". It doesn't make sense to me. 
Dec 1 
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Computer software for manipulating loop groups or matrices with polynomial entries
I've written Python code that gives a (specific) loop groupvalued function: it returns, for any input in the domain, a lambdafunction. You can manipulate these somewhat. Note you could always work with approximations to elements of the loop group in your sense, just truncate that Laurent series. Otherwise, if you have a formula for the coefficients, you could work with the functions $\mathbb{Z} \to \mathbb{C}$ as function objects. 
Dec 1 
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Proof in analytic geometry using vector multiplication
Well, given that you wrote (or rather, that's what I think you meant to write, and I edited in) the scalar product of vectors, then it definitely has something with it. I think though that this is not a researchlevel question, and so offtopic. With appropriate background and explanation, this would probably be ontopic at math.stackexchange.com 
Dec 1 
revised 
Proof in analytic geometry using vector multiplication
LaTeX 
Dec 1 
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Which ordered fields are homeomorphic to their power?
+Emil whoops, I read it as isomorphic :S 
Nov 30 
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Which ordered fields are homeomorphic to their power?
To condense Johannes Hahn's question: isomorphic in what category? 
Nov 30 
awarded  Announcer 