Mike Shulman
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Registered User
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I am currently a member of the Institute for Advanced Study. My research is in category theory, higher category theory, and homotopy theory.
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Apr 20 |
awarded | ● Nice Answer |
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Mar 26 |
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Extracting internal sites of definition Are sections C2.4-2.5 of Sketches of an Elephant relevant? |
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Mar 19 |
awarded | ● Nice Answer |
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Mar 17 |
answered | When does the direct image functor nicely push past the power/exists functor? |
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Mar 17 |
awarded | ● Good Question |
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Mar 9 |
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Connections between topos theory and topology You should probably verify that before asserting it as gospel truth; my memory is a bit hazy. (Ieke might also have more answers to your question if you asked him directly....) |
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Mar 8 |
answered | Connections between topos theory and topology |
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Mar 7 |
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An explicit description of injective fibrations Pick whatever $C$ you like! (Although yes, a generalized Reedy category would also not be very helpful...) |
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Mar 5 |
accepted | Cartesian cubes and groupoids |
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Mar 5 |
revised |
Are exponentials in categorical models of linear logic harmful? fixed grammar |
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Mar 1 |
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Is hypercompletion functorial? In 1-category theory, a universal property of objects like that expressed by 6.5.2.13 implies a unique way to extend the operation to a functor that is a right adjoint. See Theorem IV.1.2 in "Categories for the working mathematician". I presume the same must be true for $(\infty,1)$-categories. |
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Mar 1 |
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Is hypercompletion functorial? According to 6.5.2.13 in Higher Topos Theory, hypercompletion is right adjoint to the inclusion of hypercomplete $\infty$-topoi into all $\infty$-topoi. Doesn't that make it functorial? |
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Feb 27 |
awarded | ● Popular Question |
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Feb 14 |
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How many proofs that $\pi_n(S^n)=\mathbb{Z}$ are there? @unknown, Chris mentioned that already in his answer. |
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Feb 10 |
awarded | ● Nice Answer |
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Feb 8 |
awarded | ● Nice Question |
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Feb 8 |
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How many proofs that $\pi_n(S^n)=\mathbb{Z}$ are there? Re: your first paragraph: ah, yes, I should have known that. |
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Feb 8 |
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How many proofs that $\pi_n(S^n)=\mathbb{Z}$ are there? @Tom: Is there a way to generalize that argument to $n>2$? |
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Feb 8 |
asked | How many proofs that $\pi_n(S^n)=\mathbb{Z}$ are there? |
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Feb 7 |
answered | Why is Set, and not Rel, so ubiquitous in mathematics? |
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Feb 7 |
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Why is Set, and not Rel, so ubiquitous in mathematics? Although you have to be a little careful in formulating the enriched Yoneda lemma when the enriching category is not complete, since in that case enriched functor categories do not in general exist. |
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Feb 5 |
awarded | ● Necromancer |
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Feb 4 |
answered | Is there a tricategory of bicategories and biprofunctors? |
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Feb 3 |
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Basic questions on the homotopy category @Adam: replace each group $G$ with its classifying space $B G$. |
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Feb 3 |
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Basic questions on the homotopy category @Tom: The shape of the argument is "Suppose there were a limit in the homotopy category. Then ... and so we have a contradiction." In the course of deriving the contradiction, we consider also the homotopy limit of the same diagram, but I don't see how that changes the result we're proving. Is there a mistake somewhere? |
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Feb 2 |
answered | Basic questions on the homotopy category |
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Jan 31 |
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Reedy model structures on oplax limits I don't think that was the paper I'm thinking of, but thanks for pointing it out; it's certainly closely related! |
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Jan 31 |
asked | Reedy model structures on oplax limits |
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Jan 31 |
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Boolean non-hypercomplete $(\infty,1)$-toposes Thanks! Can you point me to an argument for why it is not hypercomplete? |
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Jan 30 |
asked | Boolean non-hypercomplete $(\infty,1)$-toposes |
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Jan 27 |
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Grothendieck topology for a non-small category Moreover, you can define the category of small sheaves without needing to assume a universe, giving 'small' the other meaning of 'a set' (rather than a proper class). In this case, there is generally no way to define the category of large sheaves, or that of large presheaves. That's the context in which I meant my statement that small sheaves are not particular (small) presheaves. |
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Jan 27 |
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Grothendieck topology for a non-small category If you define 'small' with reference to a universe, then yes, small sheaves are a full subcategory of large sheaves and also of large presheaves. My point is that a small sheaf may not necessarily be a small presheaf, neither in the sense of "being a small colimit of representables (as a presheaf)" nor in the sense of "taking values in small sets". |
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Jan 26 |
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Grothendieck topology for a non-small category Things formally glued together from objects of the site. For instance, schemes can be identified with "small sheaves" on the large site of all rings. |
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Jan 25 |
answered | Grothendieck topology for a non-small category |
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Jan 25 |
revised |
Are $\infty$-topoi determined by their localic points ? fixed grammar in title again |
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Jan 22 |
answered | String diagrams of special monoidal categories and higher categories |
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Jan 16 |
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Vertical and Horizontal Isomorphisms in 2-categories Doesn't working in a 2-group sort of beg the question, since in a 2-group everything is invertible? |
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Jan 14 |
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Forcing in Homotopy Type Theory One other thing which I might add is that for a very simple class of cases, there is an alternative approach: arxiv.org/abs/1203.3253 |
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Jan 14 |
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Forcing in Homotopy Type Theory Nice summary, Urs. I don't know of any further developments on the semisimplicial objects front, except that Hugo Herbelin has attacked it from a similar but different direction to Voevodsky and come quite close; he might even have succeeded by now. It's not entirely clear to me whether the problem is merely a (terrifically difficult) bookkeeping one, or something more basic. |
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Jan 13 |
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Examples of Sheafification via Hypercovers You have to look at all hypercovers, not just one of them. Related question: mathoverflow.net/questions/90969/… |
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Jan 12 |
accepted | A (too?) simple notion of “closed multicategory” |
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Jan 11 |
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Characterising categories of vector spaces @Martin: I don't know, whatever extra structure David had in mind. (-: |
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Jan 11 |
answered | A (too?) simple notion of “closed multicategory” |
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Jan 9 |
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Model for the (infinity,1)-category of functors preserving certain homotopy limits Your statement is true (or, at least, follows from the axiom of universes) if you meant Fun_{U-SSet}(M, U'-SSet), if we take U' sufficiently large that M is U'-small. There's no way you're going to get a U'-complete category by taking functors into a category (like U-SSet) that's only U-complete. |
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Jan 9 |
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How would set theory research be affected by using ETCS instead of ZFC? Learning formal type theory would undoubtedly be a better preparation! |
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Jan 7 |
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Relative consistency of ETCS over the theory of a well-pointed topos with NNO It may be worth mentioning that in "On the strength of Mac Lane set theory", Mathias describes a somewhat more structural approach to the constructible universe L, which may be similar to some of what has been mentioned above. I think that would be a good place to start for anyone who wanted to understand an analogue of L in ETCS. |
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Jan 7 |
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Example of a non-closed cocomplete symmetric monoidal category Re: Q3, there are plenty of examples in topology of even closed bicomplete symmetric monoidal categories that are not locally presentable. For instance, the category of compactly generated Hausdorff spaces. |
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Jan 7 |
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Characterising categories of vector spaces I think Morita equivalent rings should have equivalent module categories that preserve all the extra structure in sight as well. |
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Jan 7 |
answered | Model for the (infinity,1)-category of functors preserving certain homotopy limits |
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Jan 7 |
answered | How would set theory research be affected by using ETCS instead of ZFC? |
