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Thom first isotopy lemma in o-minimal structures
Thanks for the reference, Serge. It's definitely a step in the right direction, and there are some nice technical results in there. It's not exactly what I need, though, as this is ordinary Morse theory done in the o-minimal context, whereas what I need is stratified Morse theory done in the o-minimal context. (NB: even without the o-minimal constraint, stratified Morse theory is much trickier than ordinary Morse theory -- the Harvard notes by Mather make that plain!) Thanks again, though.
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Thom first isotopy lemma in o-minimal structures
"Semi-algebraic implies smooth, right?" No. Recall that a semi-algebraic set is just a union of intersections of sets defined by polynomial inequalities. By definition, a map is semi-algebraic if its graph is; for example, a PL map from the interval to itself is semi-algebraic. It need not be $C^1$.
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Thom first isotopy lemma in o-minimal structures
Thanks, Serge. I did see this and skimmed through it, but wasn't able to extract what I need from it, unfortunately.
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categorical description of elements in a direct limit
I can't speak for other people's up- or down-votes, but I am in agreement with Tom Leinster that the "rules of the game" have not yet been made clear, as the present discussion seems to bear out: I don't know what background assumptions you're working with. So the question isn't yet precise enough. (Also, your protestation to Tom that the question was clear, when he was honestly trying to understand and be helpful, wasn't the most genial or collegial reaction and probably didn't win you points.) This isn't to say that you're not onto something potentially interesting though.
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categorical description of elements in a direct limit
Martin, it's false (externally) in Set/2, where 2 = 1+1. The terminal object T is id: 2 -> 2, and the unique map !: T --> T factors through neither or the subterminal objects given by the two coproduct inclusions 1 -> T. The discussion could be refined by taking into account the subtleties of internal logic in a topos, but this type of thing is not manifest in your account.
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categorical description of elements in a direct limit
I think part of the problem is that it's not clear what the background theory of sets is supposed to be here, what the axioms are that enable you to say, "it is easy to see that..." To conclude for instance that every 1 -> X + Y factors through one of the coproduct inclusions, you need something more than the fact that Set is a topos (since this statement doesn't hold for all toposes). So, what is the background theory you are using to prove these assertedly clear statements?
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Which Banach spaces have categorical duals?
Yes, that's a good point: the compact objects in Ban with all bounded linear maps as morphisms are indeed the finite-dimensional ones. This category is smc for the same reason as Ban_1 is (on general grounds, the only "extra" issue would be whether the structural constraints used for Ban_1 are natural with respect to all bounded linear maps, not just wrt the contractions, but that's obvious from linearity -- actually I think of it as the other way around: the smc structure on Ban is easy, and that the structural constraints belong to Ban_1 is a small extra bonus).
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Categorical duals in Banach spaces
While the question is good, there are several senses of "dual" used in category theory. We (authors of the nLab article) meant a pair of Banach spaces V, W equipped with an isometric isomorphism W --> hom(V, k) [where k = R or C as appropriate], where V is reflexive, so that the transpose V --> hom(W, k) is also an isometric isomorphism. I think this is a standard functional analysis sense of "V and W are dual to one another as Banach spaces", and it's also one of the meanings of "dual" used for smc cats, even if it's not the "compact closed" sense of the question above.
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Is the category of Banach spaces with contractions an algebraic theory?
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When are epimorphisms of algebraic objects surjective?
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When are epimorphisms of algebraic objects surjective?
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Iterated adjoint functors
Yes. You probably know that as a monoidal category, the simplex category is characterized by the universal property that it is initial among strict monoidal categories equipped with a monoid object. This is the key observation underlying the bar construction, among other things. The simplex 2-category has a similar 2-universal property, where we consider instead monoidal 2-categories equipped with a "KZ" monad, where the multiplication m: M@M --> M is left adjoint to the unit u@M: M --> M@M. Examples include cocompletion monads. Try googling this with Anders Kock (the K in KZ) as a key word.