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1d
comment Natural operators in differential geometry?
You might look at Kolar's book, "Naturality in Differential Geometry," which has a free PDF under Google. Naturality means vaguely that an operator is defined in a way that doesn't use properties of the object it's defined on, and has a precise definition in category theory. All in all, this question admits only a very long answer, if any.
2d
comment Extending a Field Monomorphism
Well, I would look into (A3.4).
Jul
29
comment Dense Domain: Preimage
But I still think if you take $Y=\ell^1,W$ the sequences with finite support, $X=\mathbb{R}, A(1)=(1/n^2 e_n)$ that you'll get a counterexample, for then $A^{-1}W=\{0\}$.
Jul
29
comment Dense Domain: Preimage
Oh, silly, I wasn't thinking of $W$ as a linear subspace.
Jul
29
comment Which directed graphs correspond to “algebraic” diagrams?
@Batominovski but of course you just require all starting points to satisfy the condition.
Jul
29
comment Dense Domain: Preimage
This isn't true as stated. $A$ could be the zero map.
Jul
28
comment localization of the Pontrjagin ring of an $H$-space with respect to $\pi_0$
@Rain I don't know what you're confused about-you would need to specify.
Jul
27
answered localization of the Pontrjagin ring of an $H$-space with respect to $\pi_0$
Jul
25
answered Defining a coproduct in $\mathsf{Grp}$ using group presentations
Jul
25
comment coherence of inverses in 2-groupoids
@ZhenLin and Jacques, all this is easily settled if inversion of 1-morphisms is functorial since functors preserve inverses. This falls out of a definition of a 2-groupoid as a groupoid internal to groupoids, but doesn't seem to be implied by a definition as a 2-category in which 1- and 2-morphisms happen to be invertible. No?
Jul
24
revised coherence of inverses in 2-groupoids
edited tags
Jul
24
comment coherence of inverses in 2-groupoids
Yeah, me too. I'm adding a category theory tag in case that'll draw more attention.
Jul
22
comment coherence of inverses in 2-groupoids
This is basically the question of whether we want our (strict) 2-groupoids to be 2-categories with the property that all 1- and 2-morphisms happen to be invertible, in which case there's no reason for your flip to exist, or with the structure of an inversion endofunctor, whose action on 2-morphisms would be your flip. It does seem that the most natural examples have a functorial inverse, but I don't see it in people's definitions.
Jul
22
comment Differential structure on the cone
The most obvious reason is that it's sometimes not even a topological manifold-e.g. the Kummer surface. I'm not actually sure whether one can give a smooth orbifold which is a topological manifold but not a smooth manifold.
Jul
22
answered Differential structure on the cone
Jul
21
answered Exterior derivative
Jul
21
comment Reading mathematics at the graduate level - does every single proof matter?
I'm not sure what you want here. No one's likely to say "you must read every proof in every book you pick up." You just have to decide what you need on your own, and with advice from your advisor etc.
Jul
18
comment Fixed points of diffeomorphisms: eigenvalues of the pushforward
Incidentally, this can indeed occur, although the diffeomorphism can't be analytic. Something like $(x,y)\mapsto(x,y+e^{-1/x^2}\sin(1/x))$ should give an example.
Jul
17
comment Fixed points of diffeomorphisms: eigenvalues of the pushforward
Have you tried weakening your assumption by supposing that $\gamma(t)\cap \text{Fix}_f$ merely accumulates to $x$ as $t\to t_0$?
Jul
17
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