# Tag Info

41

There are several things left unsaid. First, there is a sense in which "a vector space is naturally isomorphic to its dual" is not even wrong: the usual dual functor is contravariant, not covariant. That is, the identity functor is of the form $\mathbf{Vect} \to \mathbf{Vect}$ while the dual functor is of the form $\mathbf{Vect}^{op} \to \mathbf{Vect}$. ...

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Here is a manifestly invariant definition of an abelian category $\mathcal{C}$. It is a category with finite limits and colimits such that: (It is pointed) the map from the initial to the final object is an isomorphism; we denote by 0 any object which is both initial and final. (It is semiadditive) the map $X \amalg Y \to X\times Y$, given on $X$ by ...

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I'm going to interpret this as a request for examples of results that were announced a while ago but whose proofs have not yet appeared. In other words, people don't doubt that the result is correct and that the author(s) can prove it, and there is an expectation that the current lack of a proof will not be a permanent state of affairs (i.e., a paper with ...

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I think that Penelope Maddy's article What Do We Want a Foundation to Do? is a good starting point if you want to read some literature. I don't agree with all of Maddy's conclusions but the terminology that she introduces in this article is exceedingly helpful, as well as the very simple but often overlooked point that the concept of a "foundation of ...

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You’ve indeed proven the statement: “There exists a functor $\newcommand{\Vect}{\mathbf{Vect}}\Vect \to \Vect$, whose action on objects sends each vector space to its dual, and which is naturally isomorphic to the identity functor.” The standard theorem about double duals, stated precisely, is not just analogous to this, it’s a stronger statement: “The ...

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What you were told is wrong, for we have the following: Proposition. If two categories are equivalent and one of them is abelian, then so is the other. A proof (and some related results) can be found in Satz 16.2.4 in H. Schubert, Kategorien II, Springer, 1970 (likewise in the English version https://www.amazon.com/Categories-Horst-Schubert/dp/3642653669, ...

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The definition of a direct limit of groups was given by Pontrjagin in his 1931 paper Über den algebraischen Inhalt topologischer Dualitätssätze

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Category theory and set theory are complementary to one another, not in competition. I think this 'debate' is a bit of academic controversialising rather than an actual difference. If you've done a bit of category theory, you will realize how important the category of sets is (for Yoneda's lemma, representability, existence of generators, etc). Even if ...

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I'll give a counter-example to the claim that having a subobject classifier and being cartesian closed implies the existence of all finite limits. However, this is based on the definition of sub-object classifier given on wikipedia (linked in the comment above) that I would consider as incorrect: The wikipedia definition (at the time this is written) only ...

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I believe, this notion is closely related to the notion of the coarse structure which indeed had found many beautiful applications in geometric group theory and algebraic topology (including proofs of the Novikov conjecture for a lot of groups). For introduction, I'd recommend Lectures on coarse geometry by John Roe where in Chapter 2 he explains how to give ...

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