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 $\mathbf{Vect}^{op} \to \mathbf{Vect}$. ...


33

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 ...


32

Typically, internal homs of $\newcommand{\C}{\textbf{C}}\C$ will look different from external homs just when “elements/points of $X$” (for objects $X \in \C$) are different from “maps $I \to X$” (where $I$ is the monoidal unit); or slightly more precisely, when $\C$ comes with a canonical forgetful functor $\newcommand{\Set}{\textbf{Set}} U : \C \to \Set$, ...


31

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 ...


31

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 ...


28

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 ...


27

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, ...


27

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


27

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 ...


25

Presentable $\infty$-categories can be understood without every having to think about cardinals. An $\infty$-category is presentable iff it is equivalent to one of the form $\mathcal{P}(C,R)$, where $C$ is a small $\infty$-category, $R=\{f_i\colon X_i\to Y_i\}$ is a set of maps in $\mathrm{PSh}(C)=\mathrm{Fun}(C^\mathrm{op}, \mathrm{Gpd}_\infty)$, and $\...


25

There's a distinction that I find striking but don't know how to formalize usefully or how to evaluate its importance: In algebraic geometry, moduli spaces get compactified, and this involves adding a relatively small set to the original space. Roughly speaking, the original space parametrizes some nice objects, and the compactification adds points "at ...


23

$\newcommand{\Hom}{\mathrm{Hom}}\newcommand{\Set}{\mathrm{Set}}\newcommand{\hom}{\mathrm{hom}}$In this example one cannot really say that the internal Hom is “exotic”, but at least it is morally different from the external Hom. Let $G$ be a group and let $\Set(G)$ be the category of sets with a $G$-action. Morphisms of this category are mappings of sets ...


21

(Edited to reflect your edit to the question!) The answer to your original statement (without the "nonempty" assumption) is no because we can let $A=\varnothing$ and $B=\ast$. Their endofunctor categories are each discrete with one object, but the categories themselves are not equivalent. The question becomes a lot trickier when you add "nonempty" to the ...


21

To add to the other good answers here, there's a family of examples that could be seen as a bit trivial. But in a sense they give the simplest answer to your question. Let $X$ be a partially ordered set, viewed as a category in the usual way: the objects of the category are the elements of $X$, and $\textrm{Hom}(x, y)$ has either $1$ or $0$ elements ...


21

The answer is yes, in fact one has a lot better than bi-interpretability, as shown by the corollary at the end. It follows by mixing the comments by Martin Brandenburg and mine (and a few additional details I found on MO). The key observation is the following: Theorem: The category of co-group objects in the category of groups is equivalent to the category ...


21

As suggested by Gerald, the notion was first introduced for groups. Given a directed system of groups, their direct limit was defined as a quotient of their direct product (which was referred to as their "weak product"). The general notion is a clear generalization, although the original reference only deals with groups. As mentioned by Cameron Zwarich in ...


20

For what it's worth, the reason I made that comment was because when I gave talks expressing skepticism about the philosophical basis of ZFC, I would often get the reaction "but as long as it's consistent, what's the problem?" Some version of this attitude is also quite prevalent in the philosophical literature on the subject. I wanted to make the point ...


20

To get to the point quickly first, the OP has definitely constructed a natural isomorphism (with some steps missing that I fill in below.) However, it is misleading to call it "a natural isomorphism between a vector space and its dual" because the interest in the dual space construction on finite-dimensional vector spaces is not simply forming $V^*$ from $V$ ...


20

Assume all the spaces are connected, $\pi_1 A \to \pi_1 B$ is surjective, and $\pi_1 C$ is abelian. The pushout, $P = C \sqcup_A B$, is contractible if and only if $\pi_1(P)$ and all the $H_i(P)$ are trivial. However, by Seifert-Van-Kampen, and the hypothesis on $\pi_1 A \to \pi_1 B$, we have that $\pi_1 C$ surjects onto $\pi_1 P$. Thus $\pi_1 P$ is ...


20

Fix a compact group $G$ and consider its category of Banach representations: the objects are (complex) Banach spaces $X$ endowed with a $G$-action by automorphims (not necessarily isometries) such that the action maps $G\times X\to X$ are jointly continuous and the morphisms are bounded maps $X\to Y$ commuting with the $G$-actions. Denote the trivial ...


20

This follows from two Facts: 1) A category monadic over Set/S is always an exact category. That is it has quotient by equivalence relation that are effective and universal. It is in particular a regular category. This is showed for example here. 2) The category of categories is not a regular category. An explicit example of regularity (and hence exactness) ...


20

The work on analytic geometry is all joint with Dustin Clausen! Your main question seems a little vague to me, but let me try to get at it by answering the subquestions. See also the discussion at the nCatCafe. Also, as David Corfield comments, much of this had been observed long before: https://nforum.ncatlab.org/discussion/5473/etale-site/?Focus=43431#...


19

The answer in general is no. Let $\mathcal C$ be the category of sets, let $\mathcal D$ be the category of pointed sets (with basepoint-preserving maps), and let $f: \mathcal C \to \mathcal D$ be the functor which adds a disjoint basepoint. Then $f$ is an equivalence on underlying groupoids, but not an equivalence of categories. Moreover, the forgetful ...


19

$\require{AMScd}$ A 1974 paper of R. Lagrange, Amalgamation and epimorphisms in $\mathfrak{m}$-complete Boolean algebras (Algebra Universalis 4 (1974), 277–279, DOI link), settled this affirmatively. In the cited paper, Lagrange shows that for any infinite cardinal $\mathfrak{m}$, the category of $\mathfrak{m}$-complete Boolean algebras with $\mathfrak{m}$-...


18

In EGA 0.1.5.2-3 (from the 1971 Springer edition) the right adjoint and the left adjoint of a functor $F$ are denoted by $F^{\rm ad}$ and ${}^{\rm ad}\!F$, respectively.


17

In fact what you need is that your ∞-category is additive (i.e. that it has direct sums and that the canonical commutative monoid structure on the mapping spaces is group-like). All stable categories are additive, so I'll just prove it in this case. Step 1: The pullback $X\times_Z Y$ is equivalent to the pullback $(X\times Y)\times_{Z\times Z}Z$. This fact ...


17

Since the dual of an abelian category is also an abelian category, the question is equivalent to the same question for projective resolutions. I will show that the category $\mathbf{Ab}^{\operatorname{f.t.}}$ of finitely generated abelian groups has enough projectives, but no functorial projective cover. The idea is that multiplication by any $n \in \mathbf ...


17

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 ...


16

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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