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1 Easy Proposition Let $f:X\to Y$ be a continuous map of topological spaces, $\mathscr F$ a sheaf of abelian groups on $X$ such that $R^jf_*\mathscr F=0$ for $j>0$. Then for all $i\geq 0$ there exists a natural isomorphism $$ H^i(Y, f_*\mathscr F)\simeq H^i(X,\mathscr F) $$ Proof Apply the composition rule for the derived functors of $G=\Gamma(Y, \_ )$ ...


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Here is an example from set theory. Set theorists commonly study not only the theory $\newcommand\ZFC{\text{ZFC}}\ZFC$ and its models, but also various fragments of this theory, such as the theory often denoted $\ZFC-{\rm P}$ or simply $\ZFC^-$, which does not include the power set axiom. One can find numerous instances in the literature where authors ...


74

I like the simple slogan: homotopical algebra is the nonlinear generalization of homological algebra. Let me assume that you value and appreciate homological algebra in the broadest sense as a fundamental, successful and highly applicable tool in many areas of math (otherwise I can't conceive of an argument that would be convincing for this question). At the ...


71

As a student, I'm always looking for organizing principles in mathematics to help me keep track of all of the mathematics I learn. It's easy to get lost in a deluge of definitions unless I organize them in some way. For a long time category theory was my main organizing principle (e.g. the idea of adjoint functors alone is already a very helpful way to ...


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I'm not quite certain what Peter May had in mind 40 years ago, but probably he had in mind the fact that pushouts are a lot better behaved in CGWH than in CHG. Specifically, CHWH is closed under pushouts, one leg of which is the inclusion of a closed subspace. CHG does not have such nice behavior, and pushouts like that are used all over The Geometry of ...


70

Set theory provides a foundation for mathematics in roughly the same way that Turing machines provide a foundation for computer science. A computer program written in Java or assembly language isn't actually a Turing machine, and there are lots of good reasons not to do real programming in Turing machines - real languages have all sorts of useful higher ...


68

Higher category theory is, roughly speaking, where category theory meets homotopy coherent mathematics. It is hence relevant to those problems in which categorical structures and homotopy coherent phenomena play a significant role. Many areas of algebraic topology and algebraic geometry have this property. There are also many such areas who don't. From what ...


63

This is too long for a comment, but doesn't exactly answer the question. However, I've had enough eggnog this Christmas that I'm going to post it anyway (despite knowing almost nothing about category theory). Reading the question and skimming over the comments, I see a lot of romantic descriptions of the practice of mathematics that bear little ...


62

If $F,G : C \to D$ are functors such that $F(x) \cong G(x)$ for every $x \in C$, I would call $F,G$ "pointwise isomorphic". You ask for examples of non-isomorphic functors which are pointwise isomorphic. There are plenty natural examples. Consider the interval category $I=\{0 \to 1\}$. The category of functors $I \to C$ is isomorphic to the category of ...


61

I'm coming a bit late to this party, but I'll put in my two cents anyway because they are rather different from everything else I've heard so far. In a nutshell, my response is: Yes, I agree that this is a problem (though I do think you would have done better to post only the question and not the rant), and What you can do is be part of the solution. For ...


61

First, there is indeed nothing mathematically very deep in this observation, and I agree that the word "breakthrough" might be exaggerated. But on the other hand lots of very deep ideas look trivial once spelled out explicitly. Moreover being younger than Voevosky I have never been really exposed to the idea that categories were sets of higher dimension (but ...


58

The Universal Coefficient Theorem for, say, singular cohomology should give examples. For any abelian group $G$ and $n> 0$, the functors from spaces to abelian groups given by $$X\mapsto H^n(X;G),\qquad X\mapsto \mathrm{Ext}(H_{n-1}(X),G)\oplus\mathrm{Hom}(H_n(X),G)$$ are isomorphic, but not naturally so. See Hatcher's "Algebraic Topology", Chapter 3.1 (...


55

There are several ways I think of expressing this 'duality'. But before describing this, maybe it would help to explain a sense in which 'existence' (at least one element, or totality of a relation) is dual to 'uniqueness' (at most one element, or well-definedness of a relation). So let's consider the category whose objects are sets and whose morphisms are ...


54

The word modulus (moduli in plural, cf. radius and radii, focus and foci, locus and loci) comes from Latin as a word meaning "small measure" or "unit of measure". This is why the absolute value of a complex number z is sometimes called the modulus of z and why the word is used in physics for Young's modulus. In 1800 Gauss introduced the congruence relation $...


53

One thing that might interest you is my result with Clark Barwick which gives an axiomatiation + uniqueness result for the homotopy theory of higher categories: arXiv:1112.0040 (i.e. $(\infty,n)$-categories). This axiomatization includes several variants of $(\infty,n)$-category for finite n, such as what you mention in your question. In particular, when ...


52

Take the category of (at most) countable-dimensional vector spaces over your favourite field. Then take the quotient by the Serre subcategory of finite-dimensional vector spaces. (And take a skeletal subcategory so that it strictly has only two objects.) Then this is an abelian category with only one non-zero object, whose endomorphism ring is the ...


50

I can muddy the waters...! According to editor E. Scholz of Hausdorff’s Collected Works (2008, p. 884): In a note of 3/20/1933 (Nachlass, fasc. 449) and in a further undated note (fasc. 571), Hausdorff symbolized the functoriality property of homology (in our later terminology) with a commutative diagram of homomorphisms between the terms of two ...


50

I do not understand what the bounty on this question is for, as it seems to me that the other answers were already rather devastating. Here is a semi-reasoned technical answer. According to G. Lolli (the paper you cite) "Sergeyev is wary of the axiomatic method because he thinks that by adopting it we would be tied to the expressive power of a language in ...


46

The surreal numbers exhibit much stronger universal properties than you have mentioned, for they also exhibit very strong homogeneity and saturation properties. For example, every automorphism of a set-sized elementary substructure of the surreals extends to an automorphism of the entire surreal numbers, and every set-sized type over the surreals that is ...


45

For a simpler, but arguably more artificial, example than Mark's, take $\mathcal{C}$ to be the category with one object and two morphisms. Then the identity functor $\mathcal{C}\to\mathcal{C}$ is "unnaturally isomorphic" to the functor that sends both morphisms to the identity map.


44

From Lefschetz's obituary of Hurewicz: “At a later date (1941) and in a very short abstract of this Bulletin Hurewicz introduced the concept of exact sequence whose mushroom like expansion in recent topology is well known.” Here is the abstract transcribed in its entirety: 329. Witold Hurewicz: On duality theorems. Let $A$ be a locally compact space,...


44

This might not quite count, but if you start with a principal $G$-bundle $f:P\rightarrow B$, there are two natural ways to put a $G\times G$ structure on the bundle $P\times G\rightarrow B$ given by $(p,b)\mapsto f(p)$. Because it is standard notational practice to denote such a bundle by simply writing down the map $P\times G\rightarrow B$, there is ...


42

As already mentioned, the phrase "down to earth" is subjective. For instance, I have friends in analytic number theory who ask what the utility of categories is (not higher categories), and to some, they might sound like a mathematician asking what the utility of groups is, but this is simply a reflection of our backgrounds. Regardless, here's a brief list ...


41

There is (another ?) description of the crossed product in categorical terms. Let ${\rm Mor}(Gp)$ be the category whose objects are homomorphisms of groups and morphisms are commutative diagrams. Let $C$ be the category of "groups acting on groups" whose objects are pairs of groups $(H,G)$ together with a homomorphism $H \to {\rm Aut}(G)$. Morphisms in ...


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


40

Let me expand on Yosemite Sam's comment. Pullbacks are indeed easier to define if you view a sheaf as a local homeomorphism. On the other hand, pushforwards are easier to define if you view a sheaf as a set-valued functor. Suppose we have a continuous map $f: X \to Y$ of topological spaces. Given a sheaf $F$ on $Y$, viewed as a local homeomorphism $\...


40

$\def\ZZ{\mathbb{Z}}$ This can happen in finitely generated abelian groups. Let $p$ be prime, set $G = \ZZ/p^2 \oplus \ZZ/p$ and set $H = \ZZ/p^3 \oplus \ZZ/p^2 \oplus \ZZ/p$. Then there are two non-isomorphic short exact sequences $0 \to G \to H \to G \to 0$. The first one is the sum of the extensions: $$\begin{matrix} 0 & \to & \ZZ/p & \to &...


39

To me the obvious answer involves sheafification of a presheaf. If you look at the construction of the associated sheaf to a presheaf in, say, Hartshorne it goes through the étalé space construction without specifically telling you, and to me it makes the construction somewhat unmotivated. Namely, if $P$ is a presheaf on $X$, then taking the stalk $P_x$ ...


39

I will have another go at arguing that dagger-categories are not evil. Let’s look at a simpler case first. Consider the property “$1 \in X$” on sets. As a property of abstract sets, this is evil: it’s not invariant under isomorphism, e.g. any iso $\{1,2\} \cong \{2,3\}$. But it is manifestly non-evil as a property of, say, “sets equipped with an ...


39

This will probably not be considered a serious mistake, but maybe it counts: According to Dray, Manogue if you ask the following question to scientist: Suppose the temperature on a rectangular slab of metal is given by $T(x,y)=k(x^2+y^2)$ where $k$ is a constant. What is $T(r,\theta)$? A: $T(r,\theta)=kr^2$ B: $T(r,\theta)=k(r^2+\theta^2)$ ...


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