149

Category theory serves several purposes. On the most superficial level it provides a common language to almost all of mathematics and in that respect its importance as a language can be likened to the importance of basic set theory as a language to speak about mathematics. In more detail, category theory identifies many similar aspects in very different ...


52

There is a precise sense in which the concept of field is not algebraic like, say, the concept of ring or group or vector space etc.: it is a theorem that any kind of mathematical structure that is defined as having a set of elements and some fixed list of total operations of constant finite arity obeying some fixed list of unconditional equations gives rise ...


47

I wouldn't call it "deep", but here's an intuitive reasoning. Intersections have elements that come from both sets, so they have the properties of both sets. If, for each of the component sets, there is some element(s) guaranteed to exist within that set, then such element(s) must necessarily exist in the intersection. For example, if $A$ and $B$ are closed ...


44

Congratulations, you have reinvented the notion of a dinatural transformation (see for instance MacLane's Categories for the working mathematician, section IX.4). And your proof, that every dinatural transformation from the identity functor to the dualization functor is zero, is correct. And I agree that this is one (and perhaps the only) way to make precise ...


40

Consider the category $C$ with four objects, $a,b,c,d$ and, other than identity arrows, a single arrow $a\to b$ and a single arrow $c\to d$. Now consider the category $D$ with three objects $x,y,z$, and, aside from identity arrows, the arrows $x\to y$, $y\to z$, and $x\to z$. Now, consider the functor $F:C\to D$ with $F(a)=x$, $F(b)=F(c)=y$, and $F(d)=z$ (...


38

No, this is not possible. For instance, let $K$ be any field with a automorphism $f:K\to K$ whose order is finite and greater than $2$. Then $A(f):A(K)\to A(K)$ would be an automorphism of the same order extending $f$. But no such automorphism exists: by the Artin-Schreier theorem, any finite-order automorphism of an algebraically closed field has order ...


37

Well. That depends on whom you might ask this. Set theory might be inconsistent. In particular $\sf ZFC$ and its extension by large cardinal axioms. It's a nontrivial thing, to feel safe with these theories, and it takes a lot of practice and time until you understand that $\sf ZFC$ is self-evident to some extent, and [some] large cardinal axioms are ...


37

Equivalent categories are identical except that they might have different numbers of isomorphic "copies" of the same objects. One way of making this precise is as follows. Say a category $\mathcal{C}$ is skeletal if isomorphic objects of $\mathcal{C}$ are equal. Given any category $\mathcal{C}$, you can find an equivalent skeletal full subcategory (or "...


35

There are tons of applications! There are also many useful reformulations. The following will be more or less in increasing level of sophistication. But first, a silly picture: I've written a blog post which expands a lot upon this answer, if you're interested. As you mentioned, it can be used to prove that the fundamental group of a topological group is ...


35

I think questions like these are often asked by people who don't have a clear/coherent idea of what foundations are and what purpose they serve. This isn't meant as some kind of insult. I think many, probably the majority, of mathematicians are in this situation1. Specifically, I believe that if you asked most mathematicians which "foundations" they use, ...


33

Morita Equivalence This is a supplement to the aspect of quiver representations mentioned in Alistair's answer. Every associative finite dimensional $k$-algebra $A$ is Morita equivalent to a path algebra $kQ/I$ (this is another Gabriel's theorem). In particular, you have a very nice equivalence (so nice that the functors giving such equivalence are given by ...


33

Set theory and category theory are both foundational theories of mathematics (they explain basics), but they attack different aspects of foundations. Set theory is largely concerned with "how do we build mathematical objects (or what could we build)" while category theory is largely concerned with "what structure to mathematical objects have (or could have)"?...


32

1) Of course there is the important gluing lemma for schemes which says that certain pushouts along open immersions exist. But the category of schemes has not all colimits, see MO/9961. 2) The category of schemes has not all limits, see MO/9134 and MO/65506. 3) In contrast to 1) and 2), the category of locally ringed spaces (as well as the category of ...


32

First, why is the diagram commutative: you've got the following commutative diagram: It is commutative precisely because this is how we defined the map $X_1 \times_Y X_2 \to X_1 \times_Z X_2$. The bottom right square is used to define $Y \to Y \times_Z Y$. Now, you diagram is commutative iff the two maps $X_1 \times_Y X_2 \to Y \times_Z Y$ are equal, iff ...


32

This is my understanding of this yoga. It may not be exactly what you seek and may differ from another person's point of view. Also I apologize for my bad english. For Grothendieck, many things should have a relative version. So instead of considering just a space $X_0$, consider a morphism $f:X\rightarrow S$ thought as a family of spaces $s\mapsto X_s:=f^{-...


31

Yes, those are accurate statements. The inclusion $\mathbb{Z}\hookrightarrow \mathbb{Q}$ is both a monomorphism and epimorphism in the category $\mathsf{Ring}$ (rings and ring homomorphisms), but not an isomorphism. The inclusion $\mathbb{Q}\hookrightarrow\mathbb{R}$ is both a monomorphism and epimorphism in the category $\mathsf{Haus}$ (Hausdorff ...


31

From a formal viewpoint it is possible to study category theory within category theory, using the notion of a topos. Topos theory does many things, but one thing it provides is an alternative, category-theoretic foundation for mathematics. By augmenting topos theory with sufficient additional axioms, it is theoretically possible to re-construct all of ZFC ...


31

Spaces for which this is true are called completely regular. In fact, it is an equivalent characterization of completely regular spaces $X$ that their topology is entirely determined by the set $C(X)$ of real-valued continuous functions on them. In other words, there is a unique completely regular topology that makes all these and only these functions ...


31

There are already many excellent answers, but I want to add another perspective, already partly found in other answers, but I hope distinct enough to stand on its own. I like to explain by analogy. Consider the question, "What is a vector?" What is a vector? Well, you might get you any of the following informal definitions as a response: (a) a list of ...


30

I won't get into the exact importance of category theory, because I don't know nearly enough to answer this. But I will comment on your last sentence instead, because I find it is important enough to address. Generalization is the bread and butter of mathematics. The idea is that often there are many similarities between two seemingly unrelated objects, but ...


30

As k.stm says in the comments, usually a more general thing is true: these categories $C$ are equipped with forgetful / underlying set functors $U : C \to \text{Set}$ which tend to have a left adjoint, the "free" functor $F : \text{Set} \to C$. Whenever this is true, it follows that $U$ preserves all limits, not just products. Sometimes, but more rarely, $...


29

Here's a good hands-on example. Posets can be thought of as categories where any two objects have at most one arrow between them. If you start with a poset $(P,\leqslant)$ then you can form a category $\mathscr{C}_P$ whose object set is $P$ itself and $u,v\in P$ have an arrow $u\to v$ if and only if $u\leqslant v$. Conversely, if $\mathscr{C}$ is a small ...


29

I disagree that most branches of mathematics are just an application of set theory and logic. The fact that most areas of mathematics use set related notions and employ logic does not mean they are applications of these areas. For instance, would you say that English Literature = English Words + English Grammar? After all, every piece of English literature ...


29

"Injection" makes sense in a concrete category, namely a category $C$ equipped with a faithful functor $F : C \to \text{Set}$: a morphism $f$ is an injection if $F(f)$ is an injection (equivalently, a monomorphism). Faithful functors always reflect monomorphisms: if $F(f)$ is a monomorphism, then so is $f$. The proof is straightforward. If $fg = fh$, then $...


28

The space $\mathbb{R}^\mathbb{R}$, which is the space of all functions from $\mathbb{R}$ to itself with the topology of pointwise convergence, is not a metric space (it is not even first countable). This kind of function space arises in many areas of math. The issue is that only countable products of metric spaces need to be metric, but function spaces like ...


28

A category consists of objects and arrows between them (which can be associatively composed). We can express various constructions and theorems with the help of categories, and when we switch the direction of all arrows, we arrive to the dual constructions and dual theorems, which are usually named by the prefix 'co', and which stay valid, as the proof/...


28

Here is a very simple example: let $P = \{a,b,c\}$ where $a \le b$ and $a \le c$ ($b$ and $c$ aren't comparable); let $Q = \{x,y,z\}$ where $x \le y \le z$. And let $f : P \to Q$ be defined by $a \mapsto x$, $b \mapsto y$ and $c \mapsto z$. $f$ is obviously bijective, and since $x \le y$ and $x \le z$ it's a poset homomorphism. But it's not an isomorphism, ...


28

In my opinion, what's "really happening" with category theory is that it is the model theory of type theory. On the type theory Wikipedia page just linked John Lane Bell is quoted as saying "Roughly speaking, a category may be thought of as a type theory shorn of its syntax." Type theory (in the broad sense of the term) can be thought of as (potentially) ...


27

A simpler example: take a category with two objects and three morphisms, one of which goes from one object to the other. This last morphism is monic and epic, but not an iso.


27

First, keep in mind that the term "homomorphism" predates both the term "morphism" and the creation of category theory. "Homomorphism," roughly speaking, refers to a map between sets equipped with some kind of structure that preserves that structure. The collection of all such structured sets and homomorphisms between them thus forms a category $C$, but ...


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