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Enriched categories, topoi, abelian categories, monoidal categories, homological algebra.

87 votes

What non-categorical applications are there of homotopical algebra?

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 th …
79 votes
5 answers
5k views

Can the Lawvere fixed point theorem be used to prove the Brouwer fixed point theorem?

The Lawvere fixed point theorem asserts that if $X, Y$ are objects in a category with finite products such that the exponential $Y^X$ exists, and if $f : X \to Y^X$ is a morphism which is surjective o …
Qiaochu Yuan's user avatar
45 votes

Category theory and arithmetical identities

I don't know what counts as an "arithmetical identity" for you, but there's a rich family of interesting examples coming from groupoid cardinality. To say it very tersely, if $X$ is a groupoid with at …
Qiaochu Yuan's user avatar
42 votes
Accepted

Why are ring actions much harder to find than group actions?

First, I am not really sure what you mean by "it is hard to come across a general theory of ring actions." This is precisely module theory! If $f : R \to S$ is any ring homomorphism whatsoever, then c …
Qiaochu Yuan's user avatar
40 votes
Accepted

Can one explain Tannaka-Krein duality for a finite-group to ... a computer ? (How to make in...

$\DeclareMathOperator\Rep{Rep}\DeclareMathOperator\Vect{Vect}\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Mod{Mod}\DeclareMathOperator\GL{GL}\DeclareMathOperator\Hom{Hom}$The infinitude of the in …
Qiaochu Yuan's user avatar
38 votes
Accepted

Linear algebra in terms of abstract nonsense?

To my mind there are two classes of interesting categorical facts here, loosely speaking "additive" facts and "multiplicative" facts. Some additive facts: Finite-dimensional vector spaces over $k$ h …
Qiaochu Yuan's user avatar
37 votes
2 answers
3k views

How can I define the product of two ideals categorically?

Given a commutative ring $R$, there is a category whose objects are epimorphisms surjective ring homomorphisms $R \to S$ and whose morphisms are commutative triangles making two such epimorphisms surj …
Qiaochu Yuan's user avatar
37 votes

Why do Lie algebras pop up, from a categorical point of view?

The category of Lie algebras is equivalent to a certain category of cocommutative Hopf algebras, with the equivalence given by sending a Lie algebra $\mathfrak{g}$ to its universal enveloping algebra …
Qiaochu Yuan's user avatar
35 votes
3 answers
4k views

Is every abelian group a colimit of copies of Z?

More precisely, is every abelian group a colimit $\text{colim}_{j \in J} F(j)$ over a diagram $F : J \to \text{Ab}$ where each $F(j)$ is isomorphic to $\mathbb{Z}$? Note that this does not follow fro …
Qiaochu Yuan's user avatar
33 votes
7 answers
3k views

Do non-associative objects have a natural notion of representation?

A magma is a set $M$ equipped with a binary operation $* : M \times M \to M$. In abstract algebra we typically begin by studying a special type of magma: groups. Groups satisfy certain additional ax …
Qiaochu Yuan's user avatar
32 votes

What's a groupoid? What's a good example of a groupoid?

Personally, the reason I'm interested in groupoids is something called groupoid cardinality and some other related ideas (the link contains a lot of other links). A motivating idea here is that certa …
31 votes

Why are monadicity and descent related?

I think of monads in terms of algebraic theories. Monads are substantially more general than this intuition suggests! Here is a better intuition: monads are categorified idempotents. The point …
Qiaochu Yuan's user avatar
28 votes

What is a field [Körper] really?

Fields are the simple (no nontrivial quotients) commutative rings. Grothendieck told us to work in nice categories with nasty objects rather than nasty categories with nice objects; fields are the nic …
Qiaochu Yuan's user avatar
26 votes

Are there non-trivial infinite chains of adjoint functors?

Let $C$ be a category enriched over finite-dimensional $k$-vector spaces. A Serre functor for $C$ is a $k$-linear automorphism $S : C \to C$ such that there is a natural equivalence $$\text{Hom}(x, y …
Qiaochu Yuan's user avatar
26 votes

Why higher category theory?

Re: David Corfield's comment, this answer will mostly address "higher as in $(\infty, 1)$-categories" rather than "higher as in $n$-categories." I also do understand you need the notion of abelia …

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