**7**

votes

**6**answers

1k views

### Giving $Top(X,Y)$ an appropriate topology

I am not sure if its OK to ask this question here.
Let $Top$ be the category of topological spaces. Let $X,Y$ be objects in $Top$.
Let $F:\mathbb{I}\rightarrow Top(X,Y)$ be a function (I will ...

**1**

vote

**1**answer

84 views

### lfp property for dagger symmetric monoidal categories and their internal categories

We can define internal categories in a monoidal category like this. Let $C$ be a dagger symmetric monoidal category. Will $C$ be locally finitely presentable? Let $C_{int}$ be the category of ...

**82**

votes

**10**answers

9k views

### Is there an introduction to probability theory from a structuralist/categorical perspective?

The title really is the question, but allow me to explain.
I am a pure mathematician working outside of probability theory, but the concepts and techniques of probability theory (in the sense of ...

**18**

votes

**4**answers

3k views

### Tannakian Formalism

The Tannakian formalism says you can recover a complex algebraic group from its category of finite dimensional representations, the tensor structure, and the forgetful functor to Vect. Intuitively, ...

**30**

votes

**2**answers

2k views

### Which colimits commute with which limits in the category of sets?

Given two categories $I$ and $J$ we say that colimits of shape $I$ commute with limits of shape $J$ in the category of sets, if for any functor $F : I \times J \to \text{Set}$ the canonical map ...

**16**

votes

**1**answer

3k views

### Infinite Tensor Products

Let $A$ be a commutative ring and $M_i, i \in I$ be a infinite family of $A$-modules. Define their tensor product $\bigotimes_{i \in I} M_i$ to be a representing object of the functor of multilinear ...

**23**

votes

**2**answers

788 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 ...

**15**

votes

**2**answers

2k views

### Large cardinal axioms and Grothendieck universes

A cardinal $\lambda$ is weakly inaccessible, iff a. it is regular (i.e. a set of cardinality $\lambda$ can't be represented as a union of sets of cardinality $<\lambda$ indexed by a set of ...

**9**

votes

**4**answers

779 views

### Localizing an arbitrary additive category

Under which conditions localizing an additive category by some class S of morphisms yields and additive category? It seems easy to define certain addition on morphisms if we fix their representatives ...

**15**

votes

**3**answers

771 views

### Characterising categories of vector spaces

Consider the category $FdVect_k$ of finite dimensional $k$-vector spaces, for some given field. It is abelian, semisimple, in that each object is a finite sum of simple objects (of which there is only ...

**4**

votes

**1**answer

414 views

### Is there a standard name for a 2-category which has an object z such that, for every object x, the category Hom(x,z) has a terminal object?

Motivation
In Pursuing Stacks, Grothendieck defines what he calls a basic localizer, which is, to put it roughly, a class of functors between small categories with which one can make homotopy in ...

**4**

votes

**1**answer

469 views

### Regular monomorphisms of schemes

In the category of schemes, the equalizer of two morphisms $f,g : X \to Y$ is always a locally closed immersion into $X$ (since this is just $X \times_{Y \times Y} Y$ and $\Delta : Y \to Y \times Y$ ...

**52**

votes

**11**answers

9k views

### “Philosophical” meaning of the Yoneda Lemma

The Yoneda Lemma is a simple result of category theory, and its proof is very straightforward.
Yet I feel like I do not truly understand what it is about; I have seen a few comments here mentioning ...

**37**

votes

**9**answers

9k views

### Relating Category Theory to Programming Language Theory

I'm wondering what the relation of category theory to programming language theory is.
I've been reading some books on category theory and topos theory, but if someone happens to know what the ...

**39**

votes

**11**answers

9k views

### Most striking applications of category theory?

What are the most striking applications of category theory? I'm trying to motivate deeper study of category theory and I have only come across the following significant examples:
Joyals ...

**39**

votes

**10**answers

5k views

### What precisely Is “Categorification”?

(And what's it good for.)
Related MO questions (with some very nice answers): examples-of-categorification; can-we-categorify-the-equation $(1-t)(1+t+t^2+\dots)=1$?; categorification-request.

**32**

votes

**6**answers

3k views

### What does “quantization is not a functor” really mean?

The answers to this question do a good job of exploring, at a heuristic level, what "quantization" should be. From my perspective, quantization involves replacing a (commutative) Poisson algebra by ...

**42**

votes

**7**answers

3k views

### The main theorems of category theory and their applications

This question first arose as I wrote an answer for the question: Is there a nice application of category theory to functional/complex/harmonic analysis?; it can also be regarded as a (hopefully) more ...

**40**

votes

**5**answers

3k views

### When does Cantor-Bernstein hold?

The Cantor-Bernstein theorem in the category of sets (A injects in B, B injects in A => A, B equivalent) holds in other categories such as vector spaces, compact metric spaces, Noetherian topological ...

**25**

votes

**8**answers

3k views

### What is a metric space?

According to categorical lore, objects in a category are just a way of separating morphisms. The objects themselves are considered slightly disparagingly. In particular, if I can't distinguish ...

**52**

votes

**3**answers

2k views

### Does linearization of categories reflect isomorphism?

Given a category $C$ and a commutative ring $R$, denote by $RC$ the $R$-linearization: this is the category enriched over $R$-modules which has the same objects as $C$, but the morphism module between ...

**52**

votes

**4**answers

3k views

### Rigidity of the category of schemes

Call a category $C$ rigid if every equivalence $C \to C$ is isomorphic to the identity. I don't know if this is standard terminology. Many of the usual algebraic categories are rigid, for example ...

**17**

votes

**6**answers

1k views

### A canonical and categorical construction for geometric realization

There is a very intimate connection between categories, simplicial sets, and topological spaces. On one hand, simplicial sets are the presheaf category on the category $\Delta$ and $\Delta$ is a ...

**15**

votes

**5**answers

2k views

### A homotopy commutative diagram that cannot be strictified

By a "homotopy commutative diagram," I mean a functor $F: \mathcal{I} \to \mathrm{Ho}(\mathrm{Top})$ to the homotopy category of spaces. By a "strictification," I mean a lifting of such a functor to ...

**26**

votes

**2**answers

1k 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 ...

**25**

votes

**2**answers

2k views

### What interesting/nontrivial results in Algebraic geometry require the existence of universes?

Brian Conrad indicated a while ago that many of the results proven in AG using universes can be proven without them by being very careful (link). I'm wondering if there are any results in AG that ...

**19**

votes

**1**answer

1k views

### Analogy between the exterior power and the power set

The symmetric algebra of an object exists in every cocomplete $\otimes$-category. For the category of sets $\mathrm{Sym}(X)$ is the set of multi-subsets of $X$.
The usual definition of the exterior ...

**15**

votes

**6**answers

1k views

### How can category theory help my research in set theory?

How can category theory help my research in set theory?
I rarely use category theory as such in my current work, and one almost never sees any category theory in set-theoretic research papers or at ...

**32**

votes

**1**answer

1k views

### Categorical definition of the ideal product within the category of rings

This is an extension of this question. Let $I,J$ be ideals of a ring $R$; every ring is commutative and unital here. Is it possible to define $R \to R/(I*J)$ out of $R \to R/I$ and $R \to R/J$ in ...

**15**

votes

**6**answers

1k views

### In what sense are fields an algebraic theory?

Since there is no "free field generated by a set", it would seem that
1) there is no monad on Set whose algebras are exactly the fields
and
2) there is no Lawvere theory whose models in Set are ...

**13**

votes

**8**answers

2k views

### References for homotopy colimit

(1) What are some good references for homotopy colimits?
(2) Where can I find a reference for the following concrete construction of a homotopy colimit? Start with a partial ordering, which I will ...

**11**

votes

**8**answers

1k views

### Cogroup Objects

Pretty much anyone who does algebra is familiar with group objects in categories, but what about cogroup objects? Most of what I've been able to find about them is that they "arise naturally in ...

**10**

votes

**4**answers

2k views

### Anabelian geometry study materials?

I want to study anabelian geometry, but unfortunately I'm having difficulties in finding some materials about it. If you could offer me some books/papers/articles I would be glad.

**7**

votes

**7**answers

2k views

### What are the advantages of phrasing results in terms of exact sequences and commutative diagrams?

For example, I find the first group isomorphism theorem to be vastly more opaque when presented in terms of commutative diagrams and I've had similar experiences with other elementary results being ...

**14**

votes

**3**answers

2k views

### Conjectures in Grothendieck's “Pursuing stacks”

I read on the nLab that in "Pursuing stacks" Grothendieck made several interesting conjectures, some of which have been proved since then. For example, as David Roberts wrote in answer to this ...

**18**

votes

**3**answers

1k views

### Invariance of $Z[x]$ under a self-equivalence of the category of commutative rings with 1.

Let $\mbox{Rings}$ be the category of commutative rings with $1$.
Is there an equivalence of categories $F: \mbox{Rings} \to \mbox{Rings}$ s.t.
$$F(\mathbb{Z}[x])\not\cong \mathbb{Z}[x]?$$

**14**

votes

**1**answer

602 views

### Lawvere's fixed point theorem and the Recursion Theorem

Building off of Qiaochu's comment on my answer to a previous mathoverflow question, I would like to know: can the Recursion Theorem, $$\forall e\exists k[\Phi_e\text{ is total }\implies ...

**9**

votes

**1**answer

501 views

### Why not _co_free modules?

Let $R$ be a ring, and $R\text{-Mod}$ its category of all left modules. There is a "forgetful" functor $\operatorname{Forget}: R\text{-Mod} \to \text{AbGp}$, which is additive, continuous, and ...

**7**

votes

**4**answers

1k views

### Definition of forgetful functor

I was wondering if it is possible to make a formal definition of what it means for a functor to be forgetful. That is, using only the terminology of categories. I have seen so many examples of ...

**5**

votes

**3**answers

925 views

### Does any tensor category correspond to a bialgebra ?

I wonder how strong the power of Tanaka philosophy is,and if we accept that a tensor category is a generalized bialgebra,what difficulties we will come up against ?
edit: whether most tensor ...

**20**

votes

**2**answers

1k views

### What is the theory of local rings and local ring homomorphisms?

It is well-known that the category of local rings and ring homomorphisms admits an axiomatisation in coherent logic. Explicitly, it is the coherent theory over the signature $0, 1, -, +, \times$ with ...

**8**

votes

**2**answers

1k views

### Infinite products exist *only* for affine schemes?

Infinite products don't exist in the category of schemes (see Jonathan Wise's answer here). However, all limits of affine schemes exist in the category of schemes (and are affine). I would like to ...

**26**

votes

**2**answers

789 views

### What are the algebras for the double dualization monad?

Let $k$ be a field, and let $\mathbf{Vect}$ denote the category of vector spaces (possibly infinite-dimensional) over $k$. Taking duals gives a functor $(\ )^*\colon \mathbf{Vect}^{\mathrm{op}} \to ...

**8**

votes

**1**answer

376 views

### Is there a notion of a “model category which admits left Bousfield localization?”

At a conference not too long ago I gave a talk on (left) Bousfield localization and was asked an interesting question afterwards. The question was whether I knew any examples of model categories which ...

**8**

votes

**2**answers

2k views

### The definition of a group object is wrong?

An old MO answer by Noah Snyder makes a claim I don't completely understand, but mostly because I don't know any examples. The answer claims that in some examples of (things that one would want to ...

**8**

votes

**1**answer

486 views

### Slices of presheaf categories

Apparently it's 'well known' that if $P$ is a presheaf on $C$ then there is an equivalence $\widehat{C}/P \simeq \widehat{\int P}$, where $\int P$ is the usual category of elements and $\widehat{C} = ...

**8**

votes

**7**answers

1k views

### Can the Category of Schemes be Concretized?

If not, are there any interesting subcategories that can be concertized? If I am not mistaken, the category of reduced finite type varieties over the complex numbers would be an example, where the ...

**7**

votes

**2**answers

487 views

### Description of quasi-coherent modules on a product

Let $R$ be a ring and $X,Y$ two $R$-schemes, which you may assume to be noetherian or anything reasonable you like. Is it possible to "construct" $\text{Qcoh}(X \times_R Y)$ out of $\text{Qcoh}(X)$ ...

**6**

votes

**3**answers

561 views

### Model category structures on the category of $L_\infty$-algebras

Let $k$ be a characteristic zero field. Then it is known that the forgetful functor $dgla(k)\to chain(k)$ from differential graded Lie algebras (over $k$) to cochain complexes induces a model category ...

**4**

votes

**2**answers

803 views

### How can I understand the “groupoid” quotient of a group action as some sort of “product”?

Recall the notion of groupoid (Wikipedia, nLab). An important construction of groupoids is as "action groupoids" for group actions. Namely, let $X$ be a groupoid and $G$ a group, and suppose that ...