# Tagged Questions

**1**

vote

**1**answer

180 views

### Concise definition of subobjects

Higher category theory tells us that it is a bad idea to identify isomorphic things. Rather, the isomorphism should belong to some additional data. Also, categorification tells us that one should, ...

**25**

votes

**9**answers

1k views

### Continuous relations?

What might it mean for a relation $R\subset X\times Y$ to be continuous? In topology, category theory or in analysis? Is it possible, canonical, useful?
I have a vague idea of the possibility of ...

**84**

votes

**9**answers

4k views

### What non-categorical applications are there of homotopical algebra?

(To be honest, I actually mean something more general than 'homotopical algebra' - topos theory, $\infty$-categories, operads, anything that sounds like its natural home would be on the nLab.)
More ...

**6**

votes

**4**answers

1k views

### 'Category-theory'-free areas of pure math, 'category-theory'-loaded areas of applied math

To put it short: In which active research areas of (pure) mathematics no (or only minimal) knowledge in category theory is required ?
To put it long: I know almost nothing about category theory - but ...

**5**

votes

**1**answer

405 views

### Is Logic/Set Theory necessary for studying Topos Theory?

I have just completed a postgraduate course, in which I studied Category Theory, without having a background in Set Theory and Logic - this probably already sounds absurd to many. This did not seem to ...

**16**

votes

**3**answers

1k views

### Is there a scheme corresponding to the unit interval?

Can someone complete the following table?
$\begin{array}{cc} \text{Topology over } \mathbb{R} & \text{Topology over } \mathbb{C} & \text{Algebraic Geometry} \\\\ \hline \mathbb{R} & ...

**9**

votes

**6**answers

1k views

### Intuitionistic logic as quantization of classical logic?

A classically trained mathematician is more likely to be familiar (at least anecdotally) with an area of mathematical physics such as deformation quantization than with Intuitionistic logic. It is ...

**50**

votes

**11**answers

3k views

### Why is Set, and not Rel, so ubiquitous in mathematics?

The concept of relation in the history of mathematics, either consciously or not, has always been important: think of order relations or equivalence relations.
Why was there the necessity of singling ...

**27**

votes

**3**answers

1k views

### “Softness” vs “rigidity” in Geometry

According to common wisdom, there are structures in Geometry that have a more "topological" flavor, others that are more "geometrical", and others that are halfway between. Usually, geometries${}^*$ ...

**5**

votes

**1**answer

611 views

### Convenient definition of “category of Riemannian manifolds”?

Has a notion of "category of Riemannian manifolds" been defined and used in the literature?
For which reasons is it or would it (not) be a useful notion?
I think the objects should be all (perhaps ...

**12**

votes

**2**answers

1k views

### Examples of algorithms that came from category theory?

Generating Compiler Optimizations from Proofs is a wonderful paper. The authors say that they were faced with the problem, got stuck, then tried reasoning about it using category theory. They took ...

**17**

votes

**2**answers

1k views

### Surveys of Goodwillie Calculus

Is there a good general introduction to Goodwillie calculus out there, like a paper or publication that gives a general overview of the calculus as well as how it is useful and why we are interested ...

**33**

votes

**15**answers

6k views

### Is there a nice application of category theory to functional/complex/harmonic analysis?

[Title changed, and wording of question tweaked, by YC, because the original title asked a question which seems different from the one people want to answer.]
I've read looked at the examples in ...

**1**

vote

**0**answers

272 views

### Do Arbib and Manes describe just concrete categories?

In “Arbib, Manes. Arrows, Structures and Functors. The Categorical Imperative. 6. Structured sets.” there is an approach to formalize structures. I have a strong feeling that they describe just ...

**21**

votes

**3**answers

2k views

### Lawvere theories versus classical universal algebra

A Lawvere theory is a small category with finite products such that every object is isomorphic to a finite product of copies of a distinguished object x. A model of the theory in a category with ...

**6**

votes

**3**answers

282 views

### 2-morphisms in structured 2-categories

There are many $2$-categories, which are first specified by certain categories with extra structure; then the $1$- and $2$-morphisms are functors and natural transformations that preserve the extra ...

**19**

votes

**3**answers

2k views

### Locales and Topology.

As someone more used to point-set topology, who is unfamiliar with the inner workings of lattice theory, I am looking to learn about the localic interpretation of topology, of which I only have a ...

**8**

votes

**0**answers

681 views

### triangulated/derived categories in Physics and algebraic geometry

Why do physicists care about the triangulated/derived categories?
I mean what are the problems we want to approach using the machinery of triangulated/derived categories. e.g. in homological mirror ...

**1**

vote

**1**answer

1k views

### Classification Problems [closed]

I was thinking about the famous question in philosophy of mathematics: "When are two proofs the same?" and I was wondering if we could somehow "classify" proofs by establishing some sort of functorial ...

**12**

votes

**1**answer

680 views

### evil properties, higher category theory and well-chosen tensor products

Let's start with the following random example: If $F$ is a presheaf, then for every chain of open subsets $U \subseteq V \subseteq W$, the morphisms $F(W) \to F(V) \to F(U)$ and $F(W) \to F(U)$ ...

**0**

votes

**1**answer

134 views

### Isomorphism by classification

Are there any examples other than using dimension for vector spaces where the easiest way to show that two objects are isomorphic is by using a classification theorem and showing that they must both ...

**29**

votes

**5**answers

3k views

### Why do categorical foundationalists want to escape set theory?

This is a question that I have seen asked passively in comments relating to the separation of category theory from set theory, but I haven't seen it addressed in full.
I know that it's possible to ...

**10**

votes

**6**answers

2k views

### What do people mean by “subcategory”?

Mac Lane defines a subcategory as a subset of objects and a subset of morphisms that form a category. But the first rule of category theory is that you do not talk about equality of objects. Up to ...

**33**

votes

**13**answers

2k views

### Equality vs. isomorphism vs. specific isomorphism

This question prompted a reformulation:
What is a really good example of a situation where keeping track of isomorphisms leads to tangible benefit?
I believe this to be a serious question because ...

**27**

votes

**4**answers

3k views

### Is “all categorical reasoning formally contradictory”?

In the December 2009 issue of the newsletter of the European Mathematical Society there is a very interesting interview with Pierre Cartier. In page 33, to the question
What was the ontological ...

**23**

votes

**8**answers

3k views

### Geometric intuition for limits

I'm the sort of mathematician who works really well with elements. I really enjoy point-set topology, and category theory tends to drive me crazy. When I was given a bunch of exercises on subjects ...

**5**

votes

**4**answers

2k views

### Why are inverse images more important than images in mathematics?

Why are inverse images of functions more central to mathematics than the image?
I have a sequence of related questions:
Why the fixation on continuous maps as opposed to open maps? (Is there an ...

**14**

votes

**7**answers

3k views

### why haven't certain well-researched classes of mathematical object been framed by category theory?

Category theory is doing/has done a stellar job on Set, FinSet, Grp, Cob, Vect, cartesian closed categories provide a setting for $\lambda$-calculus, and Baez wrote a paper (Physics, Topology, Logic ...

**8**

votes

**3**answers

777 views

### What's the “correct” smooth structure on the category of manifolds?

As will become clear, this is in some sense a follow up on my earlier question Why should I prefer bundles to (surjective) submersions?. As with that one, I hope that it's not too open-ended or ...

**42**

votes

**11**answers

10k 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 ...

**11**

votes

**3**answers

1k views

### Applications of homotopy groups of spheres

The study of the homotopy groups of spheres $\pi_i(S^n)$ is a major subject in algebraic topology. One knows for example that nearly all of them are finite groups. Some are explicitly known. There is ...

**29**

votes

**7**answers

3k views

### What is DAG and what has it to do with the ideas of Voevodsky?

In Toen's and Vezzosi's article From HAG to DAG: derived moduli stacks a kind of definition of DAG is given. I am not an expert and can't see what's the relation between DAG and the motivic cohomology ...

**5**

votes

**4**answers

925 views

### Why is the concept of topos a “metamorphosis” of the concept of space?

Hi,
I recently started studying topos theory, and I am puzzled by the Grothendieck's claim that topos is a "metamorphosis" of the concept of space. Can somebody explain what he means by this?
...

**11**

votes

**4**answers

2k views

### Category theory and model theory as “natural” counterparts

I am aware of the profound discussion of the relationship between category theory and model theory (e.g. at The n-Category Café) but do wonder why category theory (CT) is not opposed to model theory ...

**10**

votes

**2**answers

418 views

### Primacy of arcs/arrows over vertices/objects

Freyd's Abelian Categories is the only textbook I know where the primacy of arrows over objects is taken seriously already in the axioms: there is no talk of objects at all. Only later one sees, that ...

**1**

vote

**6**answers

476 views

### Between abstract and concrete: What's the right way to think of specific categories?

At the risk of annoying some of the categorists I feel urged to pose this beginner-ish question:
If one talks about a specific category such as the category of sets with functions or the category of ...

**14**

votes

**11**answers

2k views

### Learning to Think Categorically

Up to this point in my education, I have had very little exposure to the language and machinery of category theory, and I would like to rectify this. My goal is to become conversant with some of the ...