# Tagged Questions

**1**

vote

**0**answers

237 views

### Categories of sheaves and Kan Extensions

This is quite a broad question regarding constructions of categories of sheaves in geometry.
Let $\textbf{Sch}$ denote the category of schemes. Let $\textbf{SchAff}$ denote the full subcategory of ...

**13**

votes

**1**answer

405 views

### Skeleton category of the category of skeleton categories?

A category is a skeleton if, roughly speaking, no two distinct objects within the category are isomorphic. To every category is associated a skeleton, and two categories are categorically "equivalent" ...

**38**

votes

**10**answers

2k 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 ...

**20**

votes

**4**answers

1k views

### When must it be sets rather than proper classes, or vice-versa, outside of foundational mathematics?

Every once in a blue moon it actually matters that some mathematical entity which might a priori only be a class is in fact a set. For clarification, here are some examples of what I do not ...

**27**

votes

**3**answers

999 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${}^*$ ...

**21**

votes

**11**answers

1k views

### Characterizing specific “concrete” mathematical objects by abstract general properties

In this note by Tom Leinster the Banach space $\mathrm{L}^1[0,1]$ is recovered by "abstract nonsense" as the initial object of a certain category of (decorated) Banach spaces. So a function space, ...

**12**

votes

**0**answers

193 views

### Is there a common framework for Tannaka and Gabriel-Ulmer reconstruction theorems?

Gabriel-Ulmer duality is a biequivalence between the 2-category of finite limit categories and the 2-category of locally finitely presentable categories. It allows for the reconstruction of a theory ...

**1**

vote

**0**answers

443 views

### What is an algebraic object? [closed]

The question is in the title. In order to give a bit more backround about the question, one knows that their are several different notions of an algebraic object. One approach is that of Lavere and ...

**7**

votes

**1**answer

413 views

### Categorical Invariants

I apologize in advance if this question seems too vague.
In many topology courses, concepts like the fundamental group and homology groups are introduced as a means of distinguishing ...

**8**

votes

**6**answers

1k views

### Is the dual notion of a presheaf useful?

It seems that there is a common theme in mathematics where, if we want to find out about a category C, then we look at $\hat{C}$ (the category of contravariant functors from $C$ to $Set$). There are ...

**11**

votes

**2**answers

452 views

### What do you use categorical glueing/sconing/Freyd covers for?

In the theory of programming languages and structural proof theory, one of the handiest techniques we have available is a method called "logical relations", in which you can prove properties of ...

**32**

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 ...

**13**

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 CafĂ©) but do wonder why category theory (CT) is not opposed to model theory ...

**42**

votes

**11**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.

**4**

votes

**7**answers

1k views

### What can't be described by categories?

I've been reading some "introduction to categories" type materials and have been impressed with the all-encompassing nature, but the skeptic in me wonders: is there any mathematical object that ...