# Tagged Questions

**12**

votes

**0**answers

344 views

### How much choice is required to prove concretizability theorems in category theory?

A concretization of a category is a faithful functor to the category of sets. A category is concretizable if there exists such a functor.
An evident necessary condition for concretizability is ...

**10**

votes

**1**answer

207 views

### Pullback-stability of internally projective objects

An object $X$ of a category $C$ is said to be projective if the hom-functor $C(X,-)$ preserves epimorphisms (or, in general, some restricted class of epimorphisms such as the regular or effective ...

**7**

votes

**2**answers

591 views

### Category and the axiom of choice

What are (if any) equivalent forms of AC (The Axiom of Choice) in Category Theory ?

**12**

votes

**0**answers

376 views

### Categorifications of Zorn's lemma

I'm wondering about categorifications of Zorn's lemma along the following lines.
Lemma: if $\mathbf{C}$ is a small category in which every directed diagram of monomorphisms has a cocone of ...

**1**

vote

**1**answer

353 views

### Counterexemple to Urysohn's lemma in a topos without denombrable choice ?

Hello !
The Urysohn's Lemma assert that in every topological spaces which is normal two closed subset may be separated by a real valued function. It's proof use axiom of countable choice (but not the ...

**6**

votes

**0**answers

261 views

### What are these sets in Freyd's model?

Recall Freyd's model of $ZF +\neg AC$ (as recounted in MacLane and Moerdijk's book Sheaves in Geometry and Logic): it arises as the Fourman interpretation of the topos of sheaves on a particular ...

**5**

votes

**2**answers

481 views

### Example of a topos that violates countable choice

At this nLab page we have the line
In contrast, any topos that violates countable choice, of which there are plenty, must also violate internal COSHEP.
It doesn't give an example, and neither ...

**5**

votes

**2**answers

476 views

### Forcing the nonexistence of a certain set

I have a certain set-theoretic axiom (WISC) which follows from Choice (this is a nuking a fly BTW), but which I suspect is independent of ZF. To show this I need to show that a certain set does not ...

**3**

votes

**1**answer

349 views

### Existence of enough projectives in the category of sets

I am talking about the principle that says that every set is the image of a projective set. For every set $x$ there is a surjection $f:y \twoheadrightarrow x$, such that for any set $u$ and function ...

**2**

votes

**6**answers

967 views

### Splitting lemma under assumption of the axiom of choice

The splitting lemma says:
Given a short exact sequence with maps $q$ and $r$:
$0 \rightarrow A \overset{q}{\rightarrow} B \overset{r}{\rightarrow} C \rightarrow 0$
then the following are ...

**10**

votes

**5**answers

854 views

### Where are some interesting places where the axiom of choice crops up in category theory?

The two that come to mind are splitting epics in Set and taking the Skel of a category. Surely there are lots of other interesting (and maybe upsetting) places where this comes up.