All Questions
Tagged with axiom-of-choice ct.category-theory
20
questions
10
votes
1
answer
363
views
Hereditarily countable sets in Antifounded ZF
A set $x$ is hereditarily countable when every membership-descendant of $x$ (including $x$ itself) is countable.
In this paper, Jech proved in ZF that the class of all hereditarily countable sets is a ...
- 1,382
9
votes
0
answers
134
views
Locally presentable and accessible categories without the axiom of choice?
Is there a good reference for the study of locally presentable and accessible categories without the axiom of choice? For instance, it seems one will need to understand:
What is a good notion of $\...
- 62.6k
6
votes
0
answers
174
views
Does a well-pointed topos with enough projectives satisfy the internal axiom of chioice?
If yes, then I am also wondering if being well-pointed can be weakened to boolean (i.e. this is in the context of using Set as our metalogic so that well pointed Topoi are boolean). If not, then any ...
- 1,843
2
votes
1
answer
316
views
Global choice and skeletons of large categories
It is stated on the nlab that the axiom of choice is equivalent to the statement that all small categories have a weak skeleton, meaning a skeletal category which is equivalent to them.
Is the axiom ...
- 9,107
5
votes
1
answer
494
views
Failure of SVC in Grothendieck toposes
The axiom SVC (for "small violations of choice") asserts that there is a set $S$ such that for every set $X$ there is a choice set $A$ such that $X$ is a subquotient of (i.e. admits a surjection from ...
- 65.8k
1
vote
0
answers
113
views
Selection in a small category
I came across the following problem. I do not know if these notions are known (I would actually be interested to know), so the names might not be the canonical ones.
Given a small category $C$, a ...
- 121
11
votes
2
answers
650
views
Non smallness of the set of anafunctors without AC?
Trying to construct a model category constructively is difficult. One often mention the fact that without the axiom of choice one cannot prove that the localization of the category of small categories ...
- 40.8k
9
votes
2
answers
485
views
Constructively, are all fibrations cloven?
A "cloven fibration" is a fibration for which we have an explicit choice of cartesian liftings; this is often phrased as, "We can pick a lifting without using the axiom of choice".
Firstly, I'm a bit ...
5
votes
0
answers
257
views
On the Axiom of Choice for Conglomerates and Skeletons
Say that $\mathcal{X}$ is a conglomerate if $\mathcal{X} = \{X_i: i \in I\}$, where each $X_i$ and $I$ are classes. The Axiom of Choice for Conglomerates is the statement: Whenever $\mathcal{X}$ and $\...
- 149
13
votes
0
answers
499
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 ...
- 2,450
14
votes
1
answer
570
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 ones)...
- 65.8k
11
votes
2
answers
2k
views
Category and the axiom of choice
What are (if any) equivalent forms of AC (The Axiom of Choice) in Category Theory ?
user30818
17
votes
1
answer
874
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 ...
- 3,909
3
votes
1
answer
845
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 ...
- 40.8k
6
votes
0
answers
299
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 ...
- 34.8k