8
votes
1answer
81 views

Zorn's lemma in categorical language

The axiom of choice, which is equivalent to Zorn's Lemmma, has a nice categorical "translation": in the category of sets, every epi is a retraction. So the axiom of choice says something about the ...
5
votes
1answer
82 views

Does the existence of products in the category of sets imply the Axiom of Choice?

If for every family $(X_i)_{i\in I}$ of sets, there exists a categorical product in the category $\mathbf{Set}$ of sets, does this imply that the set-theoretic construction $\left\{(x_i)_{i\in ...
1
vote
0answers
43 views

Is there a connection between contravariant functors and the axiom of choice?

Given that both can be seen as talking about reversing arrows between two objects: Is there a connection between contravariant functors and the axiom of choice? I'm initially motivated geometrical ...
6
votes
1answer
65 views

Under what choice assumptions is there a monoid structure on every set?

The question arose when discussing possible cardinalities of hom-sets of whether it's any weaker than the axiom of choice that there exists a monoid of every cardinality. It's well known, or at least ...
6
votes
2answers
126 views

Axiom of Choice-esque argument to show that a proof of a statement exists without actually giving a proof

What if the set of all well-formed statements in ZFC formed a kind of pseudo-category where a morphism f between objects A, B represented a formal proof that A implied B? What if that category could ...
10
votes
1answer
200 views

Stone's Representation Theorem and The Compactness Theorem

If you're working on $\mathsf {ZF}$ and you assume the compactness theorem for propositional logic, then you have the prime ideal theorem, and thus you can show that the dual of the category of ...
5
votes
4answers
229 views

Is the powerset of every Dedekind-finite set Dedekind-finite?

Is the powerset of every Dedekind-finite set Dedekind-finite? I think this statement can be written in $\textbf{Set}$: If every mono (=injection) $f: A \to A$ is iso (=bijection), then every mono ...
2
votes
1answer
183 views

The axiom of choice and connected groupoids

Recall the two definitions of equivalence of categories: Definition 1. An equivalence of categories is a quadruplet $(F, G, \alpha, \beta)$, where $F : \mathbb{C} \to \mathbb{D}$ and $G : \mathbb{D} ...
2
votes
1answer
546 views

When is the pullback of a linear injection a surjection on dual space?

Due to the contravariance of the dual space functor on vector spaces, one might expect the pullback of an injection to be a surjection, and the pullback of a surjection to be an injection. Indeed, for ...
10
votes
5answers
478 views

Natural isomorphisms and the axiom of choice

The definitions of "natural transformation", "natural isomorphism between functors", and "natural isomorphism between objects" captures - among other things - the intuitive notion of "an isomorphism ...