Tagged Questions
4
votes
0answers
58 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 ...
8
votes
1answer
138 views
Stone's Representation Theorem and The Compactness Theorem
If you're working on $ZF$ 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 Boolean algebras ...
3
votes
4answers
168 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
167 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
364 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 ...
9
votes
5answers
438 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 ...
