Intereting Posts

Showing a uniformity is complete.
Testing continuity of the function $f(x) = \lim\limits_{n \to \infty} \frac{x}{(2\sin{x})^{2n}+1} \ \text{for} \ x \in \mathbb{R}$
What is meant by parameter in this context?
Integration of exponential with square
Find limit of the following sequence?
What is the fastest way to find the characteristic polynomial of a matrix?
When can we recover a manifold when we attach a $2n$-cell to $S^n$?
Showing that a ring is a field as well for one of the provided choices.
Proving a trig infinite sum using integration
How to tell $i$ from $-i$?
Integral of a Gaussian process
Are primes randomly distributed?
Why is that $\widehat{\mathbb{R}/\mathbb{Z}}\cong\mathbb{Z}$?
How many books are in a library?
A maximal ideal is always a prime ideal?

**Question.** Let $M$ be a model of enough set theory. Then we can form a category $\mathbf{Set}_M$ whose objects are the elements of $M$ and whose morphisms are the functions in $M$. To what extent is $M$ determined by $\mathbf{Set}_M$ as a category up to equivalence?

For example, suppose $M$ and $N$ are models of ZF. Then are $\mathbf{Set}_M$ and $\mathbf{Set}_N$ equivalent as categories if and only if $M$ and $N$ are isomorphic?

I expect the answer will depend on exactly what we assume about $M$.

- Adjoint Functor Theorem
- Meaning of commutative diagram
- Leinster question on isomorphic functor categories
- A push-out of a pull-back
- Grp as a reflexive/coreflexive subcategory of Mon
- Colimits glue. What do limits do?

For instance, let $M$ be a model of ZFA and let $M’$ be the universe of pure sets in $M$. Then $M \cong M’$ if and only if $M$ has no atoms; but the inclusion $\mathbf{Set}_{M’} \hookrightarrow \mathbf{Set}_M$ is an equivalence as soon as $M$ satisfies the axiom “each set is in bijection with some pure set”, which happens if e.g. $M$ satisfies the axiom of choice.

On the other hand, suppose $M$ is a transitive model of ZF. By transitive closure / Mostowski collapse, every set in $M$ is obtained from a “ZF-tree” in $M$, i.e. a set $T$ (in $M$) equipped with a well-founded extensional binary relation $E$ and a unique $E$-maximal element. The notion of ZF-tree is one that can be formulated in the internal language of a topos, so the collection of ZF-trees is recoverable from $\mathbf{Set}_M$ up to equivalence, and hence, $M$ is (exactly!) recoverable from $\mathbf{Set}_M$ up to equivalence.

Following Benedikt Löwe, a somewhat more sophisticated version of the above should work to recover well-founded models $M$ of ZFA with ($M$-)countably many atoms from $\mathbf{Set}_M$.

But what about, say:

- Non-well-founded models of ZF(A)?
- Weaker fragments of ZF, e.g. Mac Lane set theory?
- Set theories where the category of sets is not a topos, e.g. NBG or NF(U)?

To keep the question from being too open-ended, let me say that I would be happy to know the answer just for (possibly non-well-founded) models of ZF.

- Unit and counit are close to being inverses
- How can universal quantifier manipulation rules be made redundant by the generalization rule (metatheorem)?
- proving tautologically equivalent
- Does the intersection of sets have a categorical interpretation?
- Why can't we use implication for the existential quantifier?
- How to recognize adjointness?
- What's the last step in this proof of the uniqueness of equalizers?
- What's a good motivating example for the concept of a slice category?
- What is the name of the logical puzzle, where one always lies and another always tells the truth?
- Injective Cogenerators in the Category of Modules over a Noetherian Ring

This is not an answer, but it’s just more convenient to write a big chunk of text here instead of in a comment. I mostly just have a few small comments (that might be flawed in one way or the other). First, $\textbf{Set}_{\mathcal M}\simeq\textbf{Set}_{\mathcal N}$ holds for any two countable models of ZFC $\mathcal M$ and $\mathcal N$. The reason for this is that for every $\alpha$ we can fix bijections $F_\alpha:\aleph_\alpha^{\mathcal M}\to\omega$ and $G_\alpha:\aleph_\alpha^{\mathcal N}\to\omega$, and then define a functor $e:\textbf{Set}_{\mathcal M}\to\textbf{Set}_{\mathcal N}$ as

$e(f:\aleph_\alpha^{\mathcal M}\to\aleph_\beta^{\mathcal M}):=F_\beta^{-1}\circ G_\beta\circ f\circ F_\alpha^{-1}\circ G_\alpha:\aleph_\alpha^{\mathcal N}\to\aleph_\beta^{\mathcal N}$,

which is both essentially surjective and fully faithful, so an equivalence.

But if either model is uncountable, then an equivalence $e$ would at least require that

- $\text{Card}^V(\kappa)=\text{Card}^V(e(\kappa))$ (since $\hom_{\mathcal M}(1,\kappa)\approx\hom_{\mathcal N}(1,e(\kappa))$)
- $\text{Card}^V((\kappa^\lambda)^{\mathcal M})=\text{Card}^V((e(\kappa)^{e(\lambda)})^{\mathcal N})$ (since $\hom_{\mathcal M}(\lambda,\kappa)\approx\hom_{\mathcal N}(e(\lambda),e(\kappa))$)

So, (1) at least requires that $|\mathcal M|=|\mathcal N|$. Also, $\textsf{Set}_V\simeq\textsf{Set}_L$ would at least imply $\textsf{GCH}$ by (2).

- Arc Length of Bézier Curves
- For an integer $n \geq 1$, verify the formula: $\sum\limits_{d|n} \mu (d) \lambda(d)= 2^{\omega (n)}$
- Differential Equations without Analytical Solutions
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- Transitive subgroup of symmetric group
- How to construct polynomial ring $K$ over commutative ring $K$ by making use of universal arrows.
- Showing that $X^2$ and $X^3$ are irreducible but not prime in $K$
- Prove that if $S$ is a finite set then $S$ has no limit points.
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- Find $v_p\left(\binom{ap}{bp}-\binom{a}{b}\right)$, where $p>a>b>1$ and $p$ odd prime.
- A combinatorial task I just can't solve
- Is the set of all numbers which divide a specific function of their prime factors, infinite?
- group presentation and the inverse elements of the generators
- What does $2^x$ really mean when $x$ is not an integer?