# Tagged Questions

**4**

votes

**2**answers

212 views

### Maximum cardinality of a filtered limit of finite sets

Let $(I,<)$ be a directed, partially ordered set. Consider an inverse system $(S_i)_{i \in I}$ of finite sets, i. e. a functor $S:I^{op}\to \mathbf{FinSet}$. What is the maximum possible ...

**5**

votes

**1**answer

297 views

### A categorical characterization of ordinal numbers

It's rather easy to notice that the operation of join of categories reproduces the ordinal sum once restricted to act on (iso classes of) well-ordered set; it's rather easy to see that $\alpha\star ...

**1**

vote

**3**answers

415 views

### Another adjoint pair: Definable sets and set-builder formulas

I see adjointness between the two concepts of "being a definable set" and "being a set-builder formula":
A set $X$ is definable when there is a formula $\phi(x)$ such that $X = \lbrace x : ...

**4**

votes

**2**answers

297 views

### Functor category's objects fail to be a class?

Given two categories, we can form the functor category whose objects are functors. Functors by definition consist of two mappings from in general classes to classes, which makes it fail to be a set. ...

**1**

vote

**0**answers

149 views

### A question on definable categories

One way to define a category set-theoretically might be to give four $\in$-formulas (not sets!)
$$\begin{array}{rl}
\mathsf{O}(X)&\text{(“$X$ is an object”)}\\
\mathsf{M}(X,Y,z)&\text{(“$z$ ...

**3**

votes

**0**answers

168 views

### Can one construct Freyd-Mitchell's embeddings that respect sheafifications?

For a certain presheaf $P$ with values in an abelian category $A$ satisfying AB5 and its sheafification $S$ (with respect to a small Grothendieck site) I would like to prove: $S(f):S(X)\to S(Y)$ is ...

**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 ...

**21**

votes

**0**answers

619 views

### categorical characterization of large cardinals

Question 1. Is there a categorical representation of Kunen's inconsistency result?
Question 2. Is there a categorical characterization of very large cardinals (in particular for strong and ...

**5**

votes

**0**answers

206 views

### Orthogonality relations and accessibility?

Suppose I have a pair of locally presentable categories connected by an accessible functor. Then the preimage of an accessible subcategory of the codomain is an accessible subcategory of the domain. ...

**1**

vote

**1**answer

151 views

### How would you say that a small category is embedded into functors from a large $C'$ to abelian groups?

How would you say that a small additive category $C$ embedds (contravariantly) into the category of exact functors from a 'large' abelian $C'$ into abelian groups (this is something like Yoneda's ...

**10**

votes

**3**answers

2k views

### A “mother of all groups”? What kind of structures have “mother of all”s?

For ordered fields, we have a “mother of all ordered fields”, the surreal numbers $\mathbf{No}$, a proper-class “field” which includes (an isomorphic copy of) every other ordered field as a subfield. ...

**20**

votes

**4**answers

1k views

### When must it be sets rather than proper classes, or vice-versa, outside of foundational mathematics?

Every once in a blue moon it actually matters that some mathematical entity which might a priori only be a class is in fact a set. For clarification, here are some examples of what I do not ...

**4**

votes

**2**answers

790 views

### What kinds of operations are well-defined when working with sets, classes, conglomerates, and yet higher order collections?

There are many foundations to set theory, ZFC, NBG, SEAR, to name a few, and while they differ in how sets, classes, and higher-order collections are represented as mathematical objects, they all ...

**7**

votes

**4**answers

597 views

### Grothendieck topology for a non-small category

To define a Grothendieck topology of a category, we usually require that the category is small.
Question 1: Why do we need to require the category to be small?
I thought that the problem was that ...

**7**

votes

**2**answers

596 views

### Category and the axiom of choice

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

**12**

votes

**7**answers

2k views

### Usage of set theory in undergraduate studies

I would like to ask my colleagues their thought on good practices concerning
set theorical framework in undergraduate studies. For example, have there been any attempt to use another mathematical ...

**14**

votes

**0**answers

351 views

### Relative consistency of ETCS over the theory of a well-pointed topos with NNO

Gödel's well-known proof of the implication $Con(ZF) \Rightarrow Con(ZFC)$ used the construction of the inner model $L$ in $ZF$ to get a model of $ZFC$ (and in fact much more). However such a ...

**9**

votes

**1**answer

508 views

### Finite order arithmetic and ETCS

I'm looking for a reference to the statement that Lawvere's Elementary Theory of the Category of Sets (ETCS) is equal in proof-theoretic strength to finite order arithmetic. The person who informed ...

**8**

votes

**2**answers

716 views

### Are grothendieck universes enough for the foundations of category theory?

Grothendieck universes are equivalent to ZFC+a strongly inaccessible cardinal. This is low on the large cardinal axiom list. Is it enough to place category theory on a firm foundational basis, and how ...

**0**

votes

**3**answers

521 views

### Sets = structured sets without structure

Motivation
There is presumably no single and widely accepted formal definition of structured sets = sets plus structure based on sets as primitive objects, but several approaches are around. See e.g. ...

**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 ...

**2**

votes

**2**answers

217 views

### Small categories and completeness

(1) Can a small category be cocomplete? Meaning, have all small colimits? I'd be glad to see an example.
(2) Suppose $\mathcal C$ is a small category, with $Ob(\mathcal C)$ being of cardinality ...

**10**

votes

**1**answer

843 views

### How much do universes matter in topos theory?

Suppose we fix two Grothendieck universes $\mathcal{U} \in \mathcal{V}.$ Then one has that $\mathcal{U}$-$\mathbf{Set},$ the category of $\mathcal{U}$-small sets, is a locally $\mathcal{U}$-small, ...

**6**

votes

**3**answers

1k views

### “Axiom of global choice”

In some books on category theory (for example, in J.Adámek, H.Herrlich, E.Strecker "Abstract and concrete categories...") the authors use the idea of "big sets" ("conglomerates" or "collections") ...

**18**

votes

**4**answers

872 views

### There are two slightly different notions of ultraproduct. Why is one said to be better than the other?

Let $I$ be a set and $\mathcal{U}$ an ultrafilter on $I$. Let $(X_i)_{i \in I}$ be an $I$-indexed family of sets. The ultraproduct of the family $(X_i)$ with respect to $\mathcal{U}$ is, everyone ...

**5**

votes

**2**answers

494 views

### Is the Mostowski collapse natural?

The Mostowski collapse lemma (see here for a quick ref) is one of the key basic tools in the set-theory arsenal. I wonder if the collapse is natural, in the functorial sense.
More precisely, is ...

**3**

votes

**2**answers

324 views

### The category of Boolean-valued models associated to a model of ZFC

This is related to my previous question here, and indirectly motivated by Andreas Blass intriguing answer therein:
start from a ZF transitive model $M$, and consider the category $CBA(M)$ of ...

**1**

vote

**1**answer

171 views

### Terminology: changing the codomain in nested maps (e.g. in multilinear algebra)

The context of this question is given below but I don't think it is of essence here, so I will try to formulate the question for maps between sets.
Given two sets $A$ and $B$, denote the set of all ...

**0**

votes

**1**answer

154 views

### Indicating Dots in Graphs [closed]

Dear All,
I’d appreciate very much if you could address the following question:
Given two composable functions [domain (one) = codomain (other)]: the unique function ‘i’ with empty set E as domain ...

**1**

vote

**1**answer

225 views

### Does $\bf pSet$ admit products?

The question is in the title. The category $\bf pSet$ of partial functions has sets as objects and $\hom(X,Y)$ is the set of all triples $(X,Y,f)$ such that there exists $D\subseteq X$ and $f\colon ...

**8**

votes

**2**answers

408 views

### Does the class category of ZF-algebras satisfy the Multiverse axioms?

I read with interest both Hamkins Multiverse Axioms and Joyal and Moerdijk's algebraic set theory. Both of these perspectives takes a set-theory universe as an object, and consider collections of ...

**8**

votes

**0**answers

230 views

### What about replacing $\{0,1\}$ in Stone duality with another finite set?

Basically Stone duality or more general the duality between spatial locales and sober spaces is about enriching the set of morphisms $X \to \{0,1\}$ with an additional structure and then finding ...

**6**

votes

**0**answers

262 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 ...

**4**

votes

**5**answers

635 views

### Union of a object (a set) in the Elementary Theory of the Category of Sets

I see the Todd Trimble article "Elementary Theory of the Category of Sets" on catlab.
I ask: how make (in the categorical setting) the usual union of a set $\cup X=${$y |\exists x\in X: y\in x$}?
...

**5**

votes

**2**answers

477 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 ...

**0**

votes

**2**answers

765 views

### Structural definition of “product” in set theory

At first sight there is no abstract (= structural) definition of "product" in set theory. E.g. the Cartesian product of sets $A$ and $B$ is defined as the set of all ordered pairs $(x,y)$, $x \in A$, ...

**7**

votes

**1**answer

510 views

### What is a category of sets?

One knows that many models of set theory exist. In topos theory,"the" category of sets is to play the role of the point. Since many models of set theory are around, I believe one of the following to ...

**6**

votes

**0**answers

296 views

### Maps between forcing posets

We all know that forcing can be seen (if you like things that way) as a category of sheaves over the poset of forcing conditions equipped with the double negation Grothendieck topology. As such it is ...

**29**

votes

**3**answers

2k views

### The set-theoretic multiverse as a (bi)category

Joel Hamkin's The set-theoretic multiverse has featured in MO questions before, e.g., here and here. But I was wondering about the best category theoretic angle to take on it.
In the paper Joel ...

**0**

votes

**3**answers

945 views

### How different category theories relate

Continuing about this my question.
Mac Lane "Categories for the Working Mathematician" and "Abstract and Concrete Categories. The Joy of Cats" use different set theory foundations.
How one to ...

**1**

vote

**1**answer

397 views

### A Dedekind (pseudo) finite set

Quoting the wiki:- a set A is Dedekind-infinite if some proper subset B of A is equinumerous to A. Explicitly, this means that there is a bijective function from A onto some proper subset B of A. -.
...

**3**

votes

**1**answer

490 views

### Universal Objects in Big Categories

Let $\mathcal{A}$ be a category whose collection of object forms a proper class. Then, to be able to formulate the concept of a terminal object in $\mathcal{A}$, do we have to leave ZFC? In other ...

**1**

vote

**2**answers

490 views

### Functor category

Let $\mathcal{C}$ and $\mathcal{D}$ be categories, where $\mathcal{C}$ is an abelian category. We want to say that $\mathcal{C}^\mathcal{D}$ is also an abelian category. However, if $\mathcal{C}$ and ...

**19**

votes

**1**answer

1k views

### Can ZFC → NBG be iterated?

von Neumann-Bernays-Gödel set theory (NBG) is a conservative extension of ZFC which contains "classes" (such as the class of all sets) as basic objects. "Conservative" means that anything provable in ...

**11**

votes

**1**answer

511 views

### What's an example of a locally presentable category “in nature” that's not $\aleph_0$-locally presentable?

Recall the notion of locally presentable category (nLab): $\DeclareMathOperator{\Hom}{Hom}$
Definition: Fix a regular cardinal $\kappa$; a set is $\kappa$-small if its cardinality is strictly less ...

**2**

votes

**2**answers

479 views

### “classes” with no cardinality; “classes” with no equality notion

Hello,
If we look at the class of all vector spaces over some field, we can note two things:
1) this class should not have cardinality.
2) for two elements of this class, we should not want to be ...

**11**

votes

**1**answer

526 views

### Can Vopenka's principle be violated definably?

One form of Vopenka's principle (a large cardinal axiom) states that no locally presentable category contains a full subcategory which is large (= a proper class) and discrete (= contains no ...

**3**

votes

**3**answers

705 views

### Unions of sets exist? [closed]

Hello,
Probably this questions is very stupid, but anyway: It usually said that the category of sets is cocomplete, in particular meaning that we have disjoint unions of arbitrary families of sets, ...

**2**

votes

**1**answer

197 views

### Can we define geometric morphisms (between ETCS categories) elementarily?

The ETCS axioms give conditions on a category for it to be a category of sets. These axioms can be written out in first order language, resulting in a finite axiomatisation of the category of sets. ...

**6**

votes

**1**answer

337 views

### How do we compare models of ETCS?

The elementary theory of the category of sets (nLab) gives axioms on a category such that it is a category of sets. In answering this MO question I realised that we might have trouble comparing ...