1
vote
3answers
381 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
2answers
284 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
0answers
148 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
0answers
160 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
0answers
337 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 ...
17
votes
0answers
468 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
0answers
195 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
1answer
150 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
3answers
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. ...
18
votes
4answers
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
2answers
686 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
4answers
549 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
2answers
556 views

Category and the axiom of choice

What are (if any) equivalent forms of AC (The Axiom of Choice) in Category Theory ?
12
votes
7answers
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 ...
13
votes
0answers
344 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
1answer
503 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
2answers
668 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
3answers
496 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. ...
11
votes
0answers
360 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
2answers
216 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
1answer
815 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
3answers
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
4answers
853 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
2answers
473 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
2answers
321 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
1answer
170 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
1answer
150 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
1answer
222 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
2answers
402 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 ...
7
votes
0answers
222 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
0answers
258 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
5answers
614 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
2answers
466 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
2answers
754 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
1answer
501 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
0answers
291 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
3answers
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
3answers
924 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
1answer
391 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
1answer
487 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
2answers
488 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
1answer
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 ...
9
votes
1answer
488 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
2answers
477 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
1answer
514 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
3answers
702 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
1answer
195 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
1answer
336 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 ...
4
votes
6answers
1k views

Independence and Category Theory

I'm not very experienced with respect to Category Theory. So if this question makes no sense I'm sorry. At any rate here is my question: If the existence or non-existence of specific sets can be ...
9
votes
4answers
1k views

Axiom of Replacement in Category Theory

Does the usual development of category theory (within Goedel-Bernays set theory, for example) require the axiom of replacement? I would have asserted that this was obviously true, but it seems to be ...