1
vote
0answers
33 views

Can one talk about equivalences classes in a class(which might not be a set). [duplicate]

equivalence relation on a set breaks it into disjoint sets. does 'disjointness' make sense in classes. specifically, given a category, isomorphism classes of objects makes sense or not. can it be so ...
3
votes
2answers
85 views

Distinguishing sets according to more fine-grained notions than cardinality.

I'm interested in distinguishing sets according to more fine-grained notions than cardinality. Now I don't know a thing about computability theory, but it seems to me that considering sets up to ...
1
vote
1answer
98 views

Is there any relevance between Boolean Algebras and Fields?

In some sense Boolean Algebras and Fields have same operators and constants. In both structures there are operators addition ($+$ , $\vee$), multiplication ($\times$ , $\wedge$), inverse with respect ...
2
votes
1answer
26 views

Is the class of principal $G$-bundles over $M$ a set?

Let $G$ be a Lie group and $M$ a manifold. Question: Is the class of principal $G$-bundles over $M$ a set? This question came up when I was thinking about the classifying stack of $G$. It is the ...
1
vote
1answer
41 views

Object set of a clonal category.

I read the statement that "a clonal category has a small set of objects", which I don't quite agree about. In the definition of clonal category, at least as it is given in that context, it is required ...
1
vote
0answers
49 views

The Categories $\mathrm{Set}\times\mathrm{Set}$ and $\mathrm{Set}+\mathrm{Set}$

I'm thinking about the following problem. In $\mathrm{Cat}$ I can form the product $\mathrm{Set}\times\mathrm{Set}$. Elements are tuples, say $(A,X)$. I think that inner products and coproducts are ...
1
vote
0answers
33 views

Elementary embeddings, elementary substructures,category of sets

I would like to characterize elementary embeddings AND elementary substructures in the category of sets and functions, Set. Not only characterize, but also justify this characterization.
3
votes
1answer
81 views

What is the weakest notion of “set” that we need, so that we can say the Yoneda lemma implies something about sets?

We set up a set theory axiomatically by fixing certain statements about "$\in$". There are many different set theories. The Yoneda lemma (the theorem about functors to Set) is a early but central ...
2
votes
1answer
71 views

Size of a natural transformation and the Yoneda Lemma

Appearing on the second page (under the section Digression: Size worries) of the following PDF about the Yoneda Lemma: ...
10
votes
3answers
461 views

What is category theory?

I have read the page about category theory in wikipedia carefully, but i don't really get what this theory is. Is category theory a content in ZFC-set theory? (Just like measure theory, group theory ...
1
vote
1answer
58 views

Is there a way to phrase “there does not exist a universal set” in structural language?

Both ZFC (which is a good example of a material set theory) and ETCS (which is a good example of a structural set theory) prove the sentence "there is no set having maximum cardinality" as an easy ...
1
vote
0answers
52 views

Question about accessibility of category of free abelian groups.

I've read, that the accessibility of the category of all free abelian groups is independent on basic set theory (say ZFC). What is the reason for that? And how can I interpret this result? Does it ...
2
votes
0answers
58 views

Correct Definition of Concrete Category over Set

In the text Joy of Cats, a concrete category over $Set$ is simply a pair $\langle \mathcal C, U \rangle$ consisting of a category $\mathcal C$ and a faithful functor $U\colon \mathcal C \to Set$. But ...
1
vote
1answer
51 views

Set-like permutations of models of set theory in an arbitrary topos

Motivation: One of the most interesting (I think) semantic properties of stratified set theories like NF(U) is that the theory is preserved under every permutation that is set-like, in a sense to be ...
4
votes
1answer
221 views

Construction of *ZFC

In the following paper, page 11 (Appendix), there is a construction of a model of a theory $^*ZFC$ (see the definitions in the paper included) from a model of $ZFC$. I have been trying really hard to ...
5
votes
2answers
446 views

Is maths = set theory + logic? [closed]

It seems to me that most branches of mathematics are just an application of set theory and the rules of logic. You just definite particular types of sets and study their properties. For example, in ...
2
votes
1answer
49 views

Is there a (concrete) category of superstructures?

The superstructure $V(X)$ over a set $X$ is usually defined as follows. $V_0(X)$ $V_{i+1}(X) = V_i(X) \cup P(V_i(X))$ $V(X) = ⋃_{i=0}^∞V_i(X)$ This defines a metafunction $V.$ Remark. To make ...
-1
votes
1answer
76 views

What are morphisms in the category of sets $\mathbf{Set}$?

Do i understand correctly that morphisms in the category of sets $\mathbf{Set}$ are ordered triples $(f, A, B)$ where $f$ is a function $A\to B$? It seems that it is often claimed, even in the ...
8
votes
0answers
128 views

Cantor-Schröder-Bernstein without elements

The Cantor-Schröder-Bernstein Theorem for the category of sets has several "analogues" for other categories as well, for example measurable spaces and operator algebras (but not for arbitrary ...
4
votes
0answers
47 views

Categorical description of permutation-invariance of models

One of my projects (which I should really leave to professionals, but I'm trying anyway) is trying to find a categorical description of the permutation-invariance of models of NF, if there is such a ...
4
votes
2answers
154 views

Set theory aspects of category theory

I have never learnt axiomatic set theory, but have studied it from Munkres's Topology book first chapter. So I do not understand the difference between a class and a set except in some vague sense. ...
5
votes
1answer
232 views

How best to formalize propositions suffering from “size issues”?

Suppose we want to formalize a proposition (say, from category theory or model theory) that has "size" issues. For concreteness, lets take the following statement as a fairly typical example. ...
11
votes
1answer
111 views

How sensitive is the Yoneda lemma to set-theoretic subtleties?

In it's usual form, the Yoneda lemma cannot apply to categories that are not locally small, because, for the functor of points to have codomain $\mathsf{Set}$, the hom-sets must actually be sets. ...
4
votes
1answer
82 views

Has anyone successfully axiomatized the category of finite sets? In such a way that the resulting theory is bi-interpretable with PA.

In studies of ZFC, it is conventional to take Peano arithmetic (hereafter PA) as the metatheory. However, I don't like this convention; I think a better approach would be a metatheory (like ZF-fin ...
5
votes
1answer
85 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 ...
6
votes
0answers
87 views

On locally small category

Maybe this is a very trivial question for those who are familiar with Set theory. Let $C$ be a category, and $\hat{C}$ be its presheaves. I hear that if both $C$ and $\hat{C}$ are locally small, ...
5
votes
1answer
194 views

Foundational theories, their uses, interactions and comparisons?

Until now, I heard that there are some theories for building mathematical objects (or at least it is what it seems to my poor knowledge). Some of these are: Set theory; Logic; Category theory; Type ...
12
votes
3answers
444 views

Why is it worth spending time on type theory?

Looking around there are three candidates for "foundations of mathematics": set theory category theory type theory There is a seminal paper relating these three topics: From Sets to Types to ...
8
votes
1answer
187 views

Can we found mathematics without evaluation or membership?

In some sense, composition generalizes evaluation. The trick is, instead of writing $f(x)$ for $x$ an element of the domain of $X,$ we write $f \circ x$ for $x$ a function $1 \rightarrow X$. ...
4
votes
2answers
106 views

Question on category theory

So I have an introductory knowledge of category theory but there is one concept I can't get my head around and would like some help: When my class had categories defined we said a category ...
0
votes
1answer
55 views

Why does the empty diagram exist?

Let $\mathcal C$ be a category. Why does the Functor $\mathcal F:\emptyset \to \mathcal C$ exist? In general $Ob(\mathcal C)$ is not a set, so $\mathcal F:\emptyset \to Ob(\mathcal C)$ is not a ...
6
votes
1answer
165 views

Is it possible to formalize a “universe” of categories as a one-sorted first-order theory with one binary relation and no functions?

This is a modification of a question I asked earlier. In that question, I hadn't placed any limits on the number of binary relations allowed, so my question had an affirmative answer, but a trivial ...
6
votes
1answer
213 views

Is it possible to formalize (higher) category theory as a one-sorted theory, just like we did with set theory?

Set theory is typically formalized as a one-sorted theory without urelements. Is it possible to do the same with category theory or higher category theory, formalizing the whole affair as a theory ...
9
votes
4answers
325 views

What precisely is lost when considering proper classes rather than sets?

Motivated by concerns over the foundational issues vis-a-vis category theory. What is the essential useful characteristic of sets that is lost when instead considering proper classes? Referring to ...
3
votes
2answers
248 views

Large categories? How is a set of objects/arrows not a set?

Good night, I found there are categories whose objects and arrows aren't sets, called large categories. I understand if each object isn't a set, but the set of all objects should be a set, even if ...
2
votes
1answer
189 views

Modern reference on logic-set theory-foundation

I'm looking for a modern book on logic-set theory-foundation written as the Bourbaki's set theory. I'm particularly interested in a formal exposition of ZFC axiom with logic-set Grothendieck universe. ...
7
votes
0answers
159 views

What does category theory say about chains of set inclusions $\dots\in c\in b\in a$?

I'm currently trying to understand to what extend a set theory like ZFC can be mirrored within category theory, i.e. topos theory. What appears as an obstacle to me is the axiom of regularity, which ...
11
votes
4answers
520 views

“universe” in set theory and category theory

In all applications of the theory of sets, all sets under investigation take place in the context of the universal set $U$. What exactly is the purpose of the universal set in set theory? $U$ is ...
4
votes
1answer
89 views

Presheaves over sieves and posets

I'm looking for a proof of the claim that given a poset P, the topos of presheaves over P is equivalent to the topos of presheaves over the complete Heyting algebra of sieves on elements of P. I found ...
16
votes
1answer
393 views

On the large cardinals foundations of categories

(This was cross-posted to MathOverflow.) It is well-known that there are difficulties in developing basic category theory within the confines of $\sf ZFC$. One can overcome these problems when ...
2
votes
1answer
95 views

Defining Test-Objects

In various categories one encounters what are referred to as 'test-objects'. For example, singleton set serves as test-object (in testing the equality of functions) in the category of sets. In the ...
1
vote
1answer
78 views

A question on the category of sets

Why do SET, the category of sets and functions, is a locally small category? In other words, why do the collection of functions among two fixed sets is a set, and not a proper class?
11
votes
1answer
282 views

What of the “Sets, Classes, and Categories” approach to the foundations?

A 2001 paper by F.A. Muller proposes "ARC" (a modified form of Ackermann set theory) as a foundations of mathematics, and argues that it founds category theory more naturally and/or conveniently than ...
2
votes
3answers
207 views

Beyond Goedel incompleteness and lack of soundness/completeness of higher-order logics

As I understand that there are at least two fundamental limits of the development of the mathematics: 1) Goedel incompeleteness theorems (or more clearly Church thesis) effectively says that there ...
10
votes
1answer
205 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 ...
1
vote
1answer
90 views

If we have an equivalence relation on a class, is it possible to define what it means for the collection of equivalence “to be a set”?

My background in set theory is that of a casual acquaintance that I would like to know become friends with (I am not sure set theory feels the same way). For my question, I would like to stay within ...
1
vote
1answer
168 views

Inverse of power-set functor?

In his answer to this Q: How to interpret $1 \to 0$ in $\mathbf {Set^{op}}$, and $\mathbf {Set^{op}}$ itself? Zhen Lin proposed that $\mathbf {Set^{op}}$ is naturally equivalent to the category of ...
0
votes
1answer
79 views

Sections, Transversals and Quotient Maps

Here I read: Given a quotient space $\bar X$ with quotient map $\pi\colon X \to \bar X$, a section of $\pi $ is called a transversal. I asked myself how such sections are possible. It must be a ...
6
votes
1answer
106 views

How we can understand one category is small

"A category is said to be small if its objects form a set." Now one question is in my mind and that is although we know lots of sets and always working with them, but how we can show a class of ...
7
votes
4answers
222 views

Categorial definition of subsets

I cannot see why this definition http://en.wikipedia.org/wiki/Subobject is equivalent to subsets in the category of sets. I am confused by the following facts, 1) the partial order is defined ...