# Tagged Questions

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### Absolute coequalizers in $\mathbf {Set}$

Let $A$ be a set and let $R\subseteq A\times A$ be an equivalence relation on $A$. Denote by $p, q$ the projections $R\longrightarrow A$ on the first and second factor, respectively. The ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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.
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### 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 ...
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### 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: ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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. ...
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### 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. ...
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### 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 ...