3
votes
1answer
74 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 ...
6
votes
0answers
136 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 ...
6
votes
2answers
114 views

Is there a way to axiomatize the category of sets and relations?

The system of axioms known as ETCS axiomatizes the category of sets and functions. Does anyone know of a way to axiomatize the category (and/or allegory) of sets and relations?
0
votes
1answer
57 views

Is there a systematic account of the number systems in the following 3x3 grid?

Consider the following sets of numbers, viewed as number systems with signature $(+,\times,\leq)$. Let $\mathbb{X} = \{1,2,3,\cdots\}$ denote the nonzero natural numbers. Let the completion of ...
8
votes
2answers
436 views

Relationship between Category theory and Axiomatic set theory

I've recently started learning Category theory- and I have a pondering- wondering if anyone can help. Is it possible for two categories to satisfy two different set-axiom system. Namely- is it ...