Questions tagged [categorical-logic]
The categorical-logic tag has no usage guidance.
18
questions
35
votes
3
answers
2k
views
Internal logic of the topos of simplicial sets
I am looking for a closed statement (i.e. not depending on any parameter objects) which is true in the internal logic of the topos of simplicial sets, but is not an intuitionistic tautology. Ideally, ...
15
votes
2
answers
1k
views
Does foundation/regularity have any categorical/structural consequences, in ZF?
(Prompted by reflection on this old answer, and its suggestion of the “harmlessness” of the axiom of regularity.)
In ZFC, one may justify the axiom of foundation (AF, aka the axiom of regularity) as ...
27
votes
5
answers
3k
views
Formalizations of the idea that something is a function of something else?
I'll state my questions upfront and attempt to motivate/explain them afterwards.
Q1: Is there a direct way of expressing the relation "$y$ is a function of $x$" inside set theory?
More ...
8
votes
2
answers
1k
views
Is there one binary operation foundational for set theory?
The membership relationship "$\in$" is foundational for set theory, in the sense that the axioms of any set theory are formulated in the language of "$\in$". Naturally, the ...
50
votes
5
answers
19k
views
Categorical foundations without set theory
Can there be a foundations of mathematics using only category theory, i.e. no set theory? More precisely, the definition of a category is a class/set of objects and a class/set of arrows, satisfying ...
34
votes
3
answers
3k
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 ...
54
votes
2
answers
9k
views
Lawvere's "Some thoughts on the future of category theory."
In Lecture Notes in Mathematics 1488, Lawvere writes the introduction to the Proceedings for a 1990 conference in Como.
In this article, Lawvere, the inventor of Toposes and Algebraic Theories, ...
34
votes
2
answers
3k
views
What can be expressed in and proved with the internal logic of a topos?
The title of this post expresses what I really want, which is to learn how to wield the internal logic of a topos more effectively. However, to bring it down to earth, I'll ask a few basic questions ...
32
votes
3
answers
7k
views
Category of categories as a foundation of mathematics
In
Lawvere, F. W., 1966, “The Category of
Categories as a Foundation for
Mathematics”, Proceedings of the
Conference on Categorical Algebra, La
Jolla, New York: Springer-Verlag,
1–21.
...
28
votes
2
answers
2k
views
What do coherent topoi have to do with completeness?
There is a theorem of Deligne in SGA4 that a "coherent" topos (e.g. one on a site where all objects are quasi-compact and quasi-separated) has enough points (i.e. isomorphisms can be detected via ...
14
votes
1
answer
1k
views
Is it possible for a theorem to be constructive only in a non-constructive metatheory?
There are several theorems in category-theoretic logic which say something like, "any proposition in X logic that is provable in topos logic assuming (the law of excluded middle and) the axiom of ...
12
votes
0
answers
407
views
What does the localic reflection of a classifying topos classify?
Let $\mathbb{T}$ be a geometric theory. Let $\mathrm{Set}[\mathbb{T}]$ be its classifying topos, such that geometric morphisms from any (cocomplete) topos $\mathcal{E}$ into $\mathrm{Set}[\mathbb{T}]$ ...
12
votes
1
answer
930
views
Model existence theorem in topos theory
One of most classical and somehow striking result in classical model theory states:
A consistent first order theory $T$ has a model.
Few considerations are needed.
This result is not true for ...
12
votes
0
answers
238
views
Intuitionistic proofs of propositional formulae versus natural transformations between finite sets
The setup: Given a formula $\varphi$ of intuitionistic propositional logic (i.e., made from the connectors $\Rightarrow$, $\land$, $\lor$, $\top$ and $\bot$ from propositional variables $A,B,C,\ldots$)...
11
votes
1
answer
689
views
Set-theoretical multiverses and their representation as functors? Why *the* multiverse?
In some related MO questions like The set-theoretic multiverse as a (bi)category it is discussed how one might represent the multiverse (see The set-theoretic multiverse) in a category theoretic way, ...
6
votes
2
answers
309
views
Images of complemented subobjects in toposes
Let ${f : E \rightarrow S}$ be a geometric morphism (between toposes).
For $s$ in $S$ and $x$ in $E$ let ${\pi : f^* s \times x \rightarrow x}$ be the obvious projection in $E$.
Let ${u \rightarrow f^*...
5
votes
2
answers
229
views
Why does the category of definable sets of $T^\mathrm{eq}$ have coproducts?
For each first-order theory $T$ there is an associated weak syntactic category, sometimes also called "the category of definable sets of $T$" and denoted $\mathrm{Def}(T)$.
Also, for each ...
3
votes
1
answer
149
views
Images of complemented subobjects in hyperconnected toposes over Boolean bases
Let $S$ be a Boolean topos.
Let ${f : E \rightarrow S}$ be a hyperconnected geometric morphism.
For $s$ in $S$ and $x$ in $E$ let ${\pi : f^* s \times x \rightarrow x}$ be the obvious projection in $E$...