Questions tagged [categorical-logic]

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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, ...
David Spivak's user avatar
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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 ...
user avatar
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, ...
Mike Shulman's user avatar
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 ...
David Spivak's user avatar
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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 ...
David Corfield's user avatar
33 votes
1 answer
743 views

Proof assistant for working in weaker foundations?

In some of my works I need to prove some results within the internal logic of categories with not much structures (like pretoposes or even just categories with finite limits). The kind of things I ...
Simon Henry's user avatar
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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. ...
Marc Nieper-Wißkirchen's user avatar
29 votes
2 answers
1k views

Applications of Categorical Logic to Logic

This is definitely a very open ended question. I have been studying Categorical Logic for a while now --- I've read Sheaves in Geometry and Logic, Adámek & Rosický's Presentable Categories, ...
DeadRingerAmbassador's user avatar
28 votes
5 answers
3k views

How do we construct the Gödel’s sentence in Martin-Löf type theory?

In Martin-Löf dependent type theory (MLTT), under the proposition-as-types correspondence, we sometimes say that a proposition $A$ is true if the type $A$ is inhabited. However, there is no doubt that ...
StudentType's user avatar
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 ...
Akhil Mathew's user avatar
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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 ...
Michael Bächtold's user avatar
25 votes
1 answer
2k views

A geometric theory of Blueprints? (Algebras over the field with one element)

In my attempt to tackle the various approaches of defining algebraic geometry over $\mathbb F_1$, I was just reading through Lorscheid's paper The geometry of blueprints. I certainly like the idea a ...
Georg Lehner's user avatar
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24 votes
2 answers
1k views

Precise relationship between elementary and Grothendieck toposes?

Elementary toposes form an elementary class in that they are axiomatizable by (finitary) first-order sentences in the "language of categories" (consisting of a sort for objects, a sort for morphisms, ...
user avatar
24 votes
1 answer
1k views

On Joyal's completeness theorem for first order logic

In 1978, in a series of unpublished conferences in Montréal, A. Joyal announced a remarkable theorem that unified several completeness theorems for fragments of first order logic, as well as first ...
godelian's user avatar
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23 votes
1 answer
4k views

What is the most transparent, rigorous definition of the Univalence Axiom?

I've been studying homotopy type theory and trying to grasp the Univalence Axiom. I have yet to find a concise, accessible, rigorous definition of Univalence. I have several excellent survey papers ...
antianticamper's user avatar
20 votes
4 answers
4k views

Is there a categorical proof of Gödel's incompleteness theorem?

A significant result in set theory was shown by Cohen when he showed that the continuum hypothesis was independent of ZFC using a new technique called forcing. In Topos theory, this result has a new ...
Mozibur Ullah's user avatar
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 ...
Peter LeFanu Lumsdaine's user avatar
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 ...
Zhen Lin's user avatar
  • 14.9k
14 votes
0 answers
339 views

Comparing algebraic and analytic spaces through the universal property of classifying topoi

$\newcommand\kAlg{k\mathrm{Alg}}\DeclareMathOperator\Zar{Zar}\newcommand\Mnf{\mathrm{Mnf}}$I apologize beforehand if my question is naïve. I must admit that I do not know much about analytic/smooth ...
Nico's user avatar
  • 775
14 votes
0 answers
475 views

Constructing a topos from a Heyting algebra

It is true that, given any topos $\mathcal{C}$ with a terminal object $1_\mathcal{C}$, $Sub(1_\mathcal{C})$ is a Heyting algebra. Now suppose that we start with a Heyting algebra $H$. Is it always ...
user102845's user avatar
13 votes
4 answers
6k views

Au revoir, law of excluded middle?

In what way and with what utility is the law of excluded middle usually disposed of in intuitionistic type theory and its descendants? I am thinking here of topos theory and its ilk, namely synthetic ...
lambdafunctor's user avatar
13 votes
4 answers
1k views

Two interpretations of implication in categorical logic?

I am a bit confused about the interpretation of "implication" in the standard treatment of categorical logic, for example in [Bart Jacobs 1999] "Categorical Logic and Type Theory". ...
YKY's user avatar
  • 508
13 votes
2 answers
749 views

Brouwer's Theorem in the free topos?

In Introduction to Higher-Order Categorical Logic, Lambek & Scott remark that Brouwer's Theorem (all functions $\mathbb{R}\to\mathbb{R}$ are continuous) holds in the free topos $\mathcal{T}$. ...
Jonathan Sterling's user avatar
13 votes
1 answer
543 views

Ultracategories with one object

Historically, the theory of ultracategories was invented by Makkai to prove a strong conceptual completeness theorem for first-order logic, roughly: if $T$ and $S$ are two first-order theories such ...
user480841's user avatar
13 votes
0 answers
239 views

Birkhoff's HSP theorem in categories other than $\mathbf{Set}$

Fix a category $C$ with finite products and a set $L$ of function symbols (each equipped with an arity in $\mathbb N$). An $L$-algebra in $C$, $\mathbf A=(A,(f^\mathbf{A})_{f\in L})$, is given by some ...
user176332's user avatar
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 ...
Ivan Di Liberti's user avatar
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 $A,B,C,\ldots$)...
Gro-Tsen's user avatar
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12 votes
0 answers
406 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 $\mathrm{Set}[\mathbb{T}]$ ...
Ingo Blechschmidt's user avatar
11 votes
3 answers
879 views

"Spatial (geometrical)" realization of Elementary topos?

It is well known that (Grothendieck) Topos (in fact, Model topos too) has many good geometrical properties. In many senses reflects general forms of generic geometry. Note: Grothendieck view of Topos ...
tttbase's user avatar
  • 1,700
11 votes
2 answers
376 views

Equivalence between geometric theories and frames internal to the free topos

What is a reference for "the equivalence between geometric theories and frames internal to the free topos"? [1] This seems to be an extremely interesting theorem. [1] André Joyal, “A crash ...
user1022117's user avatar
11 votes
1 answer
418 views

Are flat functors out of a finite category necessarily finite?

Note: I've originally asked this question on math stack exchange, but I have learnt that this is the better place to ask for research level questions, so I have deleted the original question there. ...
Lingyuan Ye's user avatar
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, ...
FWE's user avatar
  • 213
11 votes
1 answer
419 views

Grothendieck toposes in (very) weak foundation

There is on the nLab page "Grothendieck topos" a part about the theory of Grothendieck toposes in weak foundation. It claims that the equivalence for a category between the Giraud's axioms and being ...
Simon Henry's user avatar
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10 votes
1 answer
519 views

Which algebraic theories are co-sites?

Given a category $C$, I'll say that a set $J$ of families $\{f_i\colon A\to B_i\mid i\in I\}\;$ is a co-coverage if their opposites $\{f_i^{op}\colon B_i\to A\mid i\in I\}\;$ form a coverage on $C^{op}...
David Spivak's user avatar
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10 votes
2 answers
772 views

Category of Judgements?

I have been able to find a lot of information on the category of contexts -- for example, the page on syntactic categories at the nLab is a good starting point. However, when I try to find similar ...
Jacques Carette's user avatar
10 votes
1 answer
383 views

Examples of Kreisel-Putnam topological spaces

Let us say that a topological space $X$ is a Kreisel-Putnam space when it satisfies the following property: For all open sets $V_1, V_2$ and regular open set $W$ of $X$, if a point $x\in X$ has a ...
Gro-Tsen's user avatar
  • 29.9k
10 votes
0 answers
384 views

How do properties of a partial order $\mathbb{P}$ affect the logic of the functor category $\mathsf{Set}^\mathbb{P}$?

$\DeclareMathOperator\true{\mathsf{true}}$I am very suspicious the answer to this (family of) question(s) is well-known, but I couldn't find anything after a bit of searching so I'll ask anyway. I am ...
Neil Barton's user avatar
10 votes
0 answers
372 views

Internal logic in topos theory, monoidal categories, and quantum mechanics

To obtain the internal logic of a topos (roughly speaking), we associate a type of free variable with an object, and a statement about such a variable with a subobject of that object. Intuitively, the ...
Neuromath's user avatar
  • 397
9 votes
1 answer
431 views

Free models of finitely presented essentially algebraic theories in elementary toposes?

The following result is well-known in folklore (I think), but I’ve been unable to find a reference in the literature: Let $T$ be a finitely presented essentially algebraic theory, and $\newcommand{\...
Peter LeFanu Lumsdaine's user avatar
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 ...
Ioachim Drugus's user avatar
8 votes
2 answers
1k views

Grothendieck toposes and logic

I am searching results in which one can extract logic information from a topological (Grothendieck topos) perspective (such as Gödel's Completeness Theorem and Deligne's Theorem ("theorem by P. ...
tttbase's user avatar
  • 1,700
7 votes
1 answer
406 views

The idempotence of Mike Shulman's "Stack semantics"

I am struggling to understand lemma 7.20 of the paper Stack Semantics and the Comparison of Material and Structural Set Theories by Mike Shulman (arXiv:1004.3802). It contains formal sequents of the ...
Nico's user avatar
  • 775
6 votes
3 answers
287 views

Validity of equations in a topos

To simplify consider simple algebraic theories (universal algebra) A and L, but the question applies to geometric theories. 1) Syntactically, we can interpret L in A if we can define the operations ...
Eduardo J. Dubuc's user avatar
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^*...
Mendieta's user avatar
  • 249
6 votes
2 answers
167 views

Stable unions without stable images

A regular category is one with finite limits and pullback-stable images (i.e. (regular epi, mono) factorizations). A coherent category is a regular category that also has pullback-stable finite ...
Mike Shulman's user avatar
6 votes
1 answer
302 views

Diagrams in an Elementary Topos

Let $T$ be a sheaf topos and $I$ a small category. Then the functor category $[I,T]$ is also a sheaf topos. Now let $E$ be an elementary topos (cartesian closed category with finite limits + subobject ...
user84563's user avatar
  • 915
6 votes
1 answer
282 views

Linear logic and linearly distributive categories

I asked this question ten days ago on MathStackexchange (see here). Despite having placed a bounty on the question, I have not received any answers or comments until now. Following Nick Champion's ...
Max Demirdilek's user avatar
6 votes
0 answers
157 views

Examples of Heyting categories that are not toposes?

When explaining how Heyting categories can model first order logic it would be nice to be able to give some small example and contrast it with Set-semantics. I realized however that I don't know of ...
rosensymmetri's user avatar
5 votes
2 answers
491 views

Proof by contradiction in a topos

In a topos which is not Boolean topos, can we use proof by contradiction?
user26296's user avatar
  • 105
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 ...
user478652's user avatar