Questions tagged [categorical-logic]

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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 ...
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  • 213
13 votes
1 answer
417 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 ...
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4 votes
2 answers
169 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 ...
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9 votes
2 answers
282 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"? This seems to be an extremely interesting theorem.
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5 votes
1 answer
150 views

Internal language proof of Lawvere's fixed point theorem for cartesian closed categories

This proof of Lawvere's fixed point theorem suggests (since it uses $\lambda$ notation) that it is written in the internal language of cartesian closed categories (which is the $\lambda$-calculus, as ...
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5 votes
3 answers
383 views

What are some interesting hyperdoctrines that are not classical models?

Short version: what are some interesting hyperdoctrines for classical (not intuitionistic) first-order logic, that are not models in the traditional sense? (Where the terminal and initial ...
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4 votes
1 answer
361 views

What structure do all kinds of theories, models, interpretations, proofs and all that form?

This is a question about a structure that can be used to investigate all kind of structures that can be investigated. Many years ago with Joseph Gubeladze we discussed something similar but I only ...
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3 votes
1 answer
117 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 $E$...
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  • 220
6 votes
2 answers
287 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^*...
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  • 220
4 votes
1 answer
178 views

Universal property of the codomain fibration

Let $\mathcal{C}$ a category with pullbacks. Does $\mathsf{cod}: \mathcal{C}^{\to}\to\mathcal{C}$ have any kind of universal property in the category of (co)fibrations over $\mathcal{C}$? I'd want it ...
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12 votes
0 answers
190 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 ...
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10 votes
0 answers
362 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 ...
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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". ...
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  • 500
6 votes
0 answers
109 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 ...
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4 votes
0 answers
56 views

Relative completeness of a relative cocompletion of a subcategory

I'm going to use the language from Lack and Rosicky's Notions of Lawvere theory, but I won't be touching on actual enriched category theory. Suppose I have a category $\mathbb{C}$ with a class of ...
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1 vote
0 answers
103 views

Aggregations (e.g., cardinality, indexed sums/products) internal to a syntactic category?

Note: Expanded and rephrased, per Todd's question below. Suppose that we have a set-valued functor $S:\mathcal{C}\to\mathbf{Sets}$, and an arrow $p:Y\to X$ such that $S(p)$ has finite fibers. From ...
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8 votes
0 answers
189 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 ...
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9 votes
1 answer
358 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{\...
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10 votes
1 answer
326 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 ...
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3 votes
1 answer
218 views

How to turn a limit sketch into an essentially algebraic theory?

An essentially algebraic theoery, according to Adamek and Rosicky (second definition on nlab), consists of a many-sorted signature $\Sigma$ (consisting of function symbols on sorts $S$), a set $E$ of ...
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3 votes
1 answer
121 views

Internal equality for Eq-fibrations' morphisms

I have posted this question here on M.SE but since it received little attention and since it seems difficult to find helpfule references I reposting it here. In Jacob's Categorical logic and Type ...
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22 votes
5 answers
2k 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 ...
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11 votes
1 answer
612 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, ...
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14 votes
2 answers
992 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 ...
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32 votes
1 answer
640 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 ...
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27 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 ...
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8 votes
2 answers
900 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. ...
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  • 1,560
12 votes
1 answer
759 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 ...
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11 votes
0 answers
322 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 $\mathrm{Set}[\mathbb{T}]$ ...
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23 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, ...
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2 votes
0 answers
378 views

Geometric Theories have models in any Grothendieck Topos?

This question is linked to this one. My question is: Is it true any consistent geometric (here I mean coherent theory, different books have different standards) theory $T$ has a model in a ...
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12 votes
2 answers
645 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 $\mathcal{T}$. ...
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14 votes
0 answers
371 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 ...
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5 votes
1 answer
263 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 ...
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  • 885
10 votes
1 answer
492 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}...
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  • 7,847
6 votes
2 answers
153 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 ...
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  • 58.8k
10 votes
3 answers
821 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 ...
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  • 1,560
4 votes
0 answers
109 views

Ref request: modelling regular theories as an injectivity condition

The background to the following observation is standard material in categorical logic, and I thought this was was too — I don’t remember learning it, but I don’t think it is original — but I can’t now ...
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25 votes
1 answer
1k 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 ...
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3 votes
0 answers
368 views

Topos Theory, internal Heyting Algebra

Given a topos $\mathcal{E}$ with subobject classifier $\Omega$. If we denote by $N\Omega$ the former of all local operators on $\Omega$, that is, Lawvere–Tierney topologies of $\mathcal{E}$, it is ...
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11 votes
1 answer
373 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 ...
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20 votes
1 answer
3k 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 ...
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7 votes
2 answers
922 views

Is there one binary operation foundational for set theory?

The membership relationship "$\epsilon$" is foundational for set theory, in the sense that the axioms of any set theory are formulated in the language of "$\epsilon$". Naturally, the question arises ...
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2 votes
0 answers
112 views

Sheaves, colimits and closure

I am considering the sheaf topos $\mathbf{Sh}(\mathfrak{X})$ on a topological space $\mathfrak{X}$. Limits in this category are constructed pointwise, as in presheaves. Colimits, however, are not ...
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13 votes
1 answer
944 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 ...
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  • 13.2k
6 votes
3 answers
255 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 ...
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30 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, ...
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  • 58.8k
4 votes
1 answer
210 views

Sketches for categories of models of complete theories

In Accessible categories : the foundations of categorical model theory, chapter 3 p.58, Makkai and Paré claim that there is "an (obvious) identification of a class of sketches so that the categories ...
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  • 61
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 ...
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  • 4,647
3 votes
2 answers
470 views

Further relation between monads and theories

This question want to be a follow up of the following question. In that thread I was interested in understanding relation between various presentation of algebraic theories. In particular in Eduardo ...
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