# Questions tagged [categorical-logic]

The categorical-logic tag has no usage guidance.

52
questions

**9**

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104 views

### Finitely presentable grids and co-presentable topological spaces

In the 90s there has been some interest for the category $\mathsf{Top}^\circ,$ the dual of the category of spaces. Most of the relevant papers on the topic are co-authored by Pedicchio.
Barr, ...

**4**

votes

**1**answer

115 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 ...

**11**

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163 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 ...

**10**

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345 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 ...

**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".
...

**6**

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94 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 ...

**4**

votes

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50 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 ...

**1**

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101 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 ...

**8**

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152 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 ...

**8**

votes

**1**answer

308 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{\...

**10**

votes

**1**answer

302 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 ...

**3**

votes

**1**answer

195 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 ...

**3**

votes

**1**answer

114 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 ...

**18**

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 ...

**10**

votes

**1**answer

569 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, ...

**13**

votes

**2**answers

876 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 ...

**30**

votes

**1**answer

550 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 ...

**24**

votes

**5**answers

2k 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 ...

**8**

votes

**2**answers

824 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. ...

**12**

votes

**1**answer

684 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 ...

**10**

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**0**answers

268 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 $\mathrm{Set}[\mathbb{T}]$ ...

**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, ...

**2**

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363 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 ...

**12**

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**2**answers

546 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 $\mathcal{T}$.
...

**13**

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327 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 ...

**5**

votes

**1**answer

246 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 ...

**10**

votes

**1**answer

481 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}...

**5**

votes

**2**answers

135 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 ...

**10**

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**3**answers

779 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 ...

**4**

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104 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 ...

**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 ...

**3**

votes

**0**answers

331 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 ...

**11**

votes

**1**answer

347 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 ...

**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 ...

**7**

votes

**2**answers

903 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 ...

**2**

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111 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 ...

**13**

votes

**1**answer

814 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 ...

**6**

votes

**3**answers

241 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 ...

**29**

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, ...

**4**

votes

**1**answer

206 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 ...

**23**

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 ...

**3**

votes

**2**answers

447 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 ...

**17**

votes

**4**answers

3k 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 ...

**5**

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**2**answers

415 views

### Proof by contradiction in a topos

In a topos which is not Boolean topos, can we use proof by contradiction?

**27**

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**2**answers

2k 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 ...

**10**

votes

**2**answers

701 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 ...

**32**

votes

**3**answers

2k 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 ...

**25**

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 ...

**13**

votes

**4**answers

5k 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 ...

**51**

votes

**2**answers

8k 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, ...