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36 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
  • 66.8k
36 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
  • 8,659
29 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
  • 25.6k
26 votes
2 answers
2k 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
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
  • 2,303
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
0 answers
586 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
14 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
14 votes
2 answers
802 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
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
  • 15.9k
13 votes
4 answers
2k 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
  • 558
13 votes
1 answer
1k 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
1 answer
439 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
  • 42.4k
12 votes
0 answers
432 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
940 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
  • 1,720
11 votes
2 answers
405 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
461 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
0 answers
411 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
  • 407
10 votes
1 answer
526 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
  • 8,659
10 votes
1 answer
234 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
10 votes
0 answers
391 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
9 votes
1 answer
2k views

What does the topos of (light) condensed sets classify?

Recall that $\mathrm{Pro}(\mathbf{FinSet}) = *_{\text{proét}}$, the category of profinite sets, forms a site with finite jointly surjective families as covers, and that the category of sheaves on this ...
xuq01's user avatar
  • 1,094
9 votes
1 answer
511 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

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
  • 1,720
6 votes
3 answers
291 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
314 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
  • 401
6 votes
2 answers
171 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
  • 66.8k
6 votes
1 answer
307 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
  • 913
5 votes
2 answers
503 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
1 answer
415 views

Intuition for the "internal logic" of a cotopos

Let $\mathcal{E}$ be an elementary topos. By definition, $\mathcal{E}$ is a category that has finite limits, is Cartesian closed, and has a subobject classifier $\Omega$. This subobject classifier can ...
safsom's user avatar
  • 225
4 votes
0 answers
269 views

Morally free toposes are free?

Let $C$ be a category with finite limits. Sometimes people say that $\mathsf{Psh}(C)$ is a free topos, and indeed such a name is consistent with the framework of lex colimits by Garner and Lack, or ...
Ivan Di Liberti's user avatar
3 votes
1 answer
165 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 $E$...
Mendieta's user avatar
  • 401
3 votes
0 answers
454 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 ...
Angel Zaldívar's user avatar
2 votes
0 answers
392 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 ...
Ivan Di Liberti's user avatar
2 votes
0 answers
129 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 ...
Aleš Bizjak's user avatar