Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [topos-theory]

A topos (plural topoi, toposes) is a category that behaves like the category of sheaves of sets on a topological space. Topos theory consists of the study of Grothendieck topoi, used in algebraic geometry, and the study of elementary topoi, used in logic.

0
votes
0answers
33 views

Translation Request from French - Comparison of the reals of a topos smooth structures on an elementary topos

I've been reading on Real numbers objects in a Topos, and I want to read this article (starting page 22 on the attached PDF) the nlab references. The problem is that I don't read French, and I haven'...
3
votes
0answers
81 views

The partial map classifier $\eta_{1}: 1 \to 1_{\bot}$ is the subobject classifier

The definition for the partial map classifier $\eta_{b} : b \to b_{\bot}$I was given is that: let $b$ be a member of a topos, and let $$\{\}_{b} : b \to \Omega^{b}$$ be the exponential adjoint of $$\...
1
vote
1answer
41 views

Subobjects in Homotopy type theory

It is conjectured here that homotopy type theory is the internal language of an elementary $(\infty,1)$-topos. I have no idea what these are, but my naive understanding is the following: Homotopy ...
2
votes
0answers
93 views

Why are (Pre)sheaves more important than Co(pre)sheaves?

I'm learning Sheaf Theory, and this is an issue that's been bothering me. Fix a small category $\mathcal{C}$. A $\mathcal{V}$-valued presheaf on the small category $\mathcal{C}$ is a functor $F:\...
0
votes
0answers
51 views

Functorial comparison of Different Models of Set Theory

I'm very much a novice to Model Theory/Categorical Model Theory (though I very much would like to learn). I apologize if my question is improperly stated. Fix a small (or locally small if desired) ...
5
votes
0answers
57 views

Equivariant homotopy theory, topos theory and intuitionistic algebraic topology

This might be a very naive question, but I don't really see what would go wrong, so I'm wondering if this has already been done. The idea is the following : equivariant homotpy theory as far as I can ...
1
vote
1answer
28 views

A certain functor in Hakim's “Topos annellés et schemas relatifs” is a sheaf for the canonical topology

In M. Hakim's Topos annellés et schemas relatifs, page 43, (3.4.7), the Author wants to define a sheaf $f_{0A}^*(X)$ over a topos $T$, with respect to the canonical topology. A scheme $X$ is given, ...
2
votes
1answer
51 views

Johnstone, Topos Theory Exercise 7.3

I need to use the following result about essential points from Johnstone's Topos Theory: Let $\mathcal{E}$ be a Grothendieck toposes and $\mathcal{S}$ be a topos with a natural number object. Show ...
7
votes
3answers
741 views

What is the generally accepted pronunciation of “topoi”?

Apologies if this annoys proponents of “toposes “. It appears to me that there are three main candidates for pronunciation, all focusing on the last syllable: Top-oy (rhyming with “toy” in British ...
3
votes
0answers
109 views

Does this property of sets and functions hold for a general topos?

In the category of sets and set-theoretic functions, every arrow can be obtained by only using the universal property of the small coproducts of $1=\{ 0 \}$ applied to the coprojections that exhibit ...
1
vote
0answers
51 views

Toposes are monadic over categories

Lambek showed that Toposes are Monadic over categories. Can someone give the gist of this paper? I am assuming that the category of toposes is the eilenberg moore category for some monad on the ...
0
votes
1answer
47 views

Is the 2-category of groupoids a topos?

I have no justification for this, but I am wondering if the 2-category of groupoids is a topos.
1
vote
0answers
72 views

Alternative sources for a topos theory description of zeroth order logic

I have recently been reading Robert Goldblatt's fantastic book Topoi: The Categorial Analysis of Logic. Through chapters 6-8, Goldblatt produces a topos theoretic approach to zeroth order logic, where ...
4
votes
1answer
117 views

Forcing in sheaf models of set theory - where do the “generics” disappear to?

I am studying "Sheaves in Geometry and Logic" by Mac Lane & Moerdijk. In their construction of a topos satisfying $\lnot CH$ they work entirely in the Grothendieck topos of double negation ...
4
votes
2answers
116 views

Is a name a sheaf?

The technique of forcing, in set theory, can be expressed in topos theory as a form of reasoning about sheaves on the notion of forcing, $\mathbb{P}$, equipped with a "double negation" Grothendieck ...
3
votes
0answers
39 views

What does “topological heuristics” refer to here?

This quote is from the wikipedia page on topoi: Since the introduction of sheaves into mathematics in the 1940s, a major theme has been to study a space by studying sheaves on a space. This idea ...
1
vote
0answers
51 views

Topological space represented as a topos?

Wikipedia states that Toposes "are in a sense a generalization of point-set topology". This suggests to me that any topological space can be represented as a topos. Is this true? How do we do this?
2
votes
1answer
68 views

Well-Pointed categories vs well-pointed topoi

When reading the definition of a well-pointed category on for example Wikipedia, only the following condition is given: The terminal object 1 is a generator. However the general definition of a well-...
2
votes
1answer
76 views

What does it mean for two morphisms with different sources and targets to be isomorphic?

The standard definition of a subobject relies on the following definition: we call two morphisms $f : X \rightarrow A, g : Y \rightarrow A$ with the same target isomorphic if there exists an ...
3
votes
0answers
79 views

A question about Ščedrov's “Forcing and Classifying Topoi”

I'm trying to read Ščedrov's Forcing and Classifying Topoi, and there's a bit in 1.1 that is frequently references, but I don't quite understand its import. If I'm not missing the point, 1.1 is ...
2
votes
0answers
32 views

The big étale and Zariski topoi are generated by small sites

Grothendieck (SGA 4, Exposé VII, Corollaire 3.2) proves that if $X$ is a quasiseparated scheme then, although the étale site is not small, there exists a subsite of it which is small and has the same ...
0
votes
0answers
24 views

A certain colimit of representables sheaves, namely group actions, is a sheaf: why?

In the first answer to this post on MO, one finds that When you look at the category of sheaves on the category of finite action with the natural topology (covering are surjection of finite ...
2
votes
0answers
60 views

The small étale topos of a scheme is equivalent to the category of finite $\pi_1(X,x)$-sets for every scheme $X$ and every geometric point $x$

Recall Milne, Etale cohomology, Theorem I.5.3: Let $x$ be a geometric point of a connected scheme $X$. The functor $Hom(x,-):FEt/X\to Sets$ ($FEt/X$ being the category of $X$-schemes finite and ...
5
votes
1answer
103 views

Are topoi like topological spaces or like set theory?

Background: I have very little intuition about category theory but I’m trying to understand the motivation behind it. Wikipedia states: In mathematics, a topos [...] is a category that behaves ...
4
votes
2answers
120 views

In algebraic geometry, what kind of theory can only be described by topos but not a site?

A (Grothendieck) topos is defined to be a category equivalent to the category of sheaves on some site. The main difference for a topos and a site is about their morphisms. I noticed that some books ...
1
vote
1answer
46 views

If a Grothendieck topos has a geometric morphism to $Sets$, then it is unique

In Caramello, Theories, Sites, Toposes, I read that Every Grothendieck topos $\mathcal E$ admits a unique (up to isomorphism) geoemtric morphism $\gamma_{\mathcal E}:\mathcal E\to \bf Set$. The ...
0
votes
1answer
57 views

Clarify definition of 'enough points'

I have checked the stacks project, Sketches of an Elephant, and Sheaves in Geometry and Logic but cannot find a satisfactory answer. If a site or topos has enough points, does that mean, that there is ...
1
vote
0answers
64 views

Johnstone, Topos theory: families of arrows inducing the same sheaf condition

Johnstone, Topos theory, 0.3, page 13, asserts that, given a Grothendieck pretopology $P$, if the equalizer condition on a presheaf $F$ is satisfied for a family of arrows $R=\{U_i\to U\}$, then it is ...
2
votes
0answers
61 views

Motivation for topological categories

Define a topological category as a small category with topologies on the set of objects $C_0$ and on the set of arrows $C_1$, such that the domain map and the codomain map are continuous. One can ...
1
vote
0answers
27 views

Boolean algebra homomorphisms with adjoints

I'm interested in a particular class of maps between boolean algebras: the homomorphisms $f:A\to B$ such that there exist functions (not necessarily homomorphisms) $u,d:B\to A$ such that $$f(a)\leq b\...
3
votes
1answer
51 views

Does sheafification of bundles have a right adjoint?

Given a topological space $X$, let $\mathbf{Bundle}(X)=\mathbf{Top}/X$ be the category of bundles over $X$, and let $\mathbf{Sh}(X)$ be the category of sheaves over $X$. Then there's a sheafification ...
2
votes
1answer
64 views

Well Definedness of the Representation of the Subobject Functor

Let $\mathcal{C}$ be a category with a subobject classifier $\Omega$. I want to show that the suboject functor $Sub_{\mathcal{C}}(\cdot): \mathcal{C}^{op} \to \textbf{Set}$ is representable via $\...
4
votes
1answer
63 views

Does a sheaf of abelian groups on a scheme $X$ induce a sheaf of abelian groups on the étale site $X_{ét}$?

Fixed a scheme $X$, étale cohomology $H^\bullet(X,F)$ is defined for all sheaves of abelian groups $F$ over the étale site $X_{ét}$. Now, just to understand, I tried to see if this covers sheaves of ...
1
vote
1answer
93 views

Different descriptions of internal languages of a topos

I wanted to learn more about the internal language of of toposes and to do so I have been reading both Sheaves in Geometry and Logic (Sheaves) by Mac Lane and Moerdijk and Introduction to Higher Order ...
0
votes
0answers
27 views

How to categorically characterize the structure of all grounded first-order logic formulas?

If the notion of topos corresponds to set theory or first-order logic, then what if one is only interested in grounded logic formulas, ie, formulas that don't contain variables? In a topos one has ...
8
votes
1answer
201 views

Is ZFC a geometric theory? Does it have a classifying topos?

Given a language $L$, the set of geometric formulae in $L$ is the smallest set of formulae containing atomic formulae and closed by finite conjunction, arbitrary disjunction, and existential ...
3
votes
0answers
65 views

Topos with sets as subobject classifier

In the computer science literature I have seen cases in which they move from booleans to $\mathsf{Set}$ for obtaining a "proof-relevant" version of a concept (especially in formalisations). As the ...
2
votes
1answer
105 views

Is there a classifying topos for locales?

Is there a Grothendieck topos $F$ such that, for any Grothendieck topos $E$, the category of geometric morphisms $$E\rightarrow F$$ is equivalent to the category of locales internal to $F$? I suspect ...
3
votes
3answers
103 views

Presheaves are sheaves for the trivial topology

I've just started learning about toposes and I have a stupid question to ask. Suppose we are given a small category $\mathcal{C}$ with trivial topology $T$ on it, (where the trivial topology $T$ on ...
6
votes
1answer
116 views

Is the interpretation of the Morleyization of a first order theory in a Heyting category automatically classical?

In Johnstone's Sketches of an Elephant (D1.5.13), he describes how given a first order theory $T$ in a signature $\Sigma$, one can devise a coherent theory $T'$ over a new signature $\Sigma'$ (the ...
2
votes
0answers
34 views

“Category” of (pre)stacks over a fixed category/site

I am wondering if prestacks over a fixed category, or stacks over a fixed site, form an (interesting) category, equipped with prestacks morphisms. First of all, do they form a proper class or a set? ...
4
votes
0answers
99 views

Logic for physics

I'm looking for references on applications of logic (e.g. intuitionistic logic, toposes) to physics (especially quantum physics, as this is the area that I'm aware can be helped by logical ...
0
votes
0answers
33 views

Looking for the Deligne's example of a non trivial topos with no points

In the topos entry of wikipedia, says: there is an example due to Pierre Deligne of a nontrivial topos that has no points (see below for the definition of points of a topos). I'm looking for the ...
5
votes
2answers
163 views

Nontrivial example of topos in the context of logic

I am reading Awodey's paper Structure in Mathematics and Logic: A Categorical Perspective and he is currently (p. 228-9) talking about how propositions, that is, elements $1\rightarrow P$, correspond ...
2
votes
0answers
64 views

A topos with enough points is the topos of equivariant sheaves on a topological groupoid (Butz-Moerdijk): where do continuous actions come from?

In Butz, Moerdijk, Representing topoi by topological groupoids, it is proven that given a topos $\mathcal E$ with enough points there exists a topological groupoid $G\rightrightarrows X$ such that $\...
1
vote
1answer
62 views

The étale topos is coherent: does the scheme need to be quasicompact?

It is known that the little étale topos of a scheme $X$ is coherent, i.e. the Grothendieck topology of the étale site is generated by a basis of finite coverings. For example, Butz and Moerdijk say ...
0
votes
0answers
31 views

Adjoint for pullback of sheaves on a topological groupoid

Consider a topological groupoid $G\xrightarrow{s} X$, with the arrow representing the domain map that associates to each morphism its domain. This induces a pullback functor $s^*:Sh(X)\to Sh(G)$. I ...
2
votes
0answers
38 views

All sheaves in a Grothendieck topos are generated by subobjects of powers of a suitable sheaf

Consider a topos $\mathcal E$. Butz and Moerdijk, in Representing topoi by topological groupoids, par. 2, say that one can find an object $S\in \mathcal E$ such that the subobjects of its powers (i.e.,...
1
vote
0answers
36 views

Homotopy classes of maps inside a Grothendieck topos

Moerdijk, Classifying spaces and classifying topoi, defines the Verdier cohomology of a Grothendieck topos $\mathcal E$ by taking a limit on $HC$, the homotopy of hypercoverings in $\mathcal E$. I ...
7
votes
1answer
163 views

Meaning of Grothendieck quote: recover a topological space by its category of sheaves

In "Récoltes et Semailles"(- Grothendieck), there is a moment when the author talks about the idea of sheaves of sets over a topological space, then taking the category of sheaves (of sets over a ...