Questions tagged [topos-theory]

A topos is a category that behaves very much like the category of sets and possesses a good notion of localization. Related to topos are: sheaves, presheaves, descent, stacks, localization,...

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Synthetic vs. classical differential geometry

To provide context, I'm a differential geometry grad student from a physics background. I know some category theory (at the level of Simmons) and differential and Riemannian geometry (at the level of ...
ಠ_ಠ's user avatar
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What are the higher homotopy groups of Spec Z ?

The homotopy groups of the étale topos of a scheme were defined by Artin and Mazur. Are these known for Spec Z? Certainly π1 is trivial because Spec Z has no unramified étale covers,...
Jonathan Wise's user avatar
53 votes
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Grothendieck's manuscript on topology

Edit: Infos on the current state by Lieven Le Bruyn: http://www.neverendingbooks.org/grothendiecks-gribouillis Edit: Just in case anyone still thinks that Grothendieck's unpublished manuscripts are (...
51 votes
2 answers
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What's an example of an $\infty$-topos not equivalent to sheaves on a Grothendieck site?

My question is as in the title: Does anyone have an example (supposing one exists) of an $\infty$-topos which is known not to be equivalent to sheaves on a Grothendieck site? An $\infty$-topos is as ...
Charles Rezk's user avatar
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50 votes
5 answers
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The unification of Mathematics via Topos Theory

In her paper The unification of Mathematics via Topos Theory, Olivia Caramello says "one can generate a huge number of new results in any mathematical field without any creative effort". Is ...
Roy Maclean's user avatar
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48 votes
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Grothendieck says: points are not mere points, but carry Galois group actions

Apologies in advance if this question is too elementary for MO. I didn't find an explanation of these ideas in any algebraic geometry books (I don't know French). The following is an excerpt from ...
Arrow's user avatar
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45 votes
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The formal p-adic numbers

The real numbers can be defined in two ways (well, more than two, but let's stick to these for now): as the Cauchy completion of the metric space $\mathbb{Q}$ with its usual absolute value, or as the ...
Mike Shulman's user avatar
44 votes
5 answers
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Several Topos theory questions

Hey. I have a few off the wall questions about topos theory and algebraic geometry. Do the following few sentences make sense? Every scheme X is pinned down by its Hom functor Hom(-,X) by the ...
Steven Gubkin's user avatar
43 votes
8 answers
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How should one think about sheafification and the difference between a sheaf and a presheaf

The first time I got in touch with the abstract notion of a sheaf on a topological space $X$, I thought of it as something which assigns to an open set $U$ of $X$ something like the ring of continuous ...
roger123's user avatar
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39 votes
4 answers
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What is an $(\infty,1)$-topos, and why is this a good setting for doing differential geometry?

In this post on the n-Category Café, Urs Schreiber says that, "The theory of G-principal bundles makes sense in any $(\infty,1)$-topos." I followed the link to the nLab and tried to chase definitions, ...
Tom LaGatta's user avatar
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38 votes
6 answers
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Advantages of diffeological spaces over general sheaves

I have been playing with/thinking about diffeological spaces a bit recently, and I would like understand something rather crucial before going further. First a little background: Diffeological spaces ...
David Carchedi's user avatar
35 votes
3 answers
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Higher Topos Theory- what's the moral?

I've often seen Lurie's Higher Topos Theory praised as the next "great" mathematical book. As someone who isn't particularly up-to-date on the state of modern homotopy theory, the book seems ...
Michael Klyachman's 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
35 votes
0 answers
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Grothendieck's "List of classes of structures"

In Lawvere's article Comments on the Development of Topos Theory, the author writes: Similarly, Grothendieck and others unerringly recognized which kinds of mathematical structures are 'preserved ...
Arrow's user avatar
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34 votes
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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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32 votes
3 answers
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Fundamental groups of topoi

Just yesterday I heard of the notion of a fundamental group of a topos, so I looked it up on the nLab, where the following nice definition is given: If $T$ is a Grothendieck topos arising as category ...
Lars's user avatar
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32 votes
3 answers
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What is the theory of local rings and local ring homomorphisms?

It is well-known that the category of local rings and ring homomorphisms admits an axiomatisation in coherent logic. Explicitly, it is the coherent theory over the signature $0, 1, -, +, \times$ with ...
Zhen Lin's user avatar
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30 votes
1 answer
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Cohomology of sheaves in different Grothendieck topologies

Suppose I have a sheaf $\mathcal{F}$ on the (small) étale site over $X$. By restriction, $\mathcal{F}$ is also a sheaf on $X$ (with the Zariski topology). When is it that the sheaf cohomologies (i.e. ...
Sam Derbyshire's user avatar
29 votes
0 answers
2k views

Did Grothendieck overestimate topoi?

I was reading the Russian translation of Recoltes et Semailles and in the footnote where Grothendieck lists his 12 contributions (including schemes) we find the following lines: Из этих тем ...
28 votes
3 answers
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Is there a good general definition of "sheaves with values in a category"?

Let $\mathcal{A}$ be a category. There is a common definition of "sheaves with values in $\mathcal{A}$", which is what one obtains by taking the Grothendieck-style definition of "sheaf ...
Zhen Lin's user avatar
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28 votes
2 answers
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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
2 answers
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The philosophy behind local rings

This question has been bugging me for a while and I can't seem to make sense of it on a clear conceptual level. The theory of local rings is given by taking the theory of rings and adding the axioms \...
Georg Lehner's user avatar
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27 votes
1 answer
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Motivation for relative schemes: why should one work with schemes over a ringed topos?

Recently I've been trying to learn more about relative schemes. These were developed in M. Hakim's thesis Topos annelés et schémas relatifs under Grothendieck's guidance and appear in many of later ...
Emily's user avatar
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26 votes
6 answers
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What is a topos?

According to Higher Topos Theory math/0608040 a topos is a category C which behaves like the category of sets, or (more generally) the category of sheaves of sets on a topological space. Could one ...
Ilya Nikokoshev's user avatar
26 votes
2 answers
4k views

What does an etale topos classify?

Any Grothendieck topos E is the "classifying topos" of some geometric theory, in the sense that geometric morphisms F→E can be identified with "models of that theory" internal to the topos F. ...
Mike Shulman's user avatar
26 votes
3 answers
1k views

What is the geometric significance of fibered category theory in topos theory?

Often in topos theory, one starts with a geometric morphism $f: \mathcal Y \to \mathcal X$, but quickly passes to the Grothendieck fibration $U_f: \mathcal Y \downarrow f^\ast \to \mathcal X$, which ...
Tim Campion's user avatar
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25 votes
1 answer
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Topos associated to a category

For each topos $\mathbb E$ let $\mathcal O(\mathbb E)$ be the locally presentable category of objects in $\mathbb E$. We can make $\mathcal O$ into a contravariant functor to the category of locally ...
Marc Nieper-Wißkirchen'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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25 votes
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Caramello's theory: applications

In this text, the author says (well, he says it in French, but I am too lazy to fix all the accents, so here is a Google translation): In any case, contemporary mathematics provides an example of ...
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25 votes
0 answers
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$\infty$-topos and localic $\infty$-groupoids?

It's known that every classical (Grothendieck) topos is equivalent to the topos of sheaves on a localic groupoid (a groupoid in the category of locales). For the record, this is proved by, starting ...
Simon Henry's user avatar
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24 votes
1 answer
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Is Lemma A.1.5.7 in Higher Topos Theory correct?

Hello to everyone, I am studying the properties of combinatorial model categories, following the exposition given by Jacob Lurie in Higher Topos Theory ([HTT] from now on), in section A.2.6. At some ...
Mauro Porta's user avatar
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, ...
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24 votes
1 answer
988 views

Is the opposite category of commutative von Neumann algebras a topos?

By the "category of commutative von Neumann algebras" I mean the category of all commutative von Neumann algebras with normal unital $*$-homomorphisms between them (I don't want to restrict ...
Simon Henry's user avatar
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24 votes
1 answer
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Examples of $(\infty,1)$-topoi that are not given as sheaves on a Grothendieck topology

An $(\infty,1)$-topos according to Lurie is defined as (accessible) left exact localization of a presheaf $(\infty,1)$-category $\text{P}(\mathcal C)$. Those $(\infty,1)$-topoi $\text{Sh}(\mathcal C)$ ...
Georg Lehner's user avatar
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24 votes
1 answer
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Small complete categories in a Grothendieck topos

It is a classical theorem of Freyd that if a small category is complete (has all small limits—in fact, having small products suffices), then it is a preorder (has at most one morphism between ...
Mike Shulman's user avatar
24 votes
1 answer
726 views

Is Lemma D4.5.3 in the Elephant correct? (“In a topos, weakly projective implies internally projective.”)

Lemma D4.5.3 of Johnstone’s Sketches of an Elephant states: Lemma. For an object $A$ of a topos $\newcommand{\E}{\mathcal{E}}\E$, the following are equivalent: $A$ is internally ...
Peter LeFanu Lumsdaine's user avatar
23 votes
5 answers
7k views

Grothendieck's Galois Theory today

I have recently become aware of, and started to study in my free time (abundant in these summer months) Grothendieck's Galois Theory (GGT), as formulated in SGA 1 and later by Grothendieck's ...
lambdafunctor's user avatar
23 votes
5 answers
1k views

Sheafification via hypercovers

The sheafification of a presheaf on a site is often constructed in a two-step process $X^{++}$, where $X^+$ consists of matching families in $X$, is always separated, and is a sheaf if $X$ is ...
Mike Shulman's user avatar
23 votes
1 answer
2k views

Condensed vs pyknotic vs consequential

As is probably clear from my previous questions, I am coming to "condensed mathematics" from the naive perspective of a category theorist, without much knowledge of the intended applications ...
Mike Shulman's user avatar
23 votes
1 answer
2k views

Variant of Conceptual Completeness

Let $\mathcal{C}$ and $\mathcal{D}$ be pretopoi, and let $f: \mathcal{C} \rightarrow \mathcal{D}$ be a pretopos functor (that is, a functor which preserves finite coproducts, finite limits, and ...
Jacob Lurie's user avatar
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22 votes
2 answers
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An extension of the Galois theory of Grothendieck

This question is about Joyal and Tierney's famous An extension of the Galois theory of Grothendieck. One of the main results states (see the MathSciNet review by Peter Johnstone): Joyal and Tierney's ...
user1005113's user avatar
22 votes
2 answers
1k views

Toposes (topoi) as classifying toposes of groupoids

A famous theorem of Joyal and Tierney says that each Grothendieck topos is equivalent to the classifying topos of a localic groupoid. I believe that Butz and Moerdijk have shown that if the topos has ...
Benjamin Steinberg's user avatar
22 votes
1 answer
1k views

How strong is "all sets are Lebesgue Measurable" in weaker contexts than ZF?

Famously, Solovay showed that, if $\textrm{ZFC}$ plus $\textrm{IC}$ (the existence of an inaccessible cardinal) is consistent, then so is $\textrm{ZF}$ plus $\textrm{DC}$ (dependent choice) plus $\...
Alex Simpson's user avatar
22 votes
1 answer
1k views

What is classified by the (big) crystalline topos?

In his paper "Generic Galois Theory of Local Rings", G.C. Wraith states on p. 743 that the (big) crystalline topos "can be conveniently described in terms of the theory it describes". What exactly is ...
Marc Nieper-Wißkirchen's user avatar
21 votes
9 answers
7k views

Topos theory reference suitable for undergraduates

I am a third year undergraduate who has just learnt the rudimentals of category theory. My specialization is computer science, not mathematics. As part of my course work I want to write an essay on ...
21 votes
1 answer
2k views

Joyal's construction of the spectrum of a commutative ring

I am trying to understand bits and pieces of Lawvere's article Continuously Variable Sets; Algebraic Geometry = Geometric Logic. I'm not doing very well. I know this is a lot to ask, but basically, I ...
Arrow's user avatar
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21 votes
1 answer
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Surmounting set-theoretical difficulties in algebraic geometry

The category $\text{AffSch}_S$ of affine schemes over some base affine scheme $S$ is not essentially small. This lends itself to certain set-theoretical difficulties when working with a category $Sh(\...
Exit path's user avatar
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21 votes
2 answers
1k views

When do two elliptic curves have equivalent small etale toposes?

Let $X$ and $Y$ be elliptic curves over an algebraically closed field $K$. If the characteristic of $K$ is nonzero, assume both curves are ordinary or both are supersingular. Does it follow that $X$ ...
JBorger's user avatar
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21 votes
1 answer
1k views

Logical endofunctors of Set?

What set-theoretic assumptions are necessary and sufficient to ensure the existence of a nontrivial (i.e. not isomorphic to the identity) endofunctor of the category Set which is logical (i.e. ...
Mike Shulman's user avatar
21 votes
1 answer
665 views

A nice subcategory of the category of measurable spaces

Is there some notion of "nice" measurable spaces and "nice" maps between them which satisfies the following properties? The real line equipped with the Lebesgue $\sigma$-algebra is nice. Any ...
Taylor Friesen's user avatar

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