All Questions
Tagged with topos-theory ag.algebraic-geometry
113 questions
9
votes
1
answer
265
views
Is there a correction to the failure of geometric morphisms to preserve internal homs?
Given a geometric morphism $$f:\mathscr{F}\to\mathscr{E}$$ where $\mathscr{F},\mathscr{E}$ are toposes, we know that $f^*$ does not preserve internal homs, i.e. $f^*[X,Y]\ncong[f^*X,f^*Y]$. We do have ...
- 171
6
votes
1
answer
590
views
Is there an English translation of Monique Hakim's thesis?
Monique Hakim's thesis, published in 1972 as Topos annelés et schémas relatifs, has been referenced on a multitude of occasions. But I struggle to find a translation into English, even an informal one....
- 1,094
2
votes
0
answers
158
views
Topos of sheaves on a scheme considered as a functor
The spectrum of a ring $R$ can be defined as $\operatorname{Spec} R := \operatorname{Hom}(R, -)\colon \mathrm{fpRing} \to \mathrm{Set}$ ($\mathrm{fpRing}$ are commutative finitely presentable rings). ...
- 6,962
3
votes
0
answers
215
views
How to read the definition of Grothendieck Pretopology in SGA4?
In SGA4, the first axiom of a Grothendieck pretopology is given as:
PT0: Pour tout objet $X$ de $C$, les morphismes des familles de morphismes de $Cov(𝑋)$
sont quarrables. (Rappelons qu’un morphisme ...
- 237
0
votes
0
answers
443
views
Local isomorphism of condensed sets and étale condensed groupoids
Is there a notion of local isomorphism for condensed sets?
$\textbf{Motivation:}$ I am trying to define what an étale condensed groupoid would be.
A topological groupoid $\mathcal{G}$ is said to be ...
- 1,485
8
votes
1
answer
328
views
Does the Zariski spectrum of a ring arise formally from the inclusion of the big Zariski topos into the classifying topos for rings?
Let $\iota_\ast : \mathcal A \to \mathcal B$ be a geometric morphism. I'm looking for some functor
$$F_{\mathcal A \to \mathcal B} : \mathrm{Topos}_{//\mathcal B} \to \mathrm{Topos}_{//\mathcal A}$$
...
- 63.9k
2
votes
1
answer
180
views
Can we encode a torsor as a binary function on the isomorphism classes of objects?
Let $G$ be a group object in a topos $\mathcal{T}$. Then we have the notion of a $G$-torsor in $\mathcal{T}$, and the set of isomorphism classes of such objects is denoted $H^1(\mathcal{T};G)$. For ...
- 15.4k
7
votes
1
answer
255
views
Subobject classifier for sheaves on large sites with WISC
Let $\mathsf{C}$ be a possibly large category with a Grothendieck topology satisfying the Weakly Initial Set of Covers condition: there is for each $X$ a set (not a proper class) of covering families ...
- 1,996
7
votes
0
answers
173
views
Beck-Chevalley conditions for the local geometric morphisms $\pi:\mathrm{Zar}/X\to \mathrm{Sh}(X)$
$\newcommand{\Zar}{\mathrm{Zar}}\newcommand{\Sh}{\mathrm{Sh}}$The category of schemes is a full subcategory of the big Zariski topos $\Zar$. For each scheme $X$, there is a local geometric morphism $\...
- 775
12
votes
4
answers
507
views
Localic or topos-theoretic definition of $\operatorname{Spec}$
Usually, the construction of the spectrum of a commutative ring starts with defining the points of $\operatorname{Spec}(A)$, and constructing a topology with the closed sets being the "zeroes&...
- 253
8
votes
1
answer
342
views
The Grothendieck topology of closed immersions on schemes
Let $S$ be a scheme. Let's define a Grothendick topology on $\mathrm{Sch}/S$ where a covering family $\{f_i:Z_i\rightarrow X\}_{i\in I}$ on an $S$-scheme $X$ is a collection of closed immersions of $S$...
- 329
1
vote
0
answers
82
views
Do the covariant maps of a sheaf with transfer automatically satisfy a dual gluing axiom?
I'm interested in describing a notion of equivariant sheaves that uses Mackey functors on topoi. We can describe a genuine G-spectrum as a spectral Mackey functor on the topos of finite G-sets. If we ...
- 1,969
17
votes
3
answers
2k
views
Why do Grothendieck topologies used in algebraic geometry typically involve finiteness conditions?
There are many Grothendieck topologies used in algebraic geometry, with complex interrelations. Generally in one of these topologies, a cover of schemes is a family of maps which is jointly surjective ...
- 63.9k
5
votes
0
answers
165
views
Do quasi-excellent rings have a good constructive definition?
$\DeclareMathOperator\Sh{Sh}$Informally, a quasi-excellent ring is a Noetherian ring with a few technical extra regularity conditions that make it turns out to be the largest class of rings that allow ...
- 1,947
5
votes
0
answers
162
views
Nullstellensatz with nilpotents and $I=J(V(I))$
Let $R$ be the ring $$\mathbb{R}[t_1,t_2\ldots]/(t_1^2,t_2^2,\ldots)$$
Let $p_1,\ldots p_\ell$ be polynomials in $R[x_1,\ldots,x_n]$ whose constant terms are 0.
Let $f$ be a polynomial which is zero ...
user44143
16
votes
1
answer
448
views
Zorn's lemma for Grothendieck sites
In every treatment of Grothendieck sites I can find, flasque sheaves are not defined in the way one would naïvely expect from ordinary sheaf cohomology; namely instead of saying that "restriction ...
- 163
14
votes
3
answers
1k
views
Making the étale topos construction a fully faithful 2-functor from schemes to Grothendieck topoi
For nice topological spaces (say Haudorff spaces) $X$ and $Y$, there is a bijection between continuous maps $X\to Y$ and isomorphism classes of geometric morphisms $\mathrm{Sh}(X)\to \mathrm{Sh}(Y)$.
...
- 143
9
votes
4
answers
1k
views
Examples of five-adjoint systems
I'm looking into Lawvere's formulation of unities of opposites and opposites of unities, and for this I would be interested in systems of five (or more) adjoint functors $X\stackrel{\stackrel{\...
9
votes
2
answers
698
views
Non-trivial automorphisms and descent
In this expository paper by Low it says:
Roughly
speaking, a topos in the sense of Grothendieck is the category of sheaves on a
kind of generalised space whose “points” may have non-trivial ...
- 91
7
votes
1
answer
590
views
If we replace the spectrally ringed space in the definition of a spectral scheme with an arbitrary infinity-topos, what objects do we get?
I'll phrase this in terms of spectral AG, but I'm curious about the same question in the classical context.
We define a nonconnective spectral Deligne-Mumford stack to be a spectrally-ringed topos ...
- 1,969
3
votes
0
answers
530
views
Flasque sheaves on a site
This is a cross-post from MathStackexchange.
We define a flasque sheaf on a site as one whose first Čech cohomology vanishes for every covering of every object of the site. I know this definition is ...
- 153
8
votes
1
answer
910
views
Questions about SGA 4
What is Deligne's motivation in Appendice 9 of Exposé VI to prove that every coherent topos has enough points? For instance, does that have applications in étale cohomology (or other parts of ...
- 714
10
votes
0
answers
230
views
Lefschetz for topoi
The Lefschetz fixed-point theorem is a nice theorem in topology which counts the number of fixed points (counted with multiplicities) of a continuous map $f\colon X\to X$ using the traces of the ...
- 714
18
votes
1
answer
682
views
De Rham via topoi
Étale cohomology of schemes $X$ is constructed as follows: one associates to $X$ the so-called étale topos of $X$, and then one just takes the sheaf cohomology of that topos.
Is it possible to ...
- 714
7
votes
1
answer
291
views
Direct and inverse image terminology
Let $f\colon X\to Y$ be a continuous map. Then $f$ induces a geometric morphism $f^\ast\dashv f_\ast\colon \mathrm{Sh}(X)\leftrightarrows\mathrm{Sh}(Y)$, whose left adjoint is called inverse image and ...
- 723
5
votes
1
answer
323
views
On realizing a topos of sheaves as a topos of equivariant sheaves
This question is motivated by the following example : let $X$ be a variety over a field $k$, with algebraic closure $\bar{k}$. The Galois group $G_k:=\mathrm{Gal}(\bar{k}/k)$ acts on $X_{\bar{k}}:=X\...
- 617
2
votes
0
answers
504
views
Relative homology in Fargues-Scholze paper
if $f:X\to Y$ is a map of small v-stacks, Scholze and Fargues define relative homology as the left adjoint to the $f^{\star}$. They say the left adjoint exist because $f$ is a slice in $v$-site (they ...
- 1,093
3
votes
0
answers
542
views
Is the category of quasi-coherent sheaves not a topos?
There are two parts to my question:
Question #1: First a sanity-check: Am I right in that the category of quasi-coherent sheaves (over e.g. an affine scheme) is not a topos?
My reasoning is thus:
It ...
- 318
6
votes
1
answer
318
views
Hakim's definition of a locally ringed topos
In Hakim's book "Topos annelés et schémas relatifs", Chap. III, Def. 2.3 states that a ringed topos $(X,A)$ is a locally ringed topos when two equivalent conditions are satisfied:
(i) For ...
- 63.1k
10
votes
0
answers
1k
views
The "unification" of geometry via topos theory?
This question is somehow motivated by The unification of Mathematics via Topos Theory and Synthetic vs. classical differential geometry. Sorry if this is a naïve question.
There has been quite a lot ...
- 1,094
4
votes
0
answers
87
views
Meta property criterion on the internal language of a topos of sheaves for Noetherianity
Let $X$ be a topological space. Then $X$ is irreducible if and only if in the internal language of $\mathrm{Sh}(X)$, $\bot$ is not true and we have one side of De Morgan's law, that is, for all ...
- 426
9
votes
0
answers
298
views
Big Zariski topos and classifying topos of local rings
In the paper ``Using the internal language of toposes in algebraic geometry'', Blechschmidt mentions several approaches to defining the big Zariski topos over a scheme. I have two questions, which are ...
- 426
5
votes
1
answer
478
views
About an argument in Olsson's book
The following picture is from Algebraic spaces and stacks (p.54) by Martin Olsson.
I don't understand how to conclude that $\alpha$ is induced by a nonzero class in the end.
It seems that there might ...
- 1,906
4
votes
1
answer
452
views
Objects and morphisms in inverse limits of toposes?
Certain Galois toposes can be written as $\lim_{i \in I} \mathbf{PSh}(G_i)$ where $(G_i)_{i \in I}$ is an inverse system of discrete groups. (The limit is a strict limit in the 2-category of ...
- 1,329
7
votes
0
answers
212
views
Internal logic of the small étale topos of an algebraic variety
If we consider the internal logic of the small étale topos of an algebraic variety, is the variety's geometry reflected in it? For instance, how different are the internal logics for the étale topoi ...
user158636
16
votes
1
answer
984
views
Reconstruct a variety from its crystalline topos
Let $k$ be a perfect field of positive characteristic. Let $X$ be a smooth projective geometrically connected $k$-scheme with a $k$-point.
Can we reconstruct $X$ from its small crystalline topos $((X/...
user158636
5
votes
0
answers
162
views
Classifying toposes of theories of rings that aren't local rings
The standard uses of toposes in algebraic geometry come from sites that look roughly like the syntactic sites of theories of local rings that they classify. This isn't particularly surprising, since ...
- 3,816
27
votes
1
answer
1k
views
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 ...
- 11.8k
9
votes
0
answers
366
views
When is the étale topos of a fibre product the fibre product of étale toposes?
In what follows, all schemes are qcqs. Also, let $\operatorname{\acute{E}t}(X)$ denote the petit étale topos of a scheme $X$.
Let $Y\to X$ be an $X$-scheme. Say that $Y$ is a special $X$-scheme if ...
- 19.6k
13
votes
0
answers
414
views
The Barr-Boole-Galois topos; a modification of sets to play well with schemes
William Lawvere, in his 2015 CT talk "Alexander Grothendieck & the Concept of Space", introduced the "Barr-Boole-Galois topos", which plays the role that sets do for topological spaces (with ...
user30211
21
votes
1
answer
2k
views
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(\...
- 3,019
11
votes
2
answers
664
views
'Continuity' of the étale topos
In certain concrete situations, I can show that the small étale topos of an inverse limit of schemes is the inverse limit of the associated toposes, for example, if $X$ is a (qcqs) scheme relative ...
- 19.6k
11
votes
0
answers
1k
views
What are the uses of topoi in algebraic geometry today?
Since Grothendieck introduced them in the 1960s, topoi have found many applications in algebraic geometry, category theory, and logic. For instance, they appear in the development of étale and ...
- 11.8k
5
votes
1
answer
319
views
Are constant $\infty$-sheaves constant on connected components?
Let $C$ be an $\infty$-category endowed with a Grothendieck topology $J$ and consider the $\infty$-topos $\infty\text{Sh}(C, J)$. There is a
natural geometric morphism to $\infty\text{Grpd}$ whose ...
- 183
3
votes
0
answers
285
views
What is the logical progression in algebraic tools for studying spaces (varieties -> schemes, sheaves, topos etc.)?
Some algebraists (Cartier, Weil, Atiyah, etc.) sometimes speak of geometry as a long history of essentially asking the same question—"what is space, and how would one describe a space uniquely". ...
- 39
3
votes
0
answers
189
views
Simple description of a Grothendieck topology on the opposite of f.p. complex algebras
Let ${\cal A}$ be the category of finitely presented $\mathbb{C}$-algebras.
Let $J$ be the largest subcanonical Grothendieck topology on ${{\cal A}^{op}}$ such that the local algebras in $\cal A$ are ...
- 401
25
votes
0
answers
1k
views
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 ...
user140765
6
votes
1
answer
330
views
Classifying Space of "Valuation Ringed Spaces over a Topos"
The classifying topos for local rings is the big Zariski topos of $\text{Spec}(\mathbb{Z})$. Call this topos $T$. Geometric maps of topoi from a topos $T'$ to $T$ are in correspondence with sheaves of ...
user30211
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 ...
10
votes
1
answer
567
views
The 'gros' functor from schemes into (strictly) locally ringed topoi
Consider the category of finitely presented commutative rings, and equip its opposite category with either the Étale or Zariski topology. These give rise to topoi $\operatorname{Sh}(\mathbf{Et})$ and ...
- 493