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### When is a functorial coverage a sheaf, and what universal property does it have?

In The Elephant (A.2.1.9), Johnstone defines the notion of a coverage on a category $\mathcal{C}$. Quoting verbatim, a coverage on $\mathcal{C}$ is
a function assigning to each object $A$ of $\...

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**0**answers

191 views

### Characterizing flatness in terms of pulling back relative schemes

All schemes are assumed to be noetherian (and maybe separated just in case)
Here is a (possible) definition of flatness:
Definition 1 (flatness): A morphism of schemes $f: X \to Y$ is flat iff ...

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**0**answers

128 views

### Yoneda embedding and Horn sentences

The following is taken from Borceux and Bourn's Mal'cev, Protomodular, Homological, and Semi-Abelian Categories.
Metatheorem 0.1.3. Let $\mathcal P$ be a statement of the form $\varphi\implies \...

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78 views

### Internal language type of power objects

It is a basic fact that in a category with finite limits the following are equivalent
Each object $X$ has a (membership relation to a) power object $PX\times X\hookleftarrow\ni_X$ subject to the ...

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**3**answers

480 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 ...

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152 views

### Is there any notion of “smoothification” from $\mathbb{R}$-schemes to generalized smooth spaces?

I will write $\operatorname{Diff}$ to denote a category of generalized smooth spaces e.g. $Sh(\mathsf{CartSp})$. Is there a version of $\operatorname{Diff}$ for which there exists a functor $\mathcal{...

**5**

votes

**1**answer

176 views

### Is the analytification functor part of a geometric morphism of topoi?

Let $Sh(\mathsf{\mathbb{C}-fAlg}^{op})$ be the topos of zariski sheaves on finitely genertaed $\mathbb{C}$-algebras.
A complex analytic space for our purpose is a locally ringed space locally ...

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vote

**1**answer

650 views

### Problems in some parts of Monique Hakim thesis?

In "Reminiscences of Grothendieck and his school" Luc Illusie says:
"I heard from Deligne that there were problems in some parts. (of Monique Hakim thesis).
Topos annelés et schémas relatifs, ...

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**1**answer

2k views

### 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 ...

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votes

**2**answers

179 views

### Which Heyting algbras arise out of some elementary topos which satisfies the ultrafilter principle?

It is known that AC implies LEM constructively, and also that AC implies the ultrafilter principle. Is there a similar relationship between the ultrafilter principle and classical logic? In other ...

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**1**answer

172 views

### Descent of Higher categorical structures along geometric morphisms

Let $f: \mathcal{E} \rightarrow \mathcal{T}$ be a geometric morphism between two (Grothendieck) toposes (or maybe more generally a bounded geometric morphism between elementary toposes).
It is well ...

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**1**answer

415 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 ...

**16**

votes

**1**answer

366 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 ...

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**0**answers

47 views

### Resources on a smooth topos containing complex analytic/holomorphic geometry

In this question Urs Schreiber mentioned there are models in synthetic differential geometry of complex analytic geometry.
First of all: When Urs writes complex analytic geometry, does he mean ...

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votes

**1**answer

145 views

### Lie functor preserves “surjections” in synthetic differential geometry?

In classical finite-dimensional differential geometry, the Lie functor preserves surjections, sending a surjective Lie group homomorphism to a surjective Lie algebra homomorphism.
As pointed out ...

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**1**answer

263 views

### Jets in synthetic differential geometry

As I understand it from Kostecki's notes, the $k$-jet $j^k f$ of a function $f: R^n \to R^m$ should be the map $$f^{D_k(n)} : {(R^n)}^{D_k(n)} \to (R^m)^{D_k(n)},$$
where $$D_k(n) = \{(x_1, \ldots, ...

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**1**answer

479 views

### Axiom of choice as zero dimensionality

In the paper Quantifiers and Sheaves by Lawvere, at the bottom of the second page, the author writes:
"... the condition that every epi splits, which geometrically we would call 0-dimensionality ...

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**1**answer

697 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 ...

**28**

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**0**answers

738 views

### 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 ...

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**1**answer

891 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 ...

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283 views

### What is an Elementary “Homotopy, Model” Topos?

Context:
Def (Rezk): A (Grothendieck) homotopy topos is a homotopy left exact Bousfield localization of the model category of simplicial presheaves sPsh(C) on a small simplicial category C.
Thm (...

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**0**answers

118 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 ...

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143 views

### Coinduction in a presheaf topos?

Fix a presheaf topos $\mathcal{E}\triangleq \mathbf{Set}^{\mathcal{C}^\mathsf{op}}$. Then, for any object/presheaf $X:\mathcal{E}$, I can define the presheaf of relations on $X$ as $\wp(X)\triangleq\...

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**1**answer

370 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 ...

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**0**answers

63 views

### smooth topos as generalized smooth space

I'm interested in generalized smooth spaces. I know there are several spaces such as Deffeological space, Frölicher space, Chen space, etc... and there are some papers compare them. However, I haven't ...

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**1**answer

223 views

### Geometric morphism of $\infty$ topos

I have a very simple question regarding geometric morphisms of $\infty$ topoi, but have been unable to find the answer in Lurie's HTT (although it seems likely that its there somewhere and I just can'...

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**1**answer

72 views

### A confusion about covering flatness

I'm reading this entry on nLab. But I'm confusing with the notion of covering-flatness. More precisely, I meet some trouble when I try to show that the $Sets$-valued flatness is a special case of ...

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88 views

### Finite covers in complex analytic geometry

Given a complex manifold or complex analytic space, one has the standard notion of open set. There are two different Grothendieck topologies that one can define using this notion, one where covers ...

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167 views

### Making the conceptual leap from locales to Grothendieck topologies?

I find the definition for locales and sheaves on locales to be straightforward, but I'm stumbling over the idea of a Grothendieck topology. Is there a nice way to see roughly how the latter ...

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69 views

### W-types and inverse image functor

All sheaf topoi have W-types and in fact there's an explicit construction given by Benno van den Berg & Ieke Moerdijk, but the construction is quite involved.
I would like to know whether the ...

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243 views

### Why do some people adamantly insist on 'toposes' instead of 'topoi'? [closed]

I've heard that several category and topos theorists, first and foremost Johnstone (see the comments to this question) adamantly insist on 'toposes' as the plural of 'topos'. I was wondering whether ...

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424 views

### Brouwer's theorem for the Cauchy reals

Brouwer famously proved, using principles motivated by intuitionistic choice sequences, that every function $\mathbb{R}\to \mathbb{R}$ is continuous. In Sheaves in geometry and logic (section VI.9), ...

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**1**answer

199 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 ...

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**2**answers

961 views

### 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 ...

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**1**answer

106 views

### Can hypercompletion be an essential localization?

The subcategory of hypercomplete objects in an ∞-topos is a left-exact-reflective subcategory by the remarks after 6.5.2.8 of Higher topos theory, i.e. its inclusion functor has a left-exact left ...

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**1**answer

318 views

### Can hypercomplete objects be coreflective?

The subcategory of hypercomplete objects in an ∞-topos is a left-exact-reflective subcategory by the remarks after 6.5.2.8 of Higher topos theory. Can it ever happen that this subcategory is ...

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**1**answer

252 views

### Etale cohomology and topological invariance

Let $X$ be a projective scheme and $X_0 \subset X$ a subscheme defined by a nilpotent ideal. Denote by $i:X_0 \to X$ the closed immersion. Let $\mathcal{F}$ be a locally free sheaf sheaf on $X_{\...

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78 views

### a topos without an underlying boolean topos

I remember an example of a topos that does not have an underlying Boolean topos.
My problem is, I cannot find where I saw it.
The solution is supposed to have Setf and Set, and the objects of the ...

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**1**answer

316 views

### Ordinals in constructive mathematics ? (references)

I'm looking for references presenting a constructive treatment of the theory of ordinals. By constructive I mean valid in the internal logic of a topos (so no axiom of choice and no law of excluded ...

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**2**answers

490 views

### When is the category of small (pre)sheaves a(n elementary) topos?

When $C$ is essentially small, the presheaf category $[C^\mathrm{op},\mathsf{Set}]$ is the free cocompletion of $C$. The presheaf category $[C^\mathrm{op},\mathsf{Set}]$ is also a topos.
When $C$ is ...

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99 views

### is sufficient cohesion equivalent to the connectedness of subobject classifier?

I'm following Lawvere article Axiomatic Cohesion.
He states (Proposition VI.4) that sufficient cohesion is equivalent to the connectedness of subject classifier, but I can't follow the proof. I can't ...

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**0**answers

175 views

### Relationship between coherent toposes/coherent logic and coherent sheaves

I've heard it claimed that the adjective "coherent" in logic/topos theory (i.e. coherent logic, coherent toposes, coherent categories) was adopted to fit in with the terminology of coherent sheaves in ...

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82 views

### Cohomology of discrete group with compact support

This is closely related to a previous question on the topic, but hopefully adds some motivation.
Let $G_{/\mathbf Q}$ be a semisimple group, $K\subset G(\mathbf R)$ a maximal compact subgroup, and $...

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**2**answers

392 views

### Definition of internal field objects

Let $\mathcal{C}$ be a category with finite limits and a strictly initial object $\mathbf{0}$. The final object is denoted by $\mathbf{1}$. I propose the following definition of a field object ...

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325 views

### The topos for forcing in computability theory

My understanding is that forcing (such as Cohen forcing) can be described via a topos. For example this nlab article on forcing describes forcing as a "the topos of sheaves on a suitable site."
My ...

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**1**answer

603 views

### Existence of internal toposes/inner models in a topos

It has been known for some time that one can define a topos as a model of a (finitary) essentially algebraic theory (or in other words, can be defined internal to any category with finite limits). In ...

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**1**answer

117 views

### Site dependance of the Cech weak equivalences on simplicial sheaves

Let $\mathcal{T}= sh(C,J)$ a Grothendieck topos of sheaves over a (ordinary) site.
One endows the category of simplicial presheaves over $C$ with the "Rezk-Lurie" model structure: that is we start ...

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### Relationship between tangent spaces and tangent categories for smooth topoi

Let $\mathscr E$ be a smooth topos, where I am using the terminology of nLab. In particular that imples the existence of a line object $R$ and an "infinitesimally thickened point", which is an object ...

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4k views

### 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 ...

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### A dual notion to Lawvere-Tierney operators for geometric surjections?

A geometric embedding into a Grothendieck topos can be characterised by giving the Lawvere-Tierney topology that induces it. This lets us reduce questions about subtoposes to more elementary order-...