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15
votes
1answer
304 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 ...
3
votes
0answers
41 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 ...
4
votes
1answer
138 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 ...
6
votes
1answer
242 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, ...
12
votes
1answer
437 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 ...
16
votes
0answers
442 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 ...
27
votes
0answers
661 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 ...
21
votes
1answer
843 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 ...
5
votes
0answers
245 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 ...
4
votes
0answers
97 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 ...
1
vote
0answers
135 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 ...
14
votes
1answer
297 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 ...
3
votes
0answers
55 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 ...
8
votes
1answer
185 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 ...
0
votes
1answer
69 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 ...
5
votes
0answers
84 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 ...
5
votes
0answers
137 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 ...
8
votes
0answers
66 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 ...
1
vote
0answers
222 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 ...
16
votes
3answers
410 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), ...
6
votes
1answer
192 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 ...
35
votes
2answers
893 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 ...
1
vote
1answer
103 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 ...
6
votes
1answer
299 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 ...
1
vote
1answer
244 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 ...
1
vote
0answers
75 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 ...
7
votes
1answer
288 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 ...
0
votes
0answers
97 views

Representing topoi by topological groupoids

i was reading an article written by Butz and Moerdijk (https://www.math.uu.nl/publications/preprints/984.ps.gz) and i have a problem in understanding their proof of theorem $5.1$ (The one in which ...
6
votes
2answers
471 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 ...
1
vote
0answers
95 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 ...
2
votes
0answers
170 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 ...
1
vote
0answers
79 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 ...
8
votes
2answers
331 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 ...
12
votes
0answers
297 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 ...
6
votes
1answer
583 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 ...
5
votes
1answer
113 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 ...
1
vote
0answers
85 views

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 ...
36
votes
6answers
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 ...
2
votes
0answers
78 views

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 ...
13
votes
1answer
452 views

The real numbers object in Sh(Top)

If $X$ is a sober topological space, the real numbers object in the topos $\mathrm{Sh}(X)$ is the sheaf of continuous real-valued functions on $X$. This is proven very explicitly in Theorem VI.8.2 of ...
4
votes
1answer
311 views

Atlas of a manifold as a Sheaf

--Hopefully this question does not dublicate another-- In this question Tom Goodwillie pointed out, that the 'atlas part' of the definition of a smooth manifold can be redefined in terms of sheaves. ...
1
vote
0answers
112 views

right adjoint functor for closed immersion of topoi

Let $i\colon (X,A)\rightarrow (Y,B)$ be a closed immersion of ringed topoi. Does functor $i_*\colon Mod(A)\rightarrow Mod(B)$ have a right adjoint?
6
votes
1answer
166 views

Do non-subcanonical Grothendieck topologies always induce a category of fractions?

Suppose that $\mathscr{C}$ is a (possibly higher) category and $J$ is a Grothendieck topology which is not subcanonical. Denote the composite $$\mathscr{C} \hookrightarrow ...
2
votes
0answers
61 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 ...
3
votes
0answers
115 views

When does prolongation preserve sheaves?

Suppose that $(C,J)$ and $(D,K)$ are two Grothendieck (possibly $\infty$-)sites and $f:C \to D$ is a functor such that $$f^*:Psh(D) \to Psh(C)$$ sends sheaves to sheaves. Under what conditions will ...
5
votes
0answers
103 views

Non-degenerate limits of topoi

Let $\mathcal{E}$ be an elementary topos with natural numbers object $\mathbb{N}$, and let $\mathbb{C}$ be a cofiltered internal category in $\mathcal{E}$. Suppose we have a diagram of shape ...
6
votes
1answer
372 views

What information is lost in $X \to \mathrm{Sh}(X)$?

Given a topological space or site $X$. Construct $\mathrm{Sh}(X)$ - the sheaves on $X$ with values in $\mathrm{Set}$. Is it known what information is lost in this procedure? Thanks, Adrian.
9
votes
2answers
709 views

Modern versions of Verdier's hypercovering theorem?

Let $\mathcal{C}$ be a small category equipped with a terminal object $1$ and a Grothendieck topology. (Assume $\mathcal{C}$ also has pullbacks, if it is more convenient.) The following is a ...
12
votes
1answer
436 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 ...
1
vote
0answers
54 views

Validity of equations in a topos bis

I clarify my previous question with a simple example. It is also a simpler (and more general) question whose resolution resolves the previous question on interpreting a theory in another. The ...