The topos-theory tag has no usage guidance.

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### 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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118 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 ...

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182 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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53 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

157 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 ...

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**1**answer

67 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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81 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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124 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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62 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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208 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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397 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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182 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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865 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

100 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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284 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

241 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 ...

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### 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

268 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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91 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 ...

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459 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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### 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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161 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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### 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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326 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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### 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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568 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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112 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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### 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 ...

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### 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 ...

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307 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. ...

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### 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?

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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369 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.

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622 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 ...

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### 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 ...

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### 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 ...

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### Validity of equations in a topos

To simplify consider simple algebraic theories (universal algebra)
A and L, but the question applies to geometric theories.
1) Syntactically, we can interprete L in A if we can define the operations ...

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### Induced adjunctions

Suppose $F: C \rightarrow D$ is the left adjoint to a functor $G$. Then is it true that the functor $F^{\star}:[C : Sets]$ defined by prescomposing a functor $P: C \rightarrow Sets$ is still left ...

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### 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, ...

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### Do coherent toposes descend along open surjection?

Let $f:\mathcal{L} \rightarrow \mathcal{S}$ be a geometric morphism between two toposes. Let $g:\mathcal{T}\rightarrow \mathcal{S}$ be an open surjection (of toposes) and assume that the map ...

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### Is the collage of two spatial toposes a spatial topos?

Consider the collage operation along a profunctor, defined between two categories ${\bf C}, \bf D$.
Suppose now that the two categories are toposes, say $Sh(X), Sh(Y)$ for two topological spaces ...

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### Cloven Kan fibrations

For each groupoid $G$ there is a monad such that the cloven fibred groupoids over $G$ are algebras for this monad. I am looking for an infinite dimensional counterpart: a monad on simplicial sets ...

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### classifying $\infty$-toposes for topological/localic groups?

Let $G$ be a locally compact topological group (or more generally a localic group). Is there an infinity topos which classify principal $G$ bundles ?
More precisely, is there an $\infty$-topos $BG$ ...

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### Dedekind reals in heyting valued models

Let $V^{H}$ be a Heyting valued model of intuitionistic set theory. What conditions does $H$ have to satisfy in order for the following claim to hold? (where $\| \phi(u) \| \in H$ is the truth value ...

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624 views

### Analogy between topology and algebraic geometry

In topos theory, there are many generalizations of topological concepts. For example, open, closed, proper and etale morphisms between toposes. However, there are also such analogous concepts in ...