# Questions tagged [topos-theory]

The topos-theory tag has no usage guidance.

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### Toposophy vs Set theoretical multiverse philosophy

Johnstones classic topos theory book talks at some length in its introduction about how category theory/topos theory suggest that we view the 'universe' in which mathematics takes place as consisting ...

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### Universal property of sheaf category

Given a site $C$ with a Grothendieck topology and the category of presheaves $P(C)$ (either in the sense of presheaves of sets or in the $\infty$-sense), and the category $S(C)$ of sheaves with ...

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### Definition of $\in_c$ for power objects

On the nLab page for power objects, the object $\in_c$ is defined as the domain of a monomorphism $\in_c\hookrightarrow c\times\Omega^c$, and it is mentioned at the end of the article that in any ...

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### Reference Request: Cohomology and limits of coherent topoi. (Non-abelian case)

SGA 4 VI Discusses finiteness conditions one can impose on topoi to make limits behave correctly. I am not that familiar with SGA but it is my impression that this expose only discusses abelian ...

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### Topos with enough projectives

It is often observed that every presheaf topos has enough projectives, as a corollary of the result that representables are projective and every presheaf is a colimit of representables. We also have ...

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### Is there a version of the “infinitary” disjunctive normal form theorem for topoi and slice categories?

According to nLab
Proposition 2.3. A complete Boolean algebra is completely distributive iff it is atomic (a CABA), i.e., is a power set as a Boolean algebra.
This is basically an infinitary ...

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### What is a spectrum object in $\infty$-topoi?

For any spectrum $E$, there is a "discrete" topos spectrum $(Spaces / E_n)_n$. And I believe any topos spectrum is a localization of a "discrete" one. Are there any "non-discrete" topos spectra?
To ...

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294 views

### Intuition for pseudo-points and the inductive step in Johnstone's proof of Deligne's completeness theorem

In Johnstone's Topos Theory appears the following lemma.
7.41 Lemma. Let $P$ be a pseudo-point of $\mathsf C$, $X$ a $J$-sheaf on $\mathsf C$, and $x,y$ two distinct element of $P(X)$. Let $(V_j\to V)...

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### Are dualizable objects in the derived category of a ringed topos perfect?

Recall that an object $a$ in a symmetric monoidal category $(\mathcal{C}, \otimes, e)$
is dualizable if there exists an object $b$ and morphisms $\varepsilon\colon b \otimes a \to e$
and $\eta\colon e ...

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### What are the monomorphisms of ($\infty$-)toposes?

There are standard notions of "surjections" and "embeddings" of toposes. However, not every surjection is an epimorphism, and not every regular monomorphism is an embedding. (EDIT: as Alexander ...

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### Are there continua in $\infty$-topoi?

If topology were invented for algebraic geometry or logic, in ignorance of Euclidean space, we might reasonably regard connected compact Hausdorff spaces as pathological, or even doubt their existence....

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### 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 ...

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262 views

### What are the “smallest” topoi?

Yesterday I was talking to somebody from the Haskell community.
Late in the night we found ourselves discussing possible topoi.
Lets order topoi (up to equivalence, ...) by number of objects/...

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340 views

### When is the category of models of a limit theory a topos?

If $\mathcal{E}$ is a Grothendieck topos on a small base, then it is locally presentable, and hence is equivalent to the category of models of some limit theory.
Is there a characterization of ...

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276 views

### Points of the big Zariski site

It's relatively simple to show that the geometric morphisms $ \mathbf{Set} \to \mathrm{Sh}(\mathbf{CRing}^\mathrm{op}_{\mathrm{fp}}, \mathrm{Zar})$ correspond to local rings.
More precisely, since ...

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### A finiteness property of profinite sets

I would like to understand the canonical topology on the category of profinite sets. Unless I am making mistakes, this translates to the following question in point set topology:
Say $X$ is a ...

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### Internal van Kampen colimits

Let $C$ be a category with pullbacks. Recall that a colimit in $C$ is van Kampen if it is preserved by $C/(-): C^{op} \to CAT$, $c \mapsto C/c$.
We can $C$-internalize everything in sight:
Let $\...

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### What does an ideal correspond to in the internal language of sheaves?

Suppose I have a sheaf $\mathcal F$ in some topos $\mathrm{Sh}(\mathcal C)$. Then this becomes the sheaf of rings from algebraic geometry when described as a ring in the internal language of the topos....

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### The étale topos of a scheme is the classifying topos of which groupoid?

[Sent here from Math.StackExchange by suggestion of an user.]
By a theorem of Joyal and Tierney, every Grothendieck topos is the classifying topos of a localic groupoid. It has been proved (e.g. C. ...

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### Why sheaves are important and why do we care about them? [closed]

Presheaves are contravariant functors from a category $C$ to the category $Set$, that is functors $P$:
$$P:C^{op}\to Set.$$
For every topology $J$ on $C$ we can generate a reflexive subcategory
$$Sh(...

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### Categorification of spaces and models for set theory

One aspect of topos theory is that it provides an enlarged view of the classical concept of space. Indeed, one may thought that the notion of topos is a sort of categorification of the notion of space....

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### 2-natural operations on toposes

Any pseudonatural endomorphism $\Phi$ of the forgetful 2-functor $U:Topos^{coop}\to Cat$ is essentially determined by its component $\Phi_{Set}$. But which endofunctors of $Set$ induce such a $\Phi$? ...

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### Left Kan extension along Yoneda of pullback-preserving functor preserving pullbacks

Let $F\colon C \to D$ be a functor. The Kan Extension of $y_D \circ F$ along $y_C$ yields a functor $F_!: Fun(C^{op},Set) \to Fun(D^{op},Set)$. Here, $y_C$ and $y_D$ denotes the respective Yoneda ...

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### Example of non-“propositional” local operators on a topos?

Let $\mathcal{E}$ be a topos, and let $\top\colon1\to\Omega$ be its subobject classifier. We refer to global elements $P\colon 1\to\Omega$ as propositions; they form a poset, denoted $(|\Omega|,\leq)$....

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271 views

### Is there something like internal language of an abelian category?

While studying topos theory I was wondering if there is something like internal logic of an abelian category. Aparently the answer is yes
(by 7º slide in https://www.mimuw.edu.pl/~gael/xxi/files/...

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### How do (co)limits in posets of subobjects relate to (co)limits in ambient category?

Sorry if this is elementary.
Let $C$ be a category, $X$ an object in $C$ and let $S(X)$ denote the poset of subobjects of $X$.
According to the nlab entry, if $C$ has all limits and co-limits, so ...

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### Tietze transformations for sites of toposes

I have read that people think of a site as a presentation of the corresponding sheaf topos. For instance, on page 7 of this text by Caramello: as Grothendieck observed himself, a site of definition ...

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### Is there a universal way to force the Axiom of Choice to be true?

Given a model of set theory $V$ there are various ways to construct a model in which the Axiom of Choice holds, such as Gödel's constructible universe $L^V$ or by using forcing*. I'm wondering if any ...

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### epimorphism of fppf sheaves is an fppf morphism

I asked this question on math.stackexchange (https://math.stackexchange.com/questions/2693471/epimorphism-of-fppf-sheaves-is-an-fppf-morphism) but didn't get an answer. Maybe someone here can help.
...

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159 views

### Locality in Grothendieck Topologies

Let $\mathcal{C}$ be a category and $\mathcal{J}$ be a Grothendieck topology on it (i.e., $(\mathcal{C},\mathcal{J})$ is a site). Then what is a good notion of locality in it?
I came up with the ...

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### When the global section functor is a Cartesian fibration?

Given a Cartesian fibration $p : \mathbf{E} \to \mathbf{B}$ over an $\infty$-topos the paper by Marc Hoyois mentioned in his answer to this question gives some sufficient conditions for $\mathbf{E}$ ...

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### A locally presentable locally cartesian closed category that is not a quasitopos

This question asks for a locally presentable locally cartesian closed category that is not a topos. All the answers given (at least in the 1-categorical case) are quasitoposes. What is an example of ...

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### Example of a locally presentable locally cartesian closed category which is not a topos?

The only way I know to get a locally cartesian closed category which is not a topos is to start with a topos and then throw out some objects so that the category is not sufficiently cocomplete to be a ...

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### Generalizations of tangent $\infty$-topos

If $\mathbf{H}$ is an $\infty$-topos, then we can define a Cartesian fibration $p : T \mathbf{H} \to \mathbf{H}$ such that the fiber of $p$ over $X$ is the $\infty$-category of spectrum objects in $\...

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### Can the subobject classifier be covered by an object in a subtopos?

Let $\mathcal{E}$ be a topos and $\Omega$ its subobject classifier.
Is it possible to have a nonidentity local operator (a.k.a Lawvere-Tierney topology) $j\colon\Omega\to\Omega$, a $j$-sheaf $X\in\...

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### Configuration spaces, Ran spaces, free semilattices, Vietoris spaces and power objects

These are five important constructions and I would like to know how they are related.
The $n$th unordered configuration space of a space $X$ is
$$
\operatorname{UConf}_n(X):=\{\text{embeddings of $\{...

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### Formal Group Laws in a lined topos

I am aware of the following: in the context of synthetic differential geometry (SDG) one obtains a Lie algebra by exponentiating a microlinear group by a standard infinitesimal object and taking the ...

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### How are the left and the right group of a bitorsor related?

This question arose from my answer to To what extent does a torsor determine a group: it turns out that I do not know one thing about it.
Let $G$, $G'$ be groups in some nice enough category (you may ...

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### Cauchy real numbers with and without modulus

In constructive mathematics there are many possible inequivalent definitions of real numbers. The greatest variety seems to be in Dedekind-style approaches: in addition to "the" Dedekind real numbers ...

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### Reference to an explixit construction of a locale from a measurable space

In A sheaf theoretic approach to measure theory shows that measures on a measurable space are equivalent to measures on some locale whose open sets are the $\sigma$-ideals of the $\sigma$-algebra. The ...

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### Are there any useful conditions for a biclosed monoidal structure on presheaves to descend to a biclosed monoidal structure on sheaves?

Suppose $C$ is a small category with a monoidal structure. Then by the special case of the Day convolution theorem for presheaves, $\operatorname{Psh}(C)$ is equipped with a corresponding biclosed ...

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### Can we internalize topological fixed point theorems in an effective topos?

Reflective oracles are a kind of Turing oracle that give stochastic answers about the outputs of Turing machines. This works in a self-referential way, where they can answer queries about Turing ...

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### Sufficient cohesion and conservativity of “underlying stuff”

Consider a category $\mathsf C$ admitting a quadruple adjunction as below.
$$(\Pi_0 \dashv \text{disc} \dashv \Gamma \dashv \text{codisc}) :
\mathsf{C}
\stackrel{\stackrel{\longrightarrow}{\...

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### Topos properties from coverage conditions

For any category $C$ and coverage $J$ on it, let $\mathcal{E}:=\mathsf{Shv}(C,J)$ denote topos of sheaves on the site $(C,J)$. What sorts of results are known about the relationship between properties ...

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### Universal basis for graph analog of Boolean circuits?

The NAND gate is universal for Boolean circuits. That means that any function $2^n \to 2$ can be built out of NAND gates.
Let $\Omega$ be the subgraph classifier, i.e., the graph with two vertices $t$...

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### Relationship between synthetic differential geometry and differential cohesion?

I'm a big fan of synthetic differential geometry (or smooth infinitesimal analysis), as developed by Anders Kock and Bill Lawvere. It's a beautiful and intuitive geometric theory, which gives ...

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### Characterize constant objects in the internal language of a topos?

A Grothendieck topos $\mathcal{E}$ is equivalent to the category of sheaves on some site $Q$. We say a sheaf $X\colon Q^{\text{op}}\to\mathsf{Set}$ is constant if it is the sheafification of a ...

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### Examples of locally-but-not-globally (pre)sheaf toposes?

Re-reading my own recently posted question What is the total space of a stack after all? I realized that I don't know something more simple and presumably more basic.
Are there (bounded if you like) ...

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### What is the total space of a stack after all?

From my general experience I think for myself of what follows as some kind of taboo question for some reason: in my imagination, everybody wants an answer to this but somehow thinks it shall not be ...

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### What are the most general conditions under which the inverse image of sheaves of abelian groups has a left adjoint?

If $f: E \to X$ is an étale map (a local homeomorphism), then the inverse image of sheaves of abelian groups $f^{-1}$ has a left adjoint, as shown by Roland in his answer here. This subsumes as a ...