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0
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2answers
55 views

Can the equivalence between principle bundles and maps to classifying spaces be turned into an adjunction.

We have that $G-PBun(X)$, the category of topological principal bundles for a structure group $G$ is equivalent to $Top[X,BG]$ where $BG$ is the classifying space of $G$. This almost looks like an ...
2
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1answer
61 views

Reference for Cech cohomology on sites (not pre-topologies)

I'm searching for a reference dealing Cech cohomology on sites (not pre-topologies). In general, when dealing with Cech cohomology on sites, one admits that the category has finite limits so you can ...
3
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0answers
104 views

If one knows Homotopy Type Theory, what is the easiest way to learn Higher Topos Theory?

Just read Lurie's book? Do there exist any papers which explain higher topos theory with an audience of homotopy type theorists in mind?
1
vote
1answer
63 views

Proof that Beck-Chevalley holds for right adjoints iff it holds for left adjoints

I am looking at Bart Jacob's book "Categorical Logic and Type Theory". The proof of Lemma 1.9.7 is left as an exercise for the reader. It does not seem that easy to me, and i have had quite limited ...
5
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2answers
94 views

Is the map into the terminal object an epimorphism?

Let $C$ be a category with a terminal object $1$. Is the unique arrow from an object into $1$ necessarily an epimorphism? If not, is it an epimorphism if $C$ is a topos?
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0answers
46 views

Why $F^{+}$ is a monopresheaf?

I'm having difficult in proving that given a site $(C, J)$, then $F^{+}(c) = colim_{R \in J(c)} R $ is a monopresheaf, in the sense that there exists at most one lifting of $R \longrightarrow F^{+}$ ...
9
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1answer
121 views

Properties of the internal language of the category of sheaves.

Consider a simple case of sheaves on some topological space $X$, $\operatorname{Sh}(X)$ (recall that a sieve on $U$ is covering iff its $\operatorname{sup}$ is $U$). All of these are Grothendieck ...
7
votes
1answer
159 views

Is there any point in a logician studying $\infty$-categories?

My primary areas of interest lately have been set theory, logic, and category theory, so naturally topos theory has been a large part of what I'm learning (in between getting caught up on some other ...
1
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1answer
78 views

Example of a coproduct and epi preserving functor $F$ which does not preserve finite colimits

What is an example of a functor $F:T\to S$ between toposes $T$ and $S$ which preserves coproducts and epimorphisms but which does not preserve finite colimits?
5
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1answer
321 views

Volume 3 of Johnstone's “Sketches of an Elephant”

Recently, I read the Chapter 8 of Johnstone's "Topos theory" and got interested in the homotopy and cohomology theory of Grothendieck toposes. So I'm looking for the textbooks expanding these ...
2
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1answer
26 views

Question on connected and locally connected geometric morphisms.

Is the property of a geometric morphism being both connected and locally connected (in the sense of Part C of the Elephant) stable under pullbacks? I know local connectedness is, I am not sure about ...
6
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2answers
140 views

ETCS set theory: Are empty sets isomorphic?

Just a quick question about ETCS: Are any two empty sets isomorphic? Here, a set $X$ is empty if there exists no $x \in X$, i.e. no functions $x: 1 \to X$. The reason I'm asking is that I need this ...
2
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1answer
71 views

Is the subobject classifier of the sheaves the sheaffication of the one from the presheaves?

maybe this is an idiot question, but I could not solve it. Let $\Omega$ be the suboject classifier in the category $\mathbf{PSh}(X, J)$ where $(X, J)$ is a site, I know that $\Omega(U) \cong Nat(h_U, ...
2
votes
1answer
38 views

Question about Lemma D1.4.4(iii) in the Elephant - possible typo?

Given a morphism $[\theta] \colon \lbrace \bar{x}.\phi \rbrace \rightarrow \lbrace \bar{y}. \psi \rbrace$ in the syntactic category $\mathcal{C}_{\mathbb{T}}$ of a (cartesian) theory, we are told, in ...
4
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0answers
106 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 ...
7
votes
1answer
506 views

What's the best way to teach oneself both Category Theory & Model Theory?

I've done a bit of reading around both Category Theory & Model Theory (CT & MT) as a novice in each field. I'm interested in how they might combine, particularly when applied to Algebra. [So ...
0
votes
1answer
58 views

Transposition of arrows in toposes (or cartesian closed categories)

Suppose $\mathcal{E}$ is an elementary topos (take as a definition that of Mac Lane and Moerdijk "Sheaves in Geometry and Logic"). I have a problem with a fact concerning the cartesian closure of ...
2
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1answer
45 views

Elementary question about subpresheaves

I have encountered the following phrase which I do not understand. The letter $L$ is used for a locale but I guess (?) this holds for other categories as well. The terminal sheaf $1$ is such that ...
0
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1answer
54 views

Lifting of a comonad to the presheaf category

Suppose I have a comonad $G$ on a category $\mathbb{C}$. If $C$ is a preorder then I can define a comonad (i.e. interior operator) $\square$ on the set $\mathcal{P}\uparrow(C)$ of upwards closed ...
0
votes
1answer
56 views

Questions regarding a particular category with functors

I have two questions regarding categories and natural transformations. The first question is as follows. Say we have categories $A$, $B_1$ and $B_2$, with functors $F: A \to B_1$, $G: B_1 \to B_2$ as ...
0
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0answers
71 views

explicit proof of pullback stability of epics in a topos

As in Explicit construction of a initial object in a topos I'm looking an elementary proof of the fact that, in a topos, epimorphisms are stable under pullback or, equivalently, that images are ...
2
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2answers
84 views

Characterization of epimorphisms of sheaves on a site

I'm stuck with a detail in the proof of the characterization of epimorphism of sheaves on a site in the Mac Lane & Moerdijk book "Sheaves in Geometry and Logic". I want to prove that: "If ...
1
vote
2answers
102 views

Explicit construction of a initial object in a topos

Let $\mathcal E$ be a topos as in Mac Lane and Moerdijk. A initial object in $\mathcal E$ can be obtained as the domain of the equalizer of the morphisms $P!,\epsilon P1:P1\to P^31$, where $1$ is a ...
2
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0answers
103 views

Is there a topos in which the natural numbers object are the finite dimensional vector spaces?

I recall reading somewhere that there is a topos in which the Dedekind reals are exactly the measurable functions. Now vector spaces are prominently characterised by dimensionality. This prompted the ...
2
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2answers
45 views

Is the cartesian product of objects in an elementary topos cancellative?

My question is the internalization of this question to an elementary topos $C$. Is it true that: For objects $X,Y$ and $Z$ in an elementary topos $C$ with $X\times Y\cong X\times Z$, then also ...
1
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1answer
77 views

What is an object classifier and how does give a natural numbers object?

According to a history of topos theory by McLarty, Blass (1989) showed that the existence of an object classifier over a given topos implies that the topos has a natural number object. What is an ...
10
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3answers
324 views

Is there a category whose internal logic is paraconsistent?

The internal language of topoi is higher-order typed intuitionistic logic. Now according to wikipedia, the dual of intuitionistic logic, in some sense is paraconsistent. They say Intuitionistic ...
6
votes
1answer
248 views

What was the Lawveres explanation of adjoint functors in terms of Hegelian Philosophy?

I was contemplating asking this question on Philsophy.SE but felt it was better directed here as there are a dearth of category theorists there. According to the wikipedia entry on Categorical Logic: ...
1
vote
2answers
90 views

Do 'nice' first order logics have universal models?

A first-order logic is interpreted in a model where sentences of the logic can be said to be true or false. There may be more than one model, and we can identify morphisms between models. Do we have ...
8
votes
1answer
371 views

Examples of mathematical statements made with adjoint functors

I am wondering if it is possible to use the adjoint functors in topos theory for statements in analysis. Any examples would be warmly welcomed. Though I would prefer simpler, atomic, lemmas or ...
1
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0answers
88 views

Can curvature be defined in Topos Theory?

I'm by no means an expert in either category theory or topos theory, but I'm trying to gain some perspective on traditional geometric ideas in this context. Topos theory claims that it is a geometric ...
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1answer
745 views

Does “cheap nonstandard analysis” take place in a topos?

Terence Tao's A cheap version of nonstandard analysis describes a way to do analysis halfway between ordinary analysis and nonstandard analysis which, if I'm not mistaken, cashes out to working in the ...
2
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0answers
28 views

Are lax kernel pairs stable under change of base?

The question is immediate for kernel pairs, but when we work in a category enriched in posets, like the category of locales is, are the lax kernel pairs stable under change of base, or since perhaps ...
2
votes
2answers
85 views

Definition of ordinals in Grothendieck toposes

What is a useful definition of an ordinal in Grothendieck toposes. Useful in this context means that I would like to construct fixed points of monotone maps on complete lattices using iteration from ...
2
votes
1answer
81 views

Projective Objects in a Topos

I'm working a problem in MacLane and Moerdijk, and am not sure how to proceed. The end goal is to show that if $Sh(X)$ has enough projectives and $X$ is $T_1$, then $X$ has a basis of clopen sets. I ...
4
votes
1answer
265 views

Showing that the sheaf-functor $\epsilon: \tilde{\sf C} \to \tilde{\tilde{\sf C}}$ is an equivalence

Let $(\mathsf C,J)$ be a site. Then we have the category of sheaves $\tilde{\mathsf C}$ and the category $\tilde{\tilde{\sf C}}$ of sheaves over $\tilde{\sf C}$ (both considered with the canonical ...
7
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0answers
175 views

What does category theory say about chains of set inclusions $\dots\in c\in b\in a$?

I'm currently trying to understand to what extend a set theory like ZFC can be mirrored within category theory, i.e. topos theory. What appears as an obstacle to me is the axiom of regularity, which ...
2
votes
1answer
99 views

Does this right adjoint of a geometric morphism preserve directed colimits?

Let $E$ be a sheaf topos $E=\operatorname{Shv}(C)$ and $\{x_i:\operatorname{Sets}\to E\}$ be a set of geometric morphisms $x_i:Sets\to E$ such that the morphism $$ x^*\colon E\to \prod_i ...
4
votes
1answer
67 views

Sites for étale $G$-spaces

This is a problem Sheaves in Geometry and Logic by MacLane and Moerdijk (problem 3.10) Suppose we have a space $X$ on which a discrete group $G$ acts by homeomorphisms. Given an étale space $p:E ...
4
votes
1answer
139 views

Is the internal language of a topos complete, sound and effective?

The internal language of a topos is higher order intuitionistic typed logic. Now according to this article in wikipedia higher order classical logic with full semantics is never complete, sound or ...
5
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1answer
109 views

Presheaves over sieves and posets

I'm looking for a proof of the claim that given a poset P, the topos of presheaves over P is equivalent to the topos of presheaves over the complete Heyting algebra of sieves on elements of P. I found ...
5
votes
1answer
173 views

Grothendieck topology on pre/sheaves

Given a Grothendieck topology $T$, say subcanonical, on a category $C$, we are able to talk about sheaves in $(C, T)$. Since pre/sheaves can be viewed as generalized objects of $C$ (via Yoneda ...
14
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4answers
419 views

What does a proof in an internal logic actually look like?

The nLab has a lot of nice things to say about how you can use the internal logic of various kinds of categories to prove interesting statements using more or less ordinary mathematical reasoning. ...
1
vote
2answers
80 views

Plus construction of a presheave factors every sheaf-valued morphism.

I'm having some trouble understanding the correctness of some proof in Sheaves in Geometry and Logic (Mac Lane, Moerdijk). It concerns the lemma III.5.3 : If $F$ is a sheaf and $P$ a presheaf, ...
0
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1answer
93 views

Commuting square of functors

Let $\mathcal{E}$ be a complete and cocomplete category. Given a functor $i: \mathcal{C} \to \mathcal{D}$ between small categories, there is a triple of adjoint functors between their respective ...
2
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1answer
37 views

Commutativity of a sheaf of groups from an epimorphism

Let $F$ and $G$ be sheaves of groups $\mathcal{S}^{op}\to Groups$ and $f:F\to G$ an epimorphism (of sheaves of sets). If $F$ is a sheaf of commutative groups, is $G$ also a sheaf of commutative ...
4
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1answer
209 views

Pullback of sheaves and pullback of schemes

Let $\mathbb{G}_m$ the multiplicative group, with coordinate ring $\mathbb{C}[x^{\pm 1}]$, and considered as a sheaf of abelian groups over $\mathrm{Spec}\,\mathbb{C}$ in the Zariski topology. Let $X$ ...
0
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1answer
119 views

Two questions about monomorphisms and pullbacks in a Grothendieck topos

Let $\mathbf{C}$ be a Grothendieck-topos and $f:Y\to X$ a morphism. Is the pullback $W\times_X Y\to Y$ of a split monomorphism $g:W\to X$ along $f$ again a split monomorphism? I don't think this ...
1
vote
0answers
91 views

Plus construction of sheafification as a colimit of presheaves.

In Sheaves in Geometry and Logic, Moerdijk and Mac Lane construct the associated sheaf on a site $(\mathcal C, J)$ of a presheaf $P$ as $$ a(P) = (P^+)^+ ,$$ where $P^+$ is defined pointwise as ...
5
votes
1answer
72 views

What functors preserve subobject classifiers?

My question is exactly that of the title. Given a topos $T$, is there a natural sufficient condition for a functor $f$ from $T$ to another topos, $S$, to the subobject classifier? (By this I mean of ...