Questions tagged [topos-theory]
A topos (plural topoi, toposes) is a category that behaves like the category of sheaves of sets on a topological space. Topos theory consists of the study of Grothendieck topoi, used in algebraic geometry, and the study of elementary topoi, used in logic.
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Introduction Of Mitchell-Bénabou-Language
I'm confused about the introduction of the Mitchell-Bénabou-language in Sheaves in Geometry and Logic:
Here are the inductive clasues which simultaneously define the terms of the language and their ...
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Constructing injective resolution on big sites.
Let's work over a category of varieties on some field. Let $\mathcal{F}$ be a presheaf on varieties that satisfies Zariski/etale sheaf conditions whenever restricted to the opens of a specific variety....
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Counterexample in topos theory
Motivation: The definition of an elementary topos requires both a subobject clasiffier and either power objects or exponentiation. But also, if a category has power objects and a terminal object $\...
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Not understanding concretely the Wikipedia or text definition of "Subobject classifier"
I have two major questions.
In the following diagrams, only some of the arrows make sense to me or are explained. Others I have never seen and are not brought up in any of the webpages or books that I ...
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Deduce that a category is a topos (and locally cartesian closed) from a dense subcategory of powers of one object
I am trying to understand the paper Classical Lambda Calculus in Modern Dress. On page 7, it states
Proposition 2.9 shows inter alia that the products of the universal are dense, and so familiar ...
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$k:A\to 0$ implies $A\cong 0$
$
\def\aut{\text{Aut}}
\def\hom{\text{Hom}}
\def\id{\text{id}}
\def\true{\mathsf{true}}
\def\xra{\xrightarrow}
$In the SiGaL book, they prove that in an elementary topos, every $k:A\to 0$ is an iso, ...
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Is The Power Object Functor Faithful?
In an elementary topos, can we conclude that the power object functor $P:\mathcal{E}\to\mathcal{E}^\text{op}$ is faithful, i.e. if $Pf=Pg:PY\to PX$ then $f=g:X\to Y$?
Scroll down for (incomplete) ...
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Fundamental Description Of Algebras Over The Monad $PP^{op}:\text{Set}\to\text{Set}$
$
\newcommand{\eps}{\epsilon}
\newcommand{\op}{^\text{op}}
\newcommand{\id}{\text{id}}
\newcommand{\set}{\text{Set}}
\newcommand{\xra}{\xrightarrow}
$Can someone help me to understand how algebras ...
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Interpreting Diagrams About Power Object Functor
I'm reading Sheaves In Geometry And Logic and this diagram confuses me:
First, do those diagrams really live in $\mathcal{E}$ resp. $\mathcal{E}^\text{op}$ or rather in $[J^\text{op},\mathcal{E}]$ ...
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Internally Inhabited Object Which Is Not Externally Inhabited
What is the simplest example of a topos with an object $X$ which is internally ($X\to \mathbf{1}$ is epi) but not externally (no $\mathbf{1}\to X$) inhabited?
I think that for all categories $C$, in ...
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Condition for an equivalence of functor categories
Given small categories $\mathcal{C}$ and $\mathcal{D}$, we have that $[\mathcal{C}^\text{op},\textbf{Set}]\simeq[\mathcal{D}^\text{op},\textbf{Set}]$ if and only if the Cauchy-completions of $\...
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Regular functors $F\colon\mathcal C\to\mathcal E$ factor essentially uniquely through a topos $\tilde C$.
I want to show the following statement:
Given a small regular category $\mathcal C$ there exists a topos $\widetilde{\mathcal C}$ and a regular functor $F\colon\mathcal C\to\widetilde{\mathcal{C}}$ ...
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Indexed Yoneda in the Elephant
I've been trying to gather definitions and results to understand the internal Diaconescu theorem, for I wish to read Joyal-Tierney's An Extension of the Galois Theory of Grothendieck. My question ...
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What is the connection between algebraic groups and topoi?
I have a longstanding interest in topos theory. (See this MSE search of my questions about topos theory.) I am studying for a postgraduate research degree in linear algebraic group theory. Naturally, ...
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Presheaf problem
I need to show that the global element of a presheaf $P\in \widehat{\mathcal{O}(X)}$ is equivalent to global section of bundle of germs $\Lambda_P\xrightarrow{p}X$. In particular, I need help to show ...
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Is every topos a De Morgan topos?
I think I'm getting something wrong, but I don't see what that is. Does anyboy see where is the error in the following argument.
Let $\mathcal{E}$ be a nondegenerate topos. The topos $\mathcal{E}_{\...
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Sheaf topos is localisation at covering sieve inclusion
I know that the sheafification functor $a:Pr(C) \to Sh(C,J)$ is up to equivalence the localisation $Pr(C) \to Pr(C)[W^{-1}]$ at the class of those morphisms which $a$ inverts. But is it also the ...
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Stackification of finite categories
Assume I have a base category $\mathscr S$ with finite limits and a geometric morphism $\gamma :\mathscr S\to Fin$ into the category of finite sets (for example because $\mathscr S$ is positive ...
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Partial arrow classifier as an (adjunction-related) universal arrow
Goldblatt gives a brief overview of adjunctions in his "Topoi", and one of the exercises asks to characterise the partial arrow classifier in terms of some universal arrow.
Well, I gave it ...
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What does exactly mean "the spectral algebra $W_{A}$"?
I'm trying to study an introduction to topos quantum theory but I keep finding this concept that I don't understand fully. In "A Topos Perspective on the Kochen-Specker Theorem:
I. Quantum States ...
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subtoposes induced by generating collections
(I will use the terminology in the book {\em Sheaves in Geometry and Logic} by Mac Lane and Moerdijk.)
A collection ${\{ G_{\xi} \}}$ of objects in a category $\mathbf{A}$ is said to {\em generate} $\...
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Is the functor category of a small category into a topos itself a topos?
I am reading the book "Sheaves in Geometry and Logic" by MacLane and Moerdijk.
Early in the book, the authors show that $Sets^{\mathscr{C}^{op}}$ where $\mathscr{C}$ is a small category is ...
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On Atomistic Categories
Being impressed by Barr and Diaconescu paper entitled "Atomic Toposes" (https://www.math.mcgill.ca/barr/papers/atom.top.pdf), I would like to ask whether it makes sense to investigate "...
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Reconstructing isomorphisms via the bijection between the corresponding posets of subobjects - Part 2
In this post
Reconstructing isomorphisms via the bijection between the corresponding posets of subobjects
I asked for the possibility of constructing an isomorphism via the order-preserving bijection ...
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Order relations on natural number objects in topoi, and symmetry
Goldblatt in "Topoi" defines the relation ≤ on an NNO $N$ as the subobject $\langle \pi_1, \oplus \rangle : N \times N \rightarrowtail N \times N$, where $\oplus : N \times N \rightarrow N$ ...
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Topos isomorphic to the sheaf topos of the circle
When we abstract the notion of space, we get more and more isomorphisms. For example, the triangle, square and circle are not isometrical, but they are homeomorphic. If we take one step further to the ...
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Is there an reciprocity in topology?
Let $K$ be a global field. As explained in this nLab page, Artin reciprocity is an isomorphism between two abelian groups
$$K^\times \backslash \mathbb{I}_K / \mathcal{O} \xrightarrow{\sim} \...
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Morphisms of frames induce morphisms of sites
One can associate a site with an arbitrary frame by defining coverages with suprema. According to Johnstone (Sketches of an Elephant, 2.3.20), "If $A$ and $B$ are frames, made into sites via ...
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What if the sub-object classifier is the terminal object?
The title says it all. Essentially, I've seen problems where you prove that the sub-object classifier is the terminal object. However, I was wondering what that would actually look like when you start ...
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Equivalence of two definitions of sheaves over a Lawvere-Tierney topology
In an older text Toposes, Triples, and Theories that served as my first introduction to the general theory of (logical as opposed to Grothendieck) topoi, the definition used for a sheaf with respect ...
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codomain fibration of a topos is a stack
In fact, the condition can be weakened into these versions as in the forum post:
https://nforum.ncatlab.org/discussion/560/codomain-stacks/
Is there any proof of this fact (or some proofs that can be ...
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Condition for partial maps being representable [closed]
Let $\operatorname{Part}(-,Y)$ be the presheaf which takes an element $X$ to the set $\operatorname{Part}(X,Y)$ of partial maps into $Y$. We say that partial maps are representable if $\operatorname{...
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Limits and colimits in the category of presheaves
We know that the category of presheaves (i.e. $Fct(C^{Op}, Set)$ ) is an elementary topos and in particular it is finitely complete and finitely cocomplete and so it has all finite limits and colimits....
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What does "full first-order logic" mean?
I see this phrase in the categorical logic literature, but always undefined. What does the "full" mean in "full first-order logic"?
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What is a Boolean functor?
According to nlab, a coherent functor is a morphism of coherent categories.
And a Boolean category is a coherent category where subobject posets are always Boolean algebras. Is a Boolean functor a ...
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Is there a sheaf model where the Weak Markov's principle fails?
We define a real number $x$ to be pseudopositive if $\forall y \in \mathbb{R}$ we have $ \neg \neg (x > y) \vee \neg \neg (y > 0) $.
The Weak Markov's Principle (WMP) is the axiom that every ...
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Stack comparison lemma
Edit. I am realising that I was asking the wrong question. I was (without knowing it) interested in stack comparison lemmata similar to the lemma $$Sh(Sh(C,J),can) = Sh(C,J)$$
for sheaves. So my ...
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How to construct a group-valued sheaf from set-valued sheaf
Suppose that I have a set-valued sheaf $S$ on a site $(\mathcal C, J)$.
Question 1.) Is there a canonical way to turn $S$ into a sheaf with Abelian group values?
I considered the following: for each ...
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Proving $\mathcal{P}(g \circ f) = \mathcal{P}(g) \circ \mathcal{P}(f)$ in a general topos
I'm going through Goldblatt's "Topoi", and I'm stuck on exercise 12.5.3. There, he suggests proving that $\Omega^{g \circ f} = \Omega^g \circ \Omega^f$, where $\Omega^f$ is the morphism $f : ...
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Epi-monic factorization of a composition as a pullback
Suppose we're working in a topos. Let $f : a \rightarrow b$ be an arbitrary morphism, and $g : c \rightarrowtail a$ be monic. Let $fg* : c \twoheadrightarrow z$ and $f[g] : z \rightarrowtail b$ be the ...
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Singleton map is always monic
Let $\mathcal{T}$ be a topos and $X,Y$ objects in $\mathcal{T}$. Let $\delta_X=\langle 1_X,1_X\rangle\colon X\to X\times X$ be the diagonal map, $\Delta_X\colon X\times X\to\Omega$ the map which ...
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What does the connection to topology provide to topoi?
What does the connection to topology provide to topoi? I've been told by a professor that the connection to topology is one of the key features of topoi, and it provides a lot of the intuition and ...
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Relationship between power object and exponential object
Let $C$ be a category where power objects and exponential objects exist, as well as the terminal object $1$, and the coproduct of $1$ with itself, denoted by $2$.
Let $X$ be an object in $C$. Is there,...
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Classifying Topos of a Classical First-order Theory
This nlab page shows a chart describing the structure of the classifying topos of a theory in various logics, but it doesn't have an entry for the classifying topos of a classical first-order theory.
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terminal in presheaf topos of smooth maps
This is a follow up to this question.
For a smooth manifold $M$ let $\mathbb{R}^M$ be the ring of smooth maps from $M$ to $\mathbb{R}$. Now let $C$ denote the category of all such rings, where arrows ...
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Finding terminal object in presheaf topos.
I am interested in finding the terminal objects of a presheaf topos, and in particular, ones constructed via Yoneda embedding $\hat{C} = \mathrm{Set}^{C^{\mathrm{op}}}$ from a category $C$.
Starting ...
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Topoi of algebras vs co-algebras, right adjointness compared to left exactness
I am asking about topoi of coalgebras over a comonad and algebras over a monad.
The comonad statement I am aware of is as follows:
Let $\mathcal{E}$ be a topos. Then if a comonad $T \colon \mathcal{E}...
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Uniqueness of arrows between Heyting-valued sets
I’m going through Goldblatt’s “Topoi”, in particular through the subchapter 11.9 on Heyting-valued sets, and it feels overly… concise. In short, I’m not sure how to prove uniqueness of arrows.
For ...
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How to translate a statement from internal to external logic?
In the definition of predicative topoi by van den Berg we encounter the following definition of a collection square, for an ambient category that is assumed to be locally cartesian closed, lextensive ...
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Olsson's Algebraic Spaces and Stacks Paragraph 2.3.1
The paragraph reads:
Let $T$ be a topos, and let $\Lambda$ be a ring in $T$. Denote by $\operatorname{Mod}_{\Lambda}$ the category of $\Lambda$-modules in $T$. My question is how is the category of $\...
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