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13
votes
4answers
158 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
40 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
votes
1answer
49 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
votes
1answer
29 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
votes
1answer
52 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
votes
1answer
40 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
35 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 ...
2
votes
1answer
35 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 ...
0
votes
0answers
60 views
A locally constant sheaf on a locally connected space is a covering space; Proof?
As part of my hobby i'm learning about sheaves from Mac Lane and Moerdijk. I have a problem with Ch 2 Q 5, to the extent that i don't believe the claim to be proven is actually true, currently. Here ...
1
vote
0answers
40 views
Equivalence of groupoids induced by a G-torsor
I'm struggling again with Joyal-Tierney's "Strong stacks & classifying spaces", and in the proof of the main theorem of the paper, where stacks are characterized as internal groupoids $\mathbb G$ ...
5
votes
1answer
97 views
Elephant: how do I prove Lemma 2.1.7, section C2.1?
I'm referring (also for notations and terminology) to P. Johnstone, Sketches of an Elephant. A Topos Theory Compendium. Volume I. Clarendon Press. Oxford, 2002. The Lemma can be found at page 540.
I ...
6
votes
1answer
57 views
Glueing sheaves on Grothendieck sites
Let $X$ be a topological space and $\{V_i\}$ a cover of $X$. Let $F_i:\mathsf{Open}(V_i)^{\mathrm{op}}\to \mathsf{Sets}$ be a family of sheaves. One can glue this family to obtain a sheaf ...
5
votes
0answers
49 views
Essential geometric morphism seen topologically
I know that any geometric morphism between toposes of sheaves on spaces $f^*\colon Sh(X)\leftrightarrows Sh(Y)\colon f_*$ comes from a continuous map $f\colon X\to Y$. But what does it mean for $f$ ...
4
votes
0answers
58 views
Products of sites
Does the category of sites (i.e. small categories equipped with a Grothendieck topology) has products? Is there a connection to the product of locales (as discussed in Johnstone's Stone spaces, ...
2
votes
1answer
35 views
Joyal-Tierney definition of locally isomorphic objects
I am struggling with Joyal-Tierney's paper Strong stacks and classifying spaces, (appeared in "Category Theory (Como, 1990)", volume 1488 of LNM, pp. 213–236, Springer 1991).
In particular one of the ...
3
votes
2answers
57 views
If $\mathcal F$ is a sheaf, then is $\mathcal F (- \times X)$ a sheaf?
Let $\mathcal F$ be a sheaf of sets on a site. Fix an object $X$ of the underlying category of the site, which is assumed to contain a final object and have products. Define a presheaf $\mathcal G$ ...
9
votes
2answers
103 views
Natural numbers objects in topoi: Recursion in a parameter
I am currently trying to prove an exercise from Sheaves in Geometry and Logic: A First Introduction to Topos Theory by Mac Lane and Moerdijk about natural numbers objects.
First, we have the ...
3
votes
1answer
93 views
Reference for forcing using topos theory
I've just saw in Maclane and Moerdijik's book ("Sheaves in Geometry and Logic: A First Introduction to Topos Theory") about the Cohen forcing viewed in a categorical way using Topos theory. Is there ...
0
votes
1answer
41 views
Is this thing K-finite?
This is related to this question:
Freyd's Geometric Finiteness : An Example Computation
I've essentially reduced the problem to the following question:
Equip $\mathbb{N}$ with the discrete ...
4
votes
0answers
72 views
Freyd's Geometric Finiteness : An Example Computation
In his paper "Numerology in Topoi" available here:
http://www.tac.mta.ca/tac/volumes/16/19/16-19abs.html
Peter Freyd defines an object $A$ in a topos $\mathcal{E}$ to be geometrically finite if ...
2
votes
1answer
65 views
Is the category of sheaves on a site always abelian?
Let $\mathcal{C}$ be a site and $\mathcal{A}$ be an abelian category.
Suppose that the category of presheaves
$$
Psh(\mathcal{C},\mathcal{A}) = \operatorname{Fun}(\mathcal{C}^{op},\mathcal{A})
$$
is ...
1
vote
0answers
47 views
A sheaf of cumulative hierarchies
I've recently read a paper about sheaf forcing in which a sheaf of cumulative hierarchies was defined (defintion 5.3 on page 30). The same object is described in this English paper (defintion 3.1 on ...
5
votes
2answers
210 views
What fragment of ZFC do we need to prove Zorn's lemma?
It is extremely well-known that Zorn's lemma is a theorem of ZFC. My interest is in a certain finitely-axiomatisable fragment of ZFC, sometimes called RZC (restricted Zermelo with choice) or ZBQC. The ...
3
votes
1answer
66 views
What does the maximal basis for a Grothendieck topology look like?
There is the notion of Grothendieck topology on a category $\mathbf{C}$. This is a specific assignment of a collection $J(C)$ of sieves to each object $C\in \mathbf{C}$.
A Grothendieck pretopology or ...
2
votes
1answer
49 views
Does a natural transformation on sites induce a natural transformation on presheaves?
Suppose $C$ and $D$ are sites and $F$, $G:C\to D$ two functors connected by a natural transformation $\eta_c:F(c)\to G(c)$.
Suppose further that two functors $\hat F$, $\hat G:\hat C\to\hat D$ on the ...
4
votes
1answer
220 views
Logic and geometry
By delving into topos theory and sheaves one will eventually discover a "deep connection" between logic and geometry, two fields, which are superficially rather unrelated.
But what if I have not the ...
3
votes
1answer
110 views
Why do we need a pullback for the definition or classification of subobjects?
Regarding the subobject classifier construction, why do we need the pullback?
Monos from $U$ to $X$ are called subobjects, but I see that there might be injections which just have elements of the X ...
6
votes
3answers
183 views
Does the category framework permit new logics?
It appears to me that a topos permits a broader concept of subsets than the yes/no decission of a characteristic function in a set theory setting. Probably because the subobject classifier doesn't ...
2
votes
1answer
40 views
Example of a functor from the slice topos into the whole topos which does not preserve monomorphisms
For an object $c$ of a site $C$ with terminal object $*$ there is a functor
\begin{equation}
f:\operatorname{Sh}(C)/c\to \operatorname{Sh}(C)/*=\operatorname{Sh}(C)
\end{equation}
from the slice topos ...
1
vote
1answer
45 views
Right adjoint to the product in an over category
Let $\mathbb{C}$ be a small category and $X\in \mathbb{C}$ an object. The category $\widehat{\mathbb{C}}$ of presheaves on $\mathbb{C}$ is cartesian closed, i.e. each product
\begin{equation}
-\times ...
3
votes
0answers
146 views
Where else has Proposition B1.3.17 in the Elephant been proved?
This is a sort of reference request. Proposition B1.3.17 in Johnstone's Elephant reads:
Proposition 1.3.17 Let $\mathcal{S}$ and $\mathcal{T}$ be categories with pullbacks, $F \colon \mathcal{S} ...
1
vote
1answer
64 views
Question on the relation between sheaves over an object and sheaves on a category over that object
The Proposition appearing in the wonderful answer of Zhen Lin to this other question states that for a small category $\mathbb{C}$ and an object $X\in \mathbb{C}$
(*)\begin{equation}
...
1
vote
2answers
186 views
What is a Lawvere-Tierney topology?
I've read some articles and books for the definition and use of Lawvere-Tierney topologies, but I still don't understand their role.
Some people introduce these topologies as modal operators for ...
10
votes
1answer
171 views
Does this notion of morphism of noncommutative rings appear in the ring theory literature?
Definition: Let $R, S$ be two rings. A classical morphism $\phi : R \to S$ is a function from elements of $R$ to elements of $S$ which restricts to a homomorphism (of rings, in the usual sense) on ...
0
votes
1answer
166 views
Sheaves in Grothendieck Topologies
Let $S$ be a scheme and the category of $S$-schemes be equipped with one of the standard Grothendieck topologies, say étale or fppf.
Let $G \rightarrow H$ be a morphism of abelian sheaves on this ...
0
votes
1answer
69 views
How to verify sheaf condition in this example?
I am learning about sheaves of sets on a site with a subcanonical topology and have a question.
$f:A\rightarrow Hom(-,X)$ is a map from a pre-sheaf $A$ (for which I want to verify sheaf condition) ...
1
vote
1answer
96 views
Composing covers with epis
I am beginner of sheaf-theory and beg your pardon for this maybe silly question.
Let $\mathcal{C}$ be a Grothendieck site and $T$ the category of sheaves on $\mathcal{C}$ and let $f:X\rightarrow Y$ ...
0
votes
1answer
108 views
Counterexample for a pullback-pushout situation
Suppose you are in the category of sets or more generally in a topos (i.e. sheaf topos) and $f:A\rightarrow C$, $g:B\rightarrow C$ are two morphisms.
There is a canonical map $u:D\to C$ from $D$ ...
5
votes
1answer
417 views
What is Mazzola's “Topos of Music” about?
Disclaimers: I am neither a musician, nor I want to discredit Mazzola's work. Corollary of the first point: please use a plain style, without technical terms in the area of Music Theory. Corollary of ...
4
votes
1answer
79 views
How to get a geometric morphism out of a section? (And general pedagogy on classifying toposes)
Let $\mathcal{E}$ and $\mathcal{F}$ be toposes, $X$ an object of $\mathcal{E}$ and $p: \mathcal{E}/X \rightarrow \mathcal{E}$ the canonical geometric morphism (whose inverse image part is pullback ...
5
votes
2answers
138 views
Exercise from Leinster's Informal introduction to topos theory
Forgive the basic question (and the typesetting!) I'm a relative novice regarding category theory, but I've recently decided to teach myself at least the rudiments of toposes. Having stumbled upon Tom ...
5
votes
0answers
113 views
Why are injective $\mathscr{O}$-modules flasque?
Let $X$ be a topological space, and let $\mathscr{O}$ be a sheaf of rings on $X$. It is easy to verify that the functor $\Gamma (U, -) : \textbf{Mod}(\mathscr{O}) \to \textbf{Ab}$ is representable, ...
2
votes
0answers
124 views
Does the restriction functor (big to small) commute with the inverse functor?
I would like to know whether the restriction functor commutes with the inverse image functor.
(I basically follow the terminology in "Topological and Smooth Stacks".)
Let $X$ be a topological space. ...
6
votes
1answer
168 views
An example of a Grothendieck topology
A Grothendieck topology on a category $\mathcal{C}$ with finite limits consists of, for each object $U$ in $\mathcal{C}$ a collection $\text{Cov}(U)$ of sets $\{ U_i \to U \}$ such that
Isomorphisms ...
4
votes
3answers
206 views
Example of a small topos
I'm currently trying to understand this article by T. Noll on the topos of triads in music theory (also, this)
However, I can't get past section 2.2 where Noll introduces the subobject classifier, ...
4
votes
1answer
102 views
Does a geometric morphism $f\colon \cal E\to F$ preserves and reflects the subobject classifier?
I'm stuck in the apparently easy exercise in the title; I tried to prove it twice but both arguments were flawed (one of the two: one can easily obtain a natural map $Sub_\mathcal E(A)\to Sub_\mathcal ...
3
votes
0answers
78 views
Is $(f_*A)\times B\to f_*(A\times f^*B)$ an iso in a elementary topos?
One can easily show by adjunction-nonsense that if we are given toposes $\cal E,F$ and a geometric morphism $f\colon \cal E\to F$ then there exist canonical arrows
$$
\begin{gather}
(f_*A)\times ...
17
votes
1answer
336 views
How to find exponential objects and subobject classifiers in a given category
In a course I'm learning about Topos theory, there are a lot of exercises which require you to prove explicitly some category is an elementary topos: i.e. to construct exponentials and a subobject ...
4
votes
2answers
198 views
Importance of 'smallness' in a category, and functor categories
I feel like, having spent a little time doing category theory now, this is probably a silly question, but I keep coming up to many things (definitions, examples etc.) where smallness is required. I ...
3
votes
2answers
123 views
Are abelian groups in a [elementary] topos $\mathcal E$ an abelian subcat of $\mathcal E$?
The title tells everything: an abelian group object in a category $\mathbf C$ with finite products is a triple $(G,m,e)$, $m\colon G\times G\to G$, $e\colon 1\to G$ such that the well known diagrams ...
