All questions with tag [math: topos-theory]


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 $f:X\rightarrow Y$ be an epic morphism in $T$ into a representable sheaf $Y$. I have a general lack of understanding how such epic morphisms look like and this leads to...
2022-07-25 20:47:06

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 $D$ (defined as the pushout of the diagram $A\leftarrow A\times_C B\rightarrow B$ consisting of the two projections) into $C$. Presumably $u$ doesn't have t...
2022-07-25 20:01:20

What is Mazzola's "Topos of Music" about?

Disclaimers : I am neither an artist, neither I intend to reject Mazzola is job. Effect of the first factor: please make use of a simple design, without technological terms in the location of Music Theory. Effect of the secondly: do not take my shock in Mazzola is job as a crime. ;) So, the inquiry is: what is Mazzola is "Topos of music&q...
2022-07-25 10:21:51

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 along $p: X \rightarrow 1$.) I am trying to figure out the correspondence between geometric morphisms $\mathcal{F} \rightarrow \mathcal{E}/X$ over $\m...
2022-07-24 06:24:46

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, and that the representation is moreover natural in $U$. An easy argument [Hartshorne, Algebraic Geometry, Ch. III, Lem. 2.4] then shows that any inje...
2022-07-16 17:39:08

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 Leinster's excellent introductory paper, I'm getting most of the concepts, but I'm still a little woolly on category theoretic proof, and I've bee...
2022-07-16 16:44:33

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. The small and big sites of $X$ are denoted $\mathrm{Op}(X)$ and $\mathbf{T}/X$ respectively. A small sheaf over $X$ is a sheaf over the s...
2022-07-14 02:00:02

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 are covers, e.g if $V \to U$ is an isomorphism then $\{ V \to U \} \in \text{Cov}(U)$ Transitivity: If $\{V_i \to U \}$ and $\{V_{ij} \to V_i \}$ a...
2022-07-12 14:50:28

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, most probably because I don't know much about topos theory. I read the definition for topoi, but I can't get the intuition behind it so my question is...
2022-07-11 06:26:32

Does a geometric morphism $f\colon \cal E\to F$ preserves and reflects the subobject classifier?

I'm embeded the evidently very easy workout in the title ; I attempted to confirm it two times yet both debates were flawed (among both: one can conveniently get an all-natural map $Sub_\mathcal E(A)\to Sub_\mathcal E(f_*A)$, yet this is hardly ever an equivalence). A close friend of mine recommended me a counterexample yet I can not fetch his e...
2022-07-11 06:22:11

Is $(f_*A)\times B\to f_*(A\times f^*B)$ an iso in a elementary topos?

One can conveniently show by adjunction - rubbish that if we are offered toposes $\cal E,F$ and also a geometric morphism $f\colon \cal E\to F$ after that there exist approved arrowheads $$ \begin{gather} (f_*A)\times B\to f_*(A\times f^*B)\\ f^*(A\times f_*B)\to (f^*A)\times B \end{gather} $$ Working in groups of sheaves (as an example $\m...
2022-07-08 20:38:52

How to find exponential objects and subobject classifiers in a given category

In a training course I'm learning more about Topos theory, there are a great deal of workouts which need you to confirm clearly some group is a primary topos: i.e. to construct exponentials and also a subobject classifier, and also to show that it has all limited restrictions. Regardless of having actually functioned my means via a variety of th...
2022-07-07 02:19:46

Importance of 'smallness' in a category, and functor categories

I seem like, having actually invested a little time doing category theory currently, this is possibly a foolish inquiry, yet I maintain coming near several points (interpretations, instances etc) where smallness is called for. I consistently fall short to see why this is: I can see why smallness (or neighborhood smallness) is a valuable buildin...
2022-07-06 03:34:26

Are abelian groups in a [elementary] topos $\mathcal E$ an abelian subcat of $\mathcal E$?

The title informs every little thing: an abelian team object in a group $\mathbf C$ with limited items is a three-way $(G,m,e)$, $m\colon G\times G\to G$, $e\colon 1\to G$ such that the popular layouts commute and also such that $m\circ \sigma_{GG}=m$, where $\sigma_{AB}\colon A\times B\to B\times A$ is the map transforming $\mathbf C$ right int...
2022-07-04 21:20:41

What are the algebras of the double powerset monad?

Let $\mathscr{P} : \textbf{Set} \to \textbf{Set}^\textrm{op}$ be the (contravariant) powerset functor, taking a set $X$ to its powerset $\mathscr{P}(X)$ and a map $f : X \to Y$ to the inverse image map $f^* : \mathscr{P}(Y) \to \mathscr{P}(X)$. By elementary abstract nonsense, we know that $\mathscr{P}$ has a right adjoint $\mathscr{P}^\textrm{o...
2022-07-01 20:15:25

Locating a copy of a thesis

Does any person have a communicable duplicate of the thesis "Logical and also cohomological facets of the room of factors of a topos" by Carsten Butz? The link given in appears to be dead, and also I can not appear to locate a duplicate in other places.
2022-06-28 23:12:24

Fuzzy logic and topos theory

Why does not one create fuzzy logic by expanding topos theory, by merely expanding the subobject classifier $\Omega$ to the device interval [0,1 ]? Have individuals done that?
2022-06-09 14:12:33

What do coherent topoi have to do with completeness?

There is a theory of Deligne that a "coherent" topos (as an example one on a website where all things are seemingly - portable and also seemingly - apart) has adequate factors (i.e. isomorphisms can be identified using geometric morphisms to the topos of collections). I've heard it claimed that this is a kind of Goedel is efficiency th...
2022-06-08 06:10:43

The category of presheaves on a possibly-large category

Suppose $\mathcal{C}$ is a group such that for every single $c \in \mathrm{Ob}(\mathcal{C})$, the piece group $\mathcal{C}/c$ amounts a tiny group. I require to show that the group of presheaves $[\mathcal{C}^\mathrm{op}, \mathbf{Set}]$ is a primary topos. I recognize the typical argument made use of when $\mathcal{C}$ is a tiny group bdsh as a...
2022-06-04 21:17:34

Where is the well-pointedness assumption of ETCS used in everyday math?

Where is the well - pointedeness presumption of the Elementary concept of the group of collections (Lawvere is group - logical axiomatization of set theory) made use of in day-to-day mathematics? Especially, if you have a topos with all-natural numbers object (think selection if you intend to), what acquainted theories do not hold? I've listene...
2019-05-29 11:33:22