Articles of topos theory

Grothendieck Topology on the Category of Elements

We are given a site $(C, J)$ for a small category $C$ and a Grothendieck topology $J$. If $F\in Sh(C, J)$, we take the natural topology $J_F$ on its category of elements $el(F)$ induced by $J$. I am wondering if $Sh(C,J)_{/F}\simeq Sh(el(F),J_F)$?

Quantificators vs pullbacks

Let $\mathscr C$ be a cartesian category with subobject classifier $\Omega$ and generic subobject $\tau:I\to\Omega$. Consider the following pullback square: $\require{AMScd} \begin{CD} P @>{\bar f}>> C\\ @V{\bar g}VV @V{g}VV \\ A @>{f}>> B \end{CD}$ I proved that for any $p:A\to\Omega$ the statement $\exists_{\bar f}(p\circ\bar g)\implies \exists_f(p)\circ g$ is true. What about the converse: $\exists_f(p)\circ g\implies\exists_{\bar […]

A Question on a claim regarding the notion of “space” in “Indiscrete Thoughts”

I’m reading Gian-Carlo Rota’s book “Indiscrete Thoughts“. In page 220 I came across a strange quotation with very few explanations: We thought that the generalizations of the notion of space had ended with topoi, but we were mistaken. We probably know less about space now than we pretended to know fifty years ago. As mathematics […]

The dense topology

The definition of the dense topology confuses me. If $C$ is a category and $X \in C$, a sieve $S$ on $X$ is a covering for the dense topology iff for every $f : Y \to X$ there is some morphism $g : Z \to Y$ such that $f \circ g \in S$ (Mac Lane, […]

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 far I’ve seen Lawvere Theories show up a lot.] What’s the best way to learn them both? [EDIT 2: What […]

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 the second: don’t take my disbelief in Mazzola’s work as an offense. 😉 So, the question is: what is Mazzola’s “Topos […]

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. However, I can’t find a single example on the nLab of what such a proof actually looks like. (The nLab has […]

What does it take to divide by $2$?

Theorem 1 [ZFC, classical logic]: If $A,B$ are sets such that $\textbf{2}\times A\cong \textbf{2}\times B$, then $A\cong B$. That’s because the axiom of choice allows for the definition of cardinality $|A|$ of any set $A$, and for $|A|\geq\aleph_0$ we have $|\textbf{2}\times A|=|A|$. Theorem 2: Theorem 1 still holds in ZF with classical logic. This is […]

How much set theory does the category of sets remember?

Question. Let $M$ be a model of enough set theory. Then we can form a category $\mathbf{Set}_M$ whose objects are the elements of $M$ and whose morphisms are the functions in $M$. To what extent is $M$ determined by $\mathbf{Set}_M$ as a category up to equivalence? For example, suppose $M$ and $N$ are models of […]

Concrete description of (co)limits in elementary toposes via internal language?

In the category of sets, limits and colimits can be concrete described respectively as subobjects of products and quotients of coproducts. It seems like these descriptions make sense in any elementary topos. I was wondering whether they actually describe the corresponding limits and colimits, and how to see this. Clarification. I’m asking about concrete element-based […]