All Questions
Tagged with topos-theory soft-question
16 questions
11
votes
3
answers
671
views
Merging single-sorted and multi-sorted theories
The general theory of single-sorted (say, algebraic) theories is very similar to the general theory of multi-sorted (algebraic) theories. Each variable gets a sort, but apart from that nothing really ...
- 63.1k
6
votes
2
answers
523
views
Is the logic of the ($\infty$-?)topos of simplicial sets "contradictory up to homotopy"?
In a sense this is a followup to my earlier question Does the (1-)topos structure on simplicial sets have any homotopy-theoretic significance?.
In the topos of simplicial sets, the subobject ...
- 18.4k
1
vote
0
answers
49
views
Defining properties of categories out of an indicial category
$\newcommand{\Hom}{\operatorname{Hom}}$Suppose we want to define the category of arrows of $S$. Below are two forms of doing it.
Definition 1: If $D$ is of the following type: $\bullet \to \bullet$, ...
- 894
3
votes
4
answers
545
views
Phenomena of topos
These days I am wandering on a wild adventure in an incredible but intimidating land. Fortunately, I could find a guide to some animals of this land
Phenomena of gerbes
But someone said to me that ...
3
votes
0
answers
186
views
The site and the space
There is a (seemingly simple) statement in the literature on sheaf theory, namely,
If $E$ is the site of opens of a topological space $X$, the notion of sheaf over $X$ coincides with that of sheaf of ...
- 894
17
votes
3
answers
2k
views
Why do Grothendieck topologies used in algebraic geometry typically involve finiteness conditions?
There are many Grothendieck topologies used in algebraic geometry, with complex interrelations. Generally in one of these topologies, a cover of schemes is a family of maps which is jointly surjective ...
- 63.9k
8
votes
1
answer
735
views
Tohoku and cohomology of toposes
In McLarty's The Rising Sea: Grothendieck on simplicity and generality I found the following quote:
The same, Grothendieck knew, would work for cases yet unimagined. He notes that Tohoku [...
- 83
7
votes
1
answer
475
views
Comparing Kripke-Joyal semantics of toposes to model-theoretic satisfaction
Let $\mathcal E$ be a topos and $\varphi$ a statement formulating a property of toposes. There are two ways of checking whether $\mathcal E$ satisfies $\varphi$:
Consider the first-order language $L$ ...
- 73
10
votes
0
answers
1k
views
The "unification" of geometry via topos theory?
This question is somehow motivated by The unification of Mathematics via Topos Theory and Synthetic vs. classical differential geometry. Sorry if this is a naïve question.
There has been quite a lot ...
- 1,094
16
votes
2
answers
2k
views
Categorification of probability theory: what does a "probability sheaf" tell us (if anything) about probability theory?
Disclaimer: I only have a superficial knowledge of what category theory and related subjects are concerned with.
So, my understanding is that category theory and related fields of higher mathematics ...
- 6,853
14
votes
1
answer
1k
views
Constructing computable synthetic differential geometry?
I'm a computer scientist, not a mathematician, so apologies if I've messed up a lot of things greatly.
I've been reading about synthetic differential geometry, and trying to formalize it in Coq. ...
- 1,221
20
votes
2
answers
1k
views
Commutative rings : Topoi = Fields :?
The following is probably a bad question, but hopefully, it might have a very good answer.
In category theory there is a quite famous analogy between topoi and commutative rings, I was never ...
- 9,111
60
votes
6
answers
11k
views
Synthetic vs. classical differential geometry
To provide context, I'm a differential geometry grad student from a physics background. I know some category theory (at the level of Simmons) and differential and Riemannian geometry (at the level of ...
- 6,025
6
votes
1
answer
703
views
Is Logic/Set Theory necessary for studying Topos Theory?
I have just completed a postgraduate course, in which I studied Category Theory, without having a background in Set Theory and Logic - this probably already sounds absurd to many. This did not seem to ...
11
votes
4
answers
2k
views
Embedding Theorem for topological spaces, and in general
There are many examples throughout mathematics of abstracting the formal properties of a "familiar" structure, but then having a theorem stating that all models of the abstract axioms embed into one ...
- 1,838
7
votes
4
answers
1k
views
Why is the concept of topos a "metamorphosis" of the concept of space?
Hi,
I recently started studying topos theory, and I am puzzled by the Grothendieck's claim that topos is a "metamorphosis" of the concept of space. Can somebody explain what he means by this?
...
- 71