22
votes
Accepted
Are the models of infinitesimal analysis (philosophically) circular?
It is not circular for us to prove the consistency of noneuclidean geometry by providing an interpretation of noneuclidean geometry within euclidean geometry, such as with the PoincarΓ© disk model of ...
- 236k
12
votes
Accepted
Query about SDG (Synthetic Differential Geometry)
In a paper by Marta Bunge and Eduardo Dubuc. "Local concepts in SDG and germ representability" (1987) certain axioms were laid down towards a synthetic theory of differential topology based on logical ...
- 136
11
votes
Accepted
The (co)tangent sheaf of a topological space
Your $ππ$ is always $0$. If $π·$ is a derivation and $π$ is a function, then for every point $π₯$ $π·π$ vanishes at $π₯$; it suffices to prove this when $π(π₯)=0$, and in that case $π=πβ$ ...
- 55.9k
8
votes
Accepted
Relationship between synthetic differential geometry and differential cohesion?
I'm a co-author on the abstract linked in the comments but I'm coming from the computer science side so I'm not an expert on the models and I know very little classical differential geometry.
I had ...
- 989
7
votes
Accepted
A complex version of the Cahiers topos
This has already been done, see the article EFC-algebra and references therein.
In particular, the paper of Pridham constructs the topos of β-sheaves on the site of (derived) Stein spaces and explores ...
- 37.8k
7
votes
Differential algebraic geometry vs Diffiety theory
Disclaimer: I don't know much about this, and I hope more knowledgeable people weigh in.
You might want to take a look at Ayoub's differential Galois theory for schemes and the foliated topology (see ...
- 16.1k
6
votes
Accepted
Constructive analysis and synthetic differential geometry
In the smooth-topos models of SDG, the situation is generally something like this. The internally-definable Cauchy real numbers $\mathbf{R}_c$ are the sheaf of locally constant $\mathbb{R}$-valued ...
- 66.8k
5
votes
Accepted
Intuition and analogue of Wraith axiom from synthetic differential geometry
Axiom W is about the behaviour of the second tangent bundle - it ensures that the vertical bundle of the tangent bundle, $V(M) \subseteq T\circ T(M)$, where $V(M) = T(p)^{-1}(0)$, decomposes as the ...
- 1,253
5
votes
"Quasi-coherent" vector spaces in Sch/S
This is not a complete answer, but it explains the affine case in a bit more elementary terms. (Also because I do not understand what is going on in Jason's answer.)
Let us first assume that $S = \...
- 63.1k
5
votes
Accepted
"Quasi-coherent" vector spaces in Sch/S
What I wrote in the first comment above is wrong. I usually work with "projective Abelian cones" rather than "Abelian cones", and projective Abelian cones (typically) do not have ...
4
votes
Are the models of infinitesimal analysis (philosophically) circular?
Yes, there is some degree of philosophical circularity, if you take the view that the only "non-circular" way to build up a subject is to start with conceptually simple primitives, and work ...
- 82.7k
4
votes
Accepted
Analogue of Kock-Lawvere axiom for power series rings?
Yes, it is consistent, it even follows from the Kock-Lawvere axiom, as follows.
We defined $\mathrm{Spf}(R[[\epsilon]]) := \mathrm{colim}_n \mathrm{Spec}(R[\epsilon]/(\epsilon^n))$, so we have
$$R^{\...
- 1,110
4
votes
"Quasi-coherent" vector spaces in Sch/S
The claim is true as long as $p$ is qc qs. This is needed so that $p_*$ preserves quasi-coherent modules. Notice that, by using the isomorphisms $\mathrm{Hom}_U(V_U,\mathbb{A}^1_U) \cong \Gamma(V_U,\...
- 63.1k
3
votes
Accepted
Semi-holonomic jets in synthetic differential geometry
According to Liebermann's Introduction to the theory of semi-holonomic jets p.177:
a local section $s:U\subset M\to J^1E$ is said to be adapted at $x\in U$ if $s(x)=j^1_x(\beta \circ s)$τ° where $\...
- 5,343
3
votes
Accepted
Constructing computable synthetic differential geometry?
I still don't quite understand what OP wants, but let me just cite a few papers that I think might be relevant to such questions. First, there are a lot of literature that describe how to work with ...
- 4,459
2
votes
Geometric intuition for $R[x,y]/ (x^2,y^2)$, kinematic second tangent bundle, and Wraith axiom
I will try to address your questions, and then point to some general cartegorical phenomena that are at play here.
Answer 1/2: In the category of smooth manifolds, or a proper model of synthetic ...
- 1,253
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