Questions tagged [synthetic-differential]

Synthetic differential geometry is an axiomatic formulation of differential geometry in smooth toposes. The axioms ensure that a well-defined notion of infinitesimal spaces exists in the topos, whose existence concretely and usefully formalizes the wide-spread but often vague intuition about the role of infinitesimals in differential geometry.

Filter by
Sorted by
Tagged with
4 votes
1 answer
188 views

A complex version of the Cahiers topos

Has anyone tried defining a complex version of the Cahiers topos? If we take the definition of $C^\infty$-rings, replace "smooth" with "holomorphic" (of course, one has to take ...
xuq01's user avatar
  • 1,054
8 votes
0 answers
185 views

A reference on a result by Steve Schanuel

In the Author Commentary section of the TAC reprint of the paper of 1968 Diagonal arguments and cartesian closed categories., Bill Lawvere wrote: ‘Nilpotent infinitesimals fall far short of even one-...
Evgeny Kuznetsov's user avatar
9 votes
2 answers
2k views

Are the models of infinitesimal analysis (philosophically) circular?

Infinitesimal analysis (by which I mean that originating from topos theory---not the nonstandard analysis of Robinson) seeks to recover the pre-limit notions of calculus (which are sufficiently useful ...
Duncan W's user avatar
  • 341
3 votes
1 answer
108 views

Analogue of Kock-Lawvere axiom for power series rings?

The Kock-Lawvere axiom for a topos $\mathcal{E}$ states that given a specified commutative ring object $R \in \mathcal{E}$, for all local Artinian $R$-algebra objects $A \in \mathcal{E}$, the morphism ...
Madeleine Birchfield's user avatar
4 votes
3 answers
477 views

"Quasi-coherent" vector spaces in Sch/S

$\DeclareMathOperator\Vec{Vec}\newcommand\Sch{\mathrm{Sch}}\DeclareMathOperator\Hom{Hom}$Let $S$ be a base scheme. Let me write $\Vec(S)$ to denote the category of $\mathbb A_S$-vector space objects ...
Nico's user avatar
  • 775
5 votes
0 answers
152 views

Nullstellensatz with nilpotents and $I=J(V(I))$

Let $R$ be the ring $$\mathbb{R}[t_1,t_2\ldots]/(t_1^2,t_2^2,\ldots)$$ Let $p_1,\ldots p_\ell$ be polynomials in $R[x_1,\ldots,x_n]$ whose constant terms are 0. Let $f$ be a polynomial which is zero ...
user avatar
10 votes
1 answer
497 views

Models of the Kock-Lawvere axioms

What are your favorite models of the KL-axioms? The motivation is having some basic models to understand the axiom scheme as presented e.g. in Synthetic Geometry of Manifolds by Kock. In that text ...
Alec Rhea's user avatar
  • 8,977
7 votes
0 answers
228 views

Constructions with Superschemes via Kan extensions

Let $\operatorname{CAlg}$ be the category of commutative rings (with unit) and $\operatorname{S-CAlg}$ the category of supercommutative $\mathbb{Z}/2$-graded rings. Then we have an adjoint triple (as ...
Markus Zetto's user avatar
6 votes
1 answer
272 views

Geometric intuition for $R[x,y]/ (x^2,y^2)$, kinematic second tangent bundle, and Wraith axiom

This is a sort of continuation of this question. In synthetic differential geometry (SDG), we have $D\subset R$ comprised of the second order nilpotents. The Kock-Lawvere axiom (KL axiom) implies that ...
Arrow's user avatar
  • 10.3k
6 votes
0 answers
262 views

Examples of connection preserving maps in differential geometry

In synthetic differential geometry and tangent categories, linear connections on the tangent bundle are treated as a sort of algebraic gadgets that incorporate the tangent bundle. Like any other ...
Ben MacAdam's user avatar
  • 1,253
0 votes
0 answers
143 views

Making area/volume calculations that use SIA rigorous

There are some intriguing "proofs" using Smooth Infinitesimal Analysis of theorems concerning areas and volumes. Some examples: A proof that $\sin'(0) = 1$. A proof that the surface area of a cone is ...
wlad's user avatar
  • 4,823
6 votes
2 answers
384 views

Intuition and analogue of Wraith axiom from synthetic differential geometry

In synthetic differential geometry, an object $M$ verifies the Wraith axiom if for all functions $\tau:D\times D\to M$ which are constant on the axes $\tau(d,0)=\tau(0,d)=\tau(0,0)$ for all $d\in D$, ...
Arrow's user avatar
  • 10.3k
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. ...
Siddharth Bhat's user avatar
9 votes
1 answer
384 views

The (co)tangent sheaf of a topological space

Let $X$ be a topological space (assume additional assumptions if needed) and denote by $\mathcal O _X$ its sheaf of $\Bbbk$-valued continuous functions where $\Bbbk$ is $\mathbb{R}$ or $\mathbb{C}$ ...
Qfwfq's user avatar
  • 22.7k
12 votes
1 answer
2k views

Differential algebraic geometry vs Diffiety theory

Algebraic geometry is said to be useful to study not only specific solutions of polynomial equations but to understand the intrinsic properties of the totality of solutions of a system of equations. ...
exchange's user avatar
  • 221
9 votes
1 answer
681 views

Constructive analysis and synthetic differential geometry

I am curious if (any of) the various inequivalent constructions of the real line in constructive mathematics can be used to build a model of Kock and Lawvere's synthetic differential geometry? In ...
ಠ_ಠ's user avatar
  • 5,933
16 votes
1 answer
884 views

Query about SDG (Synthetic Differential Geometry)

(Edited 10/17/17): With the hope of obtaining informed responses on the following intriguing remark of Marta Bunge on the status of Synthetic Differential Geometry, I have added a third question to ...
Philip Ehrlich's user avatar
11 votes
1 answer
1k views

Relationship between synthetic differential geometry and differential cohesion?

I'm a big fan of synthetic differential geometry (or smooth infinitesimal analysis), as developed by Anders Kock and Bill Lawvere. It's a beautiful and intuitive geometric theory, which gives ...
ಠ_ಠ's user avatar
  • 5,933
4 votes
1 answer
230 views

Semi-holonomic jets in synthetic differential geometry

Anders Kock's two texts on synthetic differential geometry (SDG) are a great place to get geometric intuition, especially when it comes to jets. Unfortunately, he doesn't seem to cover semi-holonomic ...
ಠ_ಠ's user avatar
  • 5,933
7 votes
0 answers
166 views

Is integration smooth?

Let $M$ be a compact manifold and $\varphi:C^\infty(M)\rightarrow \mathbb{R}$ be a function which assigns to every $f\in C^\infty(M)$ the value $\int_M fdV.$ In a smooth topos which is a well adapted ...
user avatar
2 votes
0 answers
40 views

Characterization of formally étale morphisms between microlinear objects?

The following is taken from Penon's thesis, (d) at the top of page 44. Can anybody explain how to prove this? Let $f:M\to N$ be an arrow of microlinear objects. Then the square below is a ...
Arrow's user avatar
  • 10.3k
20 votes
1 answer
3k views

What is meant by the inverse function theorem in algebraic geometry?

I heard several times the inverse function theorem fails in algebraic geometry. Now I realize I'm pretty confused by this. This question has two parts. The first part asks for the correct formulation ...
Arrow's user avatar
  • 10.3k
3 votes
1 answer
149 views

Internal characterizations of lifting properties?

This is basically a restatement of this question. Two arrows $f,g$ are orthogonal, i.e satisfy $f\perp g$, iff the square below is a pullback $$\require{AMScd} \begin{CD} \mathsf C(B,X) @>{f^\...
Arrow's user avatar
  • 10.3k
4 votes
0 answers
129 views

When are descriptions of formal unramifiedness/smoothness via lifting properties equivalent to those via induced arrows to pullbacks?

Formal unramifiedness of an arrow $f:M\rightarrow N$ in algebraic geometry or synthetic differential geometry in defined by asking the lifting problem below to have at most one solution (existence is ...
Arrow's user avatar
  • 10.3k
2 votes
0 answers
121 views

Local diffeomorphism (étale maps) in terms of infinitesimal tubular neighborhood?

In this MO question I asked for some help with several definitions of formally etale maps. The definitions I'm asking about are 'isomorphism of tangent spaces', i.e the square below is a pullback $$\...
Arrow's user avatar
  • 10.3k
7 votes
0 answers
504 views

Competing notions of formal étaleness

I'm writing some notes to myself on algebraic geometry and I'm trying to get the most conceptual definitions. Having arrived at formally étale morphisms, I am pretty desperate. Here is a list of ...
Arrow's user avatar
  • 10.3k
9 votes
1 answer
450 views

Étale morphisms in SDG

On page 70 here, Kock defines a formally étale morphism $f:M\rightarrow N$ as one for which the following square is a pullback for each $d:\mathbf 1\rightarrow D$ $$\require{AMScd} \begin{CD} M^D @>...
user89636's user avatar
15 votes
1 answer
1k views

Intuition for the Cartan connection and "rolling without slipping" in Cartan geometry

Consider a Cartan geometry $\pi: \mathcal{G} \to M$ with Cartan connection $\omega$ modelled on the Klein geometry $(G, H)$. The Cartan connection is supposed to formalize what it means to "roll ...
ಠ_ಠ's user avatar
  • 5,933
5 votes
0 answers
151 views

Resources on a smooth topos containing complex analytic/holomorphic geometry

In this question Urs Schreiber mentioned there are models in synthetic differential geometry of complex analytic geometry. First of all: When Urs writes complex analytic geometry, does he mean ...
Georg Lehner's user avatar
  • 1,943
4 votes
1 answer
293 views

Lie functor preserves "surjections" in synthetic differential geometry?

In classical finite-dimensional differential geometry, the Lie functor preserves surjections, sending a surjective Lie group homomorphism to a surjective Lie algebra homomorphism. As pointed out ...
ಠ_ಠ's user avatar
  • 5,933
7 votes
1 answer
608 views

Jets in synthetic differential geometry

As I understand it from Kostecki's notes, the $k$-jet $j^k f$ of a function $f: R^n \to R^m$ should be the map $$f^{D_k(n)} : {(R^n)}^{D_k(n)} \to (R^m)^{D_k(n)},$$ where $$D_k(n) = \{(x_1, \ldots, ...
ಠ_ಠ's user avatar
  • 5,933
60 votes
6 answers
10k 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 ...
ಠ_ಠ's user avatar
  • 5,933