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.
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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-...
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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 ...
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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
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"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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $d\in D$, ...
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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. ...
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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 $\mathbb{C}$ ...
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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.
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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
$$\require{AMScd} \begin{CD}
\mathsf C(B,X) @>{f^\...
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When are descriptions of formal unramifiedness/smoothness via lifting properties equivalent to those via induced arrows to pullbacks?
Formal unramifiedness of an arrow $f:M\rightarrow N$ in algebraic geometry or synthetic differential geometry in defined by asking the lifting problem below to have at most one solution (existence is ...
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Local diffeomorphism (étale maps) in terms of infinitesimal tubular neighborhood?
In this MO question I asked for some help with several definitions of formally etale maps. The definitions I'm asking about are 'isomorphism of tangent spaces', i.e the square below is a pullback
$$\...
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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 ...
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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 $d:\mathbf 1\rightarrow D$
$$\require{AMScd} \begin{CD} M^D @>...
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Intuition for the Cartan connection and "rolling without slipping" in Cartan geometry
Consider a Cartan geometry $\pi: \mathcal{G} \to M$ with Cartan connection $\omega$ modelled on the Klein geometry $(G, H)$.
The Cartan connection is supposed to formalize what it means to "roll ...
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Resources on a smooth topos containing complex analytic/holomorphic geometry
In this question Urs Schreiber mentioned there are models in synthetic differential geometry of complex analytic geometry.
First of all: When Urs writes complex analytic geometry, does he mean ...
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Lie functor preserves "surjections" in synthetic differential geometry?
In classical finite-dimensional differential geometry, the Lie functor preserves surjections, sending a surjective Lie group homomorphism to a surjective Lie algebra homomorphism.
As pointed out ...
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Jets in synthetic differential geometry
As I understand it from Kostecki's notes, the $k$-jet $j^k f$ of a function $f: R^n \to R^m$ should be the map $$f^{D_k(n)} : {(R^n)}^{D_k(n)} \to (R^m)^{D_k(n)},$$
where $$D_k(n) = \{(x_1, \ldots, ...
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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 ...
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