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.
Questions tagged [synthetic-differential-geometry]
83 questions
39
votes
3 answers
How is Category Theory used to study differential equations?
I know that one can use Category Theory to formulate polynomial equations by modeling solutions as limits. For example, the sphere is the equalizer of the functions
\begin{equation}
s,t:\mathbb{R}^3\rightarrow\mathbb{R},\qquad…
exchange
- 1,425
15
votes
1 answer
Motivation for the definition of an infinitesimal object
An infinitesimal object $D$ in a Cartesian closed category $\mathsf{C}$ is one for which the internal Hom functor $$(-)^D: \mathsf{C} \to \mathsf{C}$$ has a right adjoint.
I am wondering what is the motivation for this definition? Also, how would…
ಠ_ಠ
- 11,310
9
votes
6 answers
Basic Geometric intuition, context is undergraduate mathematics
At some point in your life you were explained how to understand the dimensions of a line, a point, a plane, and a n-dimensional object.
For me the first instance that comes to memory was in 7th grade in a inner city USA school district.
Getting to…
Andrew
- 405
6
votes
1 answer
$\epsilon \otimes 1 + 1 \otimes \epsilon$ is a nilcube in $\mathbb R[\epsilon] \otimes \mathbb R[\epsilon]$. What does that mean intuitively?
[EDIT: I know what the notation means, and I can easily show that $x=\epsilon \otimes 1 + 1\otimes \epsilon$ satisfies $x^3=0$ but $x^2 \neq 0$. That's not what this question is about. It might be better to restrict the question to Synthetic…
wlad
- 8,355
6
votes
0 answers
Synthetic differential geometry and algebraic geometry
I am reading here and there about basic synthetic differential geometry. One of the central ideas seems to be that it should be developed in a suitable topos, hence, in particular, a cartesian closed category. I also remember reading Lawvere was…
Arrow
- 14,390
5
votes
0 answers
What is Kock's "comprehensive axiom" in synthetic differential geometry?
Anders Kock mentions many axioms in his book about synthetic differential geometry.
It seems that, what nlab calls: "the Kock-Lawvere axiom", is called axiom $1^W_k$ in the book.
This axiom states that for a Weil algebra $W$ over $k$ we have that $R…
5
votes
1 answer
How many smooth vector space structures does $\mathbb R^n$ have?
The title is a bit misleading. I am really asking about how many ways are there to equip $\mathbb R^n$ with a smooth addition $+:\mathbb R^n \times \mathbb R^n \to \mathbb R^n$ so that the (possibly non-standard) addition-map together with the…
Nico
- 4,540
5
votes
2 answers
How should I think about this ring?
I don't know much abstract algebra or ring theory. I have come across a ring, and it would be really great if somebody could help me to understand what else in mathematics it relates to, or how to visualize what is going on in this ring (especially…
Richard Southwell
- 1,903
5
votes
0 answers
Calculi for the category theory?
Some branches of mathematics admit calculi with whom one can do syntactical (language-like, grammatical) or geometric operations to arrive at certaing conclusions. The syntactical part (proof theory) of mathematical logic is the prominent example of…
TomR
- 1,459
5
votes
1 answer
Smooth real line and Dedekind cuts
I am reading Bell's A primer of infinitesimal analysis, and the real numbers he considers have certain properties for doing synthetic differential geometry. He calls this object the smooth real line. I am not an expert in topos theory, but know some…
Quique Ruiz
- 1,132
- 9
- 20
5
votes
1 answer
Are maps $\operatorname{Spec}k[x]\times \operatorname{Spec}k[x]\to \operatorname{Spec}k[x]$ somehow bivariate polynomials?
Let $R$ be a (commutative unitary) ring object in a topos $\mathcal E$. Say $R$ is a Fermat ring if it satisfies $$\forall f:R\to R\;\exists !g:R^2\to R\;: \forall x,x^\prime \in R\;[fx^\prime -fx=g(x,x^\prime )\cdot (x^\prime -x)].$$
As Kock writes…
Arrow
- 14,390
5
votes
1 answer
Reference request: preparation for learning a little smooth infinitesimal analysis?
I'm interested in learning a little smooth infinitesimal analysis. There is a free book by Kock: Smooth Differential Geometry, http://home.imf.au.dk/kock/ . As I dive into it, I feel that I'm not quite sufficiently prepared. He seems to be assuming…
user13618
5
votes
1 answer
Some questions about synthetic differential geometry
I've been trying to read Kock's text on synthetic differential geometry but I am getting a bit confused. For example, what does it mean to "interpret set theory in a topos"? What is a model of a theory? Why does Kock use [[ ]] rather than { } for…
ಠ_ಠ
- 11,310
4
votes
1 answer
Ring Structure on the geometric line in synthetic geometry
Synthetic Differential Geometry by Kock opens with the following:
The geometric line can, as soon as one chooses two distinct points on it,
be made into a commutative ring, with the two points as respectively 0
and 1. This is a decisive structure…
Mithrandir
- 1,174
4
votes
1 answer
Must every $\mathbb Z$-shaped point of an arithmetic scheme be contained in an affine open subscheme?
Assume $X$ is a scheme over the base-ring $\mathbb Z$ and $t:\text{Spec}\,\mathbb Z \to X$ is an integer shaped point of $X$. Is there necessarily an affine open subscheme $U$ of $X$ through which $t$ factors?
Context: Richard Garner claims on page…
Nico
- 4,540