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]
74 questions
3
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1 answer
Visualizing the Nilsquare Infinitesimals $D$ of Synthetic Differential Geometry (SDG)
I've been learning synthetic differential geometry, and while the infinitesimals and microlinearity axiom make sense to me, I have not been able to grasp the geometry of these infinitesimals. In smooth infinitesimal analysis and synthetic…
Noah M
- 345
2
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1 answer
Proof that the set of infinitesimal does not consist only of the zero element in Synthetic Differential Geometry
Thanks to Z. A. K. answer I edited my question
In this lecture Synthetic Differential Geometry they have the following
Definition 4.3. An infinitesimal on $R$ is any nilsquare element of $R$, i.e. $x^{2}=0 .$ We denote the collection of…
amilton moreira
- 167
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How is it possible that area under continuous curve consist of indivisibles?
I read in article (https://plato.stanford.edu/entries/continuity/) that continuity can't consist of indivisibles, only discrete entity can:
"In a word, continua are divisible without limit or infinitely
divisible..."
"In something like the same…
Mike_bb
- 691
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1 answer
Time representation in Synthetic Differential Geometry
I found in John Bell's book "A Primer of Infinitesimal Analysis" such interpretation of time representation:
But I can't fully understand this interpretation.
Can someone explain easily what John Bell means here?
Thanks.
Mike_bb
- 691
0
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How can *$R$ be a field in Synthetic Differential Geometry or Smooth Infinitesimal analysis?
In many sources it was written that *$R$ is ring and *$R$ can't be a field because it's not possible that $d^2=0$ in field.
But in some sources it was written that *$R$ is a field.
How can *$R$ be a field in Synthetic Differential Geometry or Smooth…
Mike_bb
- 691
0
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0 answers
The turnstile in Kock's "Synthetic Differential Geometry"
I'm reading part II of A. Kock's Synthetic Differential Geometry. It's my first introduction to SDG and I'm reading it for fun.
I'm confused by the use of the "forcing" symbol $ {\vdash} $. Fix a category $ \mathcal E $, which we assume to have…
0
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1 answer
Constant derivative in SDG
In the context of Synthetic Differential geometry as in the first few sections of Kock's book, so that $R$ is ring satisfying Axiom 1. Let ${f : R \rightarrow R}$ be such that ${f' = 0}$. How do you prove that $f$ is constant?
Boogie
- 249