Questions tagged [synthetic-differential-geometry]

Question 1

Given triangle $ABC$ with altitude $AD$ (where $D \in BC$). At point $A$, construct a perpendicular to $AC$, and on the half-plane that does not contain $B$, take point $E$ such that $AE = AD$ and $AE ...

Question 2

When I read about lemmas of Smooth Infinitesimal Analysis I encountered a problem in understanding. It's following lemma: $\forall \varepsilon \in \Delta (\varepsilon\le 0\land0 \le\varepsilon)$ $\...

Question 3

As is known if $η$ $\varepsilon \in \Delta$ then $(\varepsilon\le η\landη \le\varepsilon)$ in Smooth Infinitesimal Analysis. But if $η=2\varepsilon$ then $(\varepsilon\le 2\varepsilon\land2\varepsilon\...

Question 4

Let T be a finitary algebraic theory and Set the category of sets. Write F : T → (FPT)ᵒᵖ for the canonical, full & faithful embedding into the opposite of the category FPT of finitely presented T-...

Question 5

I read in one source that $0<a\pm\varepsilon$, i.e. $a\pm\varepsilon>0$ $a$ is number and $\varepsilon$ is infinitesimal that indistinguishable from $0$ ($\neg \neg \varepsilon = 0$). From $\...

Question 6

In Bell's A primer of infinitesimal analysis, three microneighbourhoods of zero are defined: $M_1 = \{ \epsilon \ |\ \epsilon^2 = 0 \}.$ $M_2 = \{ \epsilon \ |\ \lnot \epsilon \ne 0 \}.$ $M_3 = [0, 0]...

Question 7

Consider equation from Smooth Infinitesimal Analysis: $\varepsilon^2=0$ and apply intuitionistic logic: $\neg \varepsilon = 0$ is false $\neg \neg \varepsilon = 0$ is true. It means that there is $\...

Question 8

Let $F$ and $G$ be two function spaces. If an operator is a mapping from $F$ to $G$, i.e., any function on $F$ can be mapped to a function in $G$. Then is it possible to define the differentiability ...

Question 9

In Smooth Infinitesimal Analysis $\neg \neg \varepsilon = 0$ means that all kinds of infinitesimals satisfy $\neg \neg \varepsilon = 0$. But when we proved that $\neg \neg \varepsilon = 0$ holds for $\...

Question 10

In the Appendix 4 of Models for Smooth Infinitesimal Analysis by Moerdijk and Reyes, given an object $A$ in a category $\mathbb{C}$, they want to show that the functor $(-)^A:Sh(\mathbb{C})\rightarrow ...

Question 11

In Smooth Infinitesimal Analysis: $I=\{ x \in R: \neg \neg x = 0\}$ $\Delta:=\left\{x \in R: x^{2}=0\right\}$ $\Delta\subset I$ Nilsquare infinitesimals $(x^2=0)$ in Smooth Infinitesimal Analysis are ...

Question 12

In Smooth Infinitesimal Analysis it can be proved that if $a>0$ then $a+\varepsilon>0$. $\varepsilon$ is infinitesimal that $\varepsilon^2=0$ If $a+\varepsilon>0$ then there is its inverse $(...

Question 13

I work in Smooth Infinitesimal Analysis. I want to prove that $a+\varepsilon>0$ if $a>0$ in $R$ and $\varepsilon \in \Delta$ $\Delta:=\left\{\varepsilon \in R \mid \varepsilon^{2}=0\right\}$ My ...

Question 14

I work in Smooth Infinitesimal Analysis. $\Delta:=\left\{x \in R: x^{2}=0\right\}$ $I=\{ x \in R: \neg x \neq 0\}$ $J=\{x \in R : x \le 0 \land x \ge 0\}$ Are $\varepsilon$ from I and J neighborhoods ...

Question 15

In Synthetic Differential Geometry, it is well-known that limits of microlinear spaces are microlinear. Suppose we are in a model with $\mathcal{R}$ as the line object. Then $X = \{ (x, y) \in \...

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Questions tagged [synthetic-differential-geometry]

Parallelogram formed by altitude, perpendicular construction, and angle bisector in a triangle

Problem in understanding SIA lemma $\forall \varepsilon \in \Delta (\varepsilon\le 0\land0 \le\varepsilon)$

How is it possible that $2\varepsilon\le\varepsilon$ if $\varepsilon\in\Delta$ and $(\varepsilon\le 2\varepsilon\land2\varepsilon\le\varepsilon)$ SIA?

Left‐exact extension of a T-algebra along the embedding F : T→(FPT)ᵒᵖ

Is inequality $a+\varepsilon>0$ correct in Smooth Infinitesimal Analysis?

Two microneighbourhoods coincide

Intuitionistic logic and finding roots of equation (by example of Smooth Infinitesimal Analysis)

How to define the Jacobian of an operator on function spaces? [duplicate]

Why if $\varepsilon^2=0$ then $\neg \neg \varepsilon = 0$? (Smooth Infinitesimal Analysis)

Atoms in Models for Smooth Infinitesimal Analysis

How is it possible that infinitesimals $x$ which don't satisfy $x^2=0$ in $I$ are not invertible $(\neg \neg x=0)$? (Smooth Infinitesimal Analysis)

Is there inverse $(a+\varepsilon)^{-1}$ if $a+\varepsilon>0$?

How to prove that $a+\varepsilon>0$ if $a>0$ and $\varepsilon \in \Delta$? (Smooth infinitesimal analysis)?

Are $\varepsilon$ from I and J neighborhoods included in the closed interval $[0,0]$ as $\Delta$? (Smooth Infinitesimal Analysis)

SDG Microlinear spaces strange example

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