The "homotopic"/Geometric Nature of Mathematical Induction

 Mathematical Induction for the Natural Numbers ${\mathbb N}$ can be expressed as the following Inference rule within Martin-Löf's Dependent Type Theory:

$\Gamma\,,\,x:{\mathbb N}\,\vdash\,P(x)\;Type$

$\Gamma \,\vdash\,p_0\,:\,P(0_{\mathbb N})$

$\Gamma \,\vdash\,p_s\,:\,\Pi_{(n:{\mathbb N})}\,P(n)\,\to\,P(succ_{\mathbb N}(n))$

 $-----------------\;{\mathbb N} ind$ 

$\Gamma \,\vdash\, ind_{\mathbb N}(p_0\,p_s) \,:\, \Pi_{(n:{\mathbb N})}\,P(n)$

with the computation rules -where we implicitly assume the same premises:

$ind_{\mathbb N}(p_0\,,\,p_s\,,\,0_{\mathbb N})\,\doteq\,p_0\,:\,P(0_{\mathbb N})$

$ind_{\mathbb N}(p_0\,,\,p_s\,,\, succ(n) ) \,\doteq\, p_s(n\, ,\, ind_{\mathbb N}(p_0\,,\,p_s\,,\, n )) \,:\, P(succ(n))$

Together this can be expressed graphically as  

 Thus, induction provides a way to build a function for free on an Inductive Type: If we have a way to hop from one fiber to "its neighboring" fiber, then, given a point at a starting fiber, we can built a section in the total space of that Type.  

That way of "hopping" between fibers is nothing else than the inductive step! 

First analogy, a curve can be defined by a differential equation and an initial condition. This equation can be given by a vector field that expresses the tangent to the curve at each point.

Also, in Differential Geometry we have the Covariant derivative, which expresses how we can move a tensor between infinitesimally close points in order to compare it meaningfully at those two points. Imposing the "Covariant change" to be zero along a given curve gives us a particular way to move a tensor between infinitesimally close points such that it stays parallel. This is the parallel transport of a tensor.

We can do the following: parallel transport a vector field along the very same curve it defines.  This curve is nothing other than a geodesic.

I'm definitely tempted to draw here an analogy between these Differential Geometry problems and the Induction rule, I'm just not sure what the best analogy might be. As induction gives as a curve, is Induction akin to building a geodesic through a point?

More general an analogy is the relation between induction an Sheaf theory. Paraphrasing an LLM, "both guarantee existence of global objects from local rules, though the technical frameworks differ: induction is sequential and algebraic; sheaf theory is topological and categorical".

Analogies aside, Dependent Type Theory through the lens of Homotopy Type Theory is very rich in intuitive pictures. I find this super cool!