$\mathcal Q1$ Question. I'm not entirely sure about the definition of this object. I bet that it has to be well known and has a name but I have no clue. So the question is for terminology, references for this structure or some similar structure that capture the spirit of my definition.
It should be a category that has as objects "points of a space" and as arrows "smooth paths between them". But attached to every point there has to be some additional information of the slope and "the higher slopes" so that two paths can be composed only if the first ends with the same direction, speed (and so on) of how the second starts.
$\mathcal Q2$ Side questions: I know this can be vague. $(2.a)$ If this construction has not a name (maybe because it doesn't exists) can the definition be fixed to obtain something close to the desired object (*)? $(2.b)$ Are there other categories that capture a similar idea?
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Let me try to define this kind of structure by approximations. A first good approximate picture is the fundamental groupoid $\Pi_1(X)$ but it is not exactly like that even if I don't know exactly the technical details where my definition could fail.
Take a space $X=\mathbb R^n$ and let $0,1:\bullet\to I=[0,1]$ the two endpoints of the closed unit interval, where $\bullet$ is the singleton. Notation: $!_X$ is a morphism to the singleton and $0_{\mathbb N}:\bullet\to \mathbb N$ is the inclusion of zero.
First attempt. I try to define a category $\mathfrak C_0(X)$ as follows:
${\rm Arr}_{{\mathfrak C}_0(X)}:={\mathcal C}(I,X)$ is the set of continuous functions $\gamma:I\to X$
${\rm Ob}_{{\mathfrak C}_0(X)}:=\mathcal C(\bullet,X)$
Define $\partial_0,\partial_1:{\rm Arr}_{{\mathfrak C}_0(X)}\to {\rm Ob}_{{\mathfrak C}_0(X)}$ where for a path $\gamma:I\to X$ $$\partial_i(\gamma):=\gamma\circ i=\gamma(i)$$
Define ${\rm id}:{\rm Ob}_{{\mathfrak C}_0(X)}\to {\rm Arr}_{{\mathfrak C}_0(X)}$ where for each point ${\bf x}:\bullet\to X$ $${\rm id}_{\bf x}:={\bf x}\,\,\circ \,\,!_I$$
Define composition as $\circ:{\rm Arr}_{{\mathfrak C}_0(X)}\times_{\partial_0,\partial_1} {\rm Arr}_{{\mathfrak C}_0(X)}\to {\rm Arr}_{{\mathfrak C}_0(X)}$, s.t. given $\partial_0(\beta)=\partial_1(\gamma)$
$$\beta\circ\gamma(t)=\gamma(2t)\quad \text{if } t\in[0,1/2];\\ \beta\circ\gamma(t)=\beta(2t-1) \quad \text{if } t\in[1/2,1].$$
This thing is not a category because it is not associative. As far as I understand the classical trick is to take all the arrows up to homotopy achieving associativity up to an "higher morphism of morphisms". I'd like instead, for obvious reasons, to consider two paths equivalent only if they "draw" the same image in $X$ (${\rm im} \gamma={\rm im}\beta$). I guess that the formally this amounts to modding out $\mathcal C(I,X)$ by the right action of the group of auto-homeomorhisms $\mathcal C(I,I)$ that fix the endopoints.
$$[\gamma]:=\{\beta\,:\,\exists i:I\to I.\beta\circ i=\gamma\}$$
I'm not sure everything works, probably the definition of composition should be tweaked. But I try to define the category of smooth paths now.
Second attempt. I try to define a category $\mathfrak C_\infty(X)$ as follows: define $\gamma'(0)$ to be the right hand derivative and $\gamma'(1)$ the left hand derivative.
${\rm Arr}_{{\mathfrak C}_\infty(X)}:={\mathcal C}^\infty(I,X)$ is the set of smoothfunctions $\gamma:I\to X$
${\rm Ob}_{{\mathfrak C}_\infty(X)}:=X^{\mathbb N}$
Define $\partial_0,\partial_1:{\rm Arr}_{{\mathfrak C}_\infty(X)}\to {\rm Ob}_{{\mathfrak C}_\infty(X)}$ where for a path $\gamma:I\to X$ $$\partial_i(\gamma):=(\gamma(i),\gamma'(i),\gamma''(i),...,\gamma^{[n]}(i),...)$$
At this point I'm lost because only the points of the form $k:X\to X^{\mathbb N}:{\bf x}\mapsto ({\bf x},{\bf 0},...,{\bf 0},...)$ are really points. The others are some kind of points enriched with some kind of infinitesimal information. I.e. to get a category we should manually adjoin a bunch of constant paths/morphisms that, even if constant, have non-trivial derivatives at zero. So lets assume we have now ${\rm Arr}_{{\mathfrak C}_\infty(X)}:={\mathcal C}^\infty(I,X)\sqcup (X^\mathbb N\setminus {\rm im}k)$ where we extend on it $\partial_i$ in the obvious way (identity).
- Define ${\rm id}:{\rm Ob}_{{\mathfrak C}_\infty(X)}\to {\rm Arr}_{{\mathfrak C}_\infty(X)}$ where for each point ${\bf a}:\mathbb N\to X$ if it is in ${\rm im }k$ $${\rm id}_{\bf a}:={\bf a}\,\circ 0_{\mathbb N} \circ \,\,!_I$$ if it's not then we map to its copy in ${\rm Arr}_{{\mathfrak C}_0(X)}$.
- Define composition as $\circ:{\rm Arr}_{{\mathfrak C}_\infty(X)}\times_{\partial_0,\partial_1} {\rm Arr}_{{\mathfrak C}_\infty(X)}\to {\rm Arr}_{{\mathfrak C}_\infty(X)}$, s.t. given $\partial_0(\beta)=\partial_1(\gamma)$ composition is defined as before: patching paths together when endpoints are compatible.
At this point is not clear to me if we can still mod out by the action of the "smooth automorphisms" of $I$, or something like that, and obtain associativity up to that equivalence relation.
(*) Note on the side questions: if my definition needs fixes and an improvement or a right definition is proposed, I'd like to have it spelled out.