How do you generalize "loops" from usual topology $I:[0,1] \to X$

Question

$I:[0,1] \to X$ being continuous and $I(0) = I(1)$ is usually how one defines a loop in a topological space $X$, but what if your space $X$ isn't typical. For instance what if $X$ is subspace of a free monoid? I thought about it and the closest I came to was recognizing that there are loops in regular language subsets of $X$, for instance if $X = \{a,b\}^*$ (Kleene star), then the regular subset $aab(aab)^*bba$ would have loop-looking paths in its DFA. But how is this related to "loops in the space." I don't know. The point of looking for loops is so I can then do the whole "fundamental group" development for free monoids. Any ideas come to your beautiful minds? Thanks.

Answer 1

I have no idea what topology you want to place on free monoids, but...

The point of looking for loops is so I can then do the whole "fundamental group" development for free monoids.

One keyword to look up is directed topological space. Another one is the pair "stratified space" and "exit path $\infty$-category."