Using the formalism of model categories its possible define the concept of homotopy as done here.
If we take as model category $\mathbf{Top}$ having homotopy-equivalence as weak-equivalence and fibration and co-fibration defined in the standard topological way, these type of homotopies are just homotopies as defined in basic courses of algebraic topology.
From this point of view seems that weak equivalence are what really matter, so here's my question:
Is there any way to characterize homotopy equivalence (in $\mathbf{Top}$) without using the concept of homotopy?
Edit: after
I'm wondering if there's a way to discriminate homotopy equivalence without using the answer concept of Karol Szumiło I think I've to add some specifications: what homotopy at all, meaning that I'm looking for is a characterization (if exists) criteria which can enable to see if say that a certain continuous function map $f \colon X \to Y$ is an homotopy equivalence without using looking for a morphism $g \colon Y \to X$ and continuous maps $\mathcal F \colon X \times I \to X$ and $G \colon Y \times I \to Y$ which are indeed respectively homotopies at all, not just not naming them. of $g \circ f$ with $1_X$ and $f \circ g$ with $1_Y$.
