Using the formalism of model categories its possible define the concept of homotopy as done [here](http://ncatlab.org/nlab/show/homotopy).


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:



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> Is there any way to characterize homotopy equivalence (in $\mathbf{Top}$) without using the concept of homotopy? 
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> 
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I'm wondering if there's a way to discriminate homotopy equivalence *without using the concept of homotopy at all*, meaning that I'm looking for a criteria which enable to say that a certain continuous map $f \colon X \to Y$ is an homotopy equivalence without 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 of $g \circ f$ with $1\_X$ and $f \circ g$ with $1\_Y$.



Alternative characterization of homotopy equivalence
EDIT: Now that the OP has edited his question to make clearer what he wants as an answer, I'm removing speculation about what he wanted. The answer is: yes, you can characterize homotopy equivalences as the maps which become isomorphisms after applying the localization functor to invert the weak equivalences. This answer doesn't require you to "use the notion of homotopy" because it's part of a much more general framework.


Here is a nice article on [localization](http://ncatlab.org/nlab/show/localization). One of the best reasons for studying model categories is that they let you get your hands on the maps (weak equivalences) which build homotopy equivalences. Many of the axioms for a model category are there to get around set-theoretic issues that arise when you try to localize a category with respect to an arbitrary class of morphisms. It turns out you need to localize at a class of morphisms which admits a calculus of fractions, and the class $W$ of weak equivalences does. If you read the article, you'll see how to construct $C[W^{-1}]$ in general, and you can then specialize to the case where $C=Top\_\*$ and $W$ is the class of weak equivalences.


Unfortunately, for computation, this highly general approach is not as good as just using the Path and Cylinder objects mentioned in the nLab article the OP links to. That's why most people who study model categories use those instead of this general framework: because a model category is more than just a category with a nice class $W$--it's a category where you can really get your hands on everything and compute!


