In general, I am familiar with this notion of duality (i.e. in category theory, a statement is dualized simply by "reversing all arrows" and leaving objects unchanged). There are a couple of questions I would like to ask:
I do not understand why the notion of fibration and cofibration are dual to one another. What I don't get in this explanation (which is also the argument given in the lecture where the problem arose) is why $Y \times I$ is substituted with Hom$(I,Y)$ when passing to the dual statement. Aren't the objects to be left alone, and only the arrows reversed? If not, what is the exact, general, correspondence between objects and their "duals"?
In connection with the last sentence of the question above: one commonly refers to the dual of a $\mathbb{K}-$vector space $V$ as the vector space of linear maps $V \to \mathbb{K}$. Is this a special case of some "dualizing" correspondence between objects (which can be described for a sufficiently general class of categories, for example as a functor $\mathcal{C} \to \mathcal{C}^{op}$, or $\mathcal{C} \to \mathcal{C}$) or is it just an intuitive way of referring to that particular construction, which shouldn't be taken as a broad statement? Edit as seen in this page, the construction of dual vector space is somewhat general in a categorical sense. However I still fail to see a general, formal connection between this notion of "dual object" and the notion of dual property (and in particular in the case of (co)fibrations), except for the fact that in both cases "arrows are reversed".
Some answers and comments have already been helpful, but I stil fail to see the big picture! Thanks for any further help.
For your first question duality in homotopy theory, it is precisely an example of categorical duality. A homotopy between $f,g:A\to B$ is a continuous function $H:A\wedge I\to B$ where $A\wedge I$ can be any cylinder object for $A$. It is common to take $A\times [0,1]$ as a cylinder object, but that is not required. What is important is that the cylinder object has some properties (look up 'cylinder object' to see what they are, I suggest the nlab). Now the dual notion is that of a path object $A^I$. Its definition is precisely the categorical dual of a cylinder object. Again, it is common to take $A^{[0,1]}$ but it is not required. Now the dual notion of a homotopy becomes a continuous function $H:A\to B^I$. It turns out that for ordinary homotopy these two notions of homotopy are equivalent.
As for the dual of a vector space, the object $\mathbb K$ in the category $K-Vect$ of finite dimensional vector spaces is a particular case of what is called a dualizing object. The duality here is a little different than just categorical duality. Categorical duality is just a (very powerful) tautology. Dualizing objects are different. Again, I suggest the nlab for further reading.