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According to May's A Concise Course in Algebraic Topology, the diagrams in the following represent cofibration and fibration, respectively if there exists an arrow diagonally (to the upper right object) to make the diagram commute:

$$ \require{AMScd} \begin{CD} A @>{}>> Y^I\\ @VVV @VVV \\ X @>{}>> Y \end{CD}$$

$$ \require{AMScd} \begin{CD} Y @>{}>> X\\ @VVV @VVV \\ Y\times I @>{}>> A \end{CD}$$

The author says the construction is "dual" in a certain sense. Clearly $\cdot\times I$ is left adjoint to $[I,\cdot]$, and they don't necessarily induce equivalence of categories, i.e., from $\mathbf{CGHaus}$ to itself. Therefore, existence of diagonal arrow making one of diagrams to commute doesn't necessarily imply that the same thing happens to another diagram. If we take the dual of the above diagram, then it coincides with the second one except at $Y^I$ and $Y\times I$ parts. They are similar because of adjunction, but why could I say they are dual to each other? Is this kind of "duality" rigorously defined? This one seems to me quite different from an usual dual construction such as colimit of a diagram in comparison to limit.

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The sense that $Y^I$ and $Y \times I$ are dual to each other are in the language of model categories. A model category is a category in which one has a notion of homotopy; it is a category $\mathfrak{C}$ with three classes of maps: $\mathscr{C}$ of cofibrations, $\mathscr{F}$ of fibrations, and $\mathscr{W}$ of weak equivalences such that $\mathfrak{C}$ is both complete and cocomplete (with respect to whatever $\mathscr{V}$ enriched universe of set theory you care about), $\mathscr{W}$ satisfies a "two-out-of-three" axiom (given maps $f, g,$ and $g \circ f$, if any two of $f, g$, or $g \circ f$ are in $\mathscr{W}$ then so is the third), and such that both $(\mathscr{C}, \mathscr{W} \cap \mathscr{F})$ and $(\mathscr{C} \cap \mathscr{W}, \mathscr{F})$ are weak factorization systems on $\mathfrak{C}$. The object $Y \times I$ is a way of producing an object homotopically equivalent to $Y$ through a cofibration (when $Y$ is cofibrant) that provides a location for (left) homotopies to be defined (note that in $\mathbf{Top}$ this is exactly what happens). Dually, the object $Y^I$ is an object that is homotopically equivalent to $Y$ (when $Y$ is fibrant) through a fibration that provides a different, but equally as useful, location for (right) homotopies to be defined.

If you are interested in reading more about this in a general setting, I highly recommend Dwyer and Spalinski's notes on model category theory, as well as Hovey's notes and Hirschhorn's book on the localization of model categories. If you are more interested in the homotopy of the category $\mathbf{Top}$ as approached through simplicial sets, Joyal and Tierney wrote a great set of notes on the subject, but Goerss and Jardine have a solid reference as well. All these references are good and go into a fairly detailed description of the model categorical approach to homotopy theory that is alluded to in the diagrams you have drawn above.

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  • $\begingroup$ Model category is a timely topic, since I'm currently reading Goerss and Jardine. But I didn't expect the answer for my question was due to that at all, since I thought $[I,\cdot]$ and $Y\times\cdot$ have some "duality" purely in the sense of classical category theory. I appreciate your clarification. $\endgroup$ – Math.StackExchange May 6 '16 at 8:54

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