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I'm reading Jeff Strom's book on Homotopy Theory and I am trying to make some sense of a certain exercise. On page 91, "Homotopy in Mapping Categories" we consider the category of arrows of topological spaces, that is, the objects are continuous maps $f:X \rightarrow Y$ , X and Y are topological spaces. Morphisms between two maps $f:X \rightarrow Y$, $g: Z \rightarrow W$ are pairs $\alpha = (\alpha^d,\alpha_t)$ $\alpha^d:X \rightarrow Z$, $\alpha_t : Y \rightarrow W$ making the obvious diagram commute. Let us call this category Map.

Later on, he introduces a notion of homotopy in Map. A homotopy between morphisms $\alpha:f \rightarrow g$ and $\beta:f \rightarrow g$ consists of two maps $(H^d,H_t)$, $H^d: X \times I \rightarrow Z $ , $H_t: Y \times I \rightarrow W$ , such that we have $H_t \circ (f \times id_I) = g \circ H^d $. Now, $\alpha:f \rightarrow g$ is a homotopy equivalence in Map if we can find $\beta:g \rightarrow f$ such that both $\alpha \circ \beta \simeq id_{g}$ and $\beta \circ \alpha \simeq id_{f}$. OK, so later on he explains that this is quite a strong requirement of homotopy, and one sometimes weakens it. One is pointwise homotopy equivalence in Map, which is the requirement that both $\alpha^d$ and $\alpha_t$ as above are homotopy equivalences of topological spaces. He then asks the following:

Show that if $f \simeq g:X \rightarrow Y$, then f and g are equivalent in $\textbf{h}\mathcal{T}$.

Here $\textbf{h} \mathcal{T}$ is a homotopy category of topological spaces. I am very confused by the above problem. My question is:

What does the problem ask me to do?

Does $f \simeq g$ above mean that f and g are homotopic? Or simply that they are an equivalence in Map? I am not sure what to show to be honest, and I have been trying to understand the notation for quite some time now here, so I would be beyond grateful for any help on this matter.

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    ... and then deduce from this information that $f$ and $g$ are homotopic (*as maps in* $T$!).2012-07-26

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