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Calling this category the homotopy category at this level of generality is a bit misleading. The sense in which it is a homotopy-category-as-in-morphisms-up-to-homotopy comes from, for example, taking the simplicial localization first. This is a simplicially enriched category which presents an $\infty$-category. It has a notion of homotopy between maps given ...

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The standard simplicial enrichment on $Simp(C)$ is defined as follows. First, note that since $C$ is cocomplete, it can be considered to be tensored over $Set$: If $X$ is an object of $C$ and $S$ is a set, then $X\otimes S$ is a coproduct of copies of $X$ indexed by elements of the set $S$. If $K$ is a simplicial set and $A$ is an object of $Simp(C)$, we ...

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There are three important model structures on spaces: Quillen, with Serre fibrations and weak equivalences; Strom, with Hurewicz fibrations and closed Hurewicz cofibrations and homotopy equivalences; mixed, with Hurewicz fibrations and weak equivalences. The last two are Quillen equivalent (which means they present the same notion of homotopy theory,) but ...

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It seems to me that the general pattern is that things to do with the first variable of the hom-functor – colimits, cofibrations, tensors, etc. – are thought of as "left" while things to do with the second variable of the hom-functor – limits, fibrations, cotensors, etc. – are thought of as "right". The main exception to the above rule of thumb is ...

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