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A homotopical category is category with a distinguished class of morphism called weak equivalence.

A class $W$ of morphisms in $\mathcal{C}$ is a weak equivalence if:

  • All identities are in $W$.
  • for every $r,s,t$ which compositions $rs$ and $st$ exist and are in $W$, then so are $r,s,t,rst$

Given two homotopical categories $\mathcal{C}$ and $\mathcal{D}$, there is the usual notion of functor category $\mathcal{C}^{\mathcal{D}}$. But, there is also the category of homotopical functors from $\mathcal{C}$ to $\mathcal{D}$, which I am confused about. Note that, $F:\mathcal{C} \rightarrow \mathcal{D}$ is a homotopical functor if it preserve the weak equivalence. I am quoting from page 78 of a work by Dwyer, Hirschorn, Kan and Smith; called Homotopy Limit Functors on Model Categories and ...

Homotopical functor category $\left(\mathcal{C}^{\mathcal{D}}\right)_W$ is the full subcategory of $\mathcal{C}^\mathcal{D}$ spanned by the homotopical functors. (What does it mean?)

Also, in the definition of homotopical equivalence of homotopical categories, we read that $f:\mathcal{C}\rightarrow \mathcal{D}$ and $g:\mathcal{D}\rightarrow \mathcal{C}$ are homotopical equivalence if their compositions $fg$ and $gf$ are naturally weakly equivalent, i.e. can be connected by a zigzag of natural weak equivalence.

I do not know what the authors mean by a zigzag of natural equivalence. Please, help me with these concepts and tell me where I can learn more about them.

Thank you.

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up vote 4 down vote accepted

A zig-zag of natural equivalences from a functor $F$ to a functor $G$ means a sequence of functor $F_1,\cdots, F_n$, with $F_1=F$ and $F_n=G$, and natural equivalences (in the context above, these are natural transformations whose components belong to the class of weak equivalences) $\alpha_1 ,\cdots ,\alpha_{n-1}$ such that, for each $\alpha_i$, either the domain of $\alpha_i$ is $F_i$ and the codomain is $F_{i+1}$ or the other way around.

So, the sequence of natural equivalences doesn't all have to point in the same direction (if they did, you can just compose all of them and get a single natural equivalence).

It is perhaps better to think of such zig-zag constructions in the context of the process of the localization of a category with respect to a subclass of its morphisms declared to be weak equivalences. When you formally invert those weak equivalences that don't actually have inverses you essentially allow such zig-zags of weak equivalences. In more detail, the localization of $C$ with respect to $W$ is going to have the same objects as $C$ but arrows are going to be equivalence classes of zig-zags. Of course, a lot of care is needed to ensure that such a localization even exists as the zig-zags can get out of control.

A much more thorough discussion with many examples of localizations is to be found here.

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Thank you.. I need to think to localization more. I see what universal property is satisfied by localization, but the proof, i.e. the construction of the localized category is yet artificial in my eyes. – user34942 Jan 26 '13 at 7:50

Maybe you're beginning your journey through model, localized, homotopy categories by a steep way. I would try this short paper first: W. G. Dwyer and J. Spalinski.

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Totally agree. This one sounds more instructive. – user34942 Jan 26 '13 at 7:50

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