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Functors and morphisms of functors (aka natural transformations) have become powerful tools in all areas of pure (and meanwhile also applied) mathematics. There are lots of nontrivial constructions of functors and morphisms of functors. But I think I don't know any example where it is hard to prove that something already constructed (I will make this precise) is a functor, or a morphism of functors. Therefore my question is:

What are interesting and natural examples for categories $C,D$ and functions $F : \mathrm{Ob}(C) \to \mathrm{Ob}(D)$, $F : \mathrm{Mor}(C) \to \mathrm{Mor}(D)$ with $F(f : X \to Y) : F(X) \to F(Y)$ where the proof that $F$ is a functor is nontrivial and involves interesting mathematics? Also, the proof should not just replace $F$ by a different function coming from an obvious functor.

What are interesting and natural examples of functors $F,G : C \to D$ and functions $\eta : C \to \mathrm{Mor}(D)$ with $\eta(x) : F(x) \to G(x)$ where the proof that $\eta$ is a morphism of functors is nontrivial and involves interesting mathematics?

I have put "natural" here because for example I am not interested in the case where $C,D$ are just plain monoids / groups (although there might be interesting functions between monoids where it is hard to prove that they are homomorphisms). Further non-examples include the functoriality of hom-sets, (co)limits and most of the basic cohomology and homotopy theories.

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This probably isn't quite what you want, but it might be of some interest: – Adam Saltz Jan 9 '13 at 4:46
Are you asking about when it is difficult to verify that a given pair of maps $\operatorname{ob} \mathcal{C} \to \operatorname{ob} \mathcal{D}$ and $\operatorname{mor} \mathcal{C} \to \operatorname{mor} \mathcal{D}$ is a functor? In that case I would say it boils down to how easy it is to compute the composition and equality of morphisms in $\mathcal{C}$ and $\mathcal{D}$. This can be extremely non-trivial, but the best example I know of is the bicategory of polynomials. – Zhen Lin Jan 9 '13 at 7:51
Yes this is my question. Could you please add this example (polynomials) and explain a little bit what makes the proof interesting and nontrivial? – Martin Brandenburg Jan 12 '13 at 3:02
@Adam: Why should this fit to the question? Usually in papers one just writes down the definition of $F$ on objects and on morphisms, and then claims without proof that $F$ is a functor, because it is always(?) trivial. I think this also happens in the paper you linked. – Martin Brandenburg Jan 26 '13 at 17:54

A whole class of examples is coming from the theory of Quillen model categories. For a concrete example, consider the category $Top$ of topological spaces and the category $sSet$ of simplicial sets. Recall that a localization of a category $C$ by a class of morphisms $W$ is the category resulting from $C$ by freely inverting the arrows in $W$. Keeping track of the resulting category can be extremely difficult.

Consider the localization of $Top$ at the weak equivalences and consider the localization of $sSet$ at those morphisms that the geometric realization turns into weak equivalences.

You can now consider the processes between these localizations that are inherited by the geometric realization functor and the singular complex functor. Of course, these will be messy to describe. Verifying directly that these will be functors is hard. In a sense, the whole point of Quillen model structures is to avoid working directly with the localizations and instead working with the original category. Then one can show that with respect to the appropriate model structures each of the functors I mentioned is a Quillen functor which then implies that it does indeed pass to a functor between the localizations.

Of course, the preservation of hard work is in place. We don't get anything non-trivial for free here since all the hard work goes into proving that model structures.

I should mention though that probably the 'right' way to think of Quillen model categories is not so much as tools to work with the localization indirectly but rather as presentations of $\infty $-categories: .

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