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I have never seen a definition of functoriality, but this word keeps coming out in texts on category theory. Wikipedia redirects to the page about functor if I search functoriality on it, but I'm not sure whether functoriality is exactly the functor.

What is functoriality? (I'm only familiar with basic terminologies on category theory such as natural transformation, adjoint functor etc, so please do not use complicated terminologies.) Thank you in advance!

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    $\begingroup$ Functoriality means that something is a functor. E.g. when someone asks about the functoriality of some construction, they're asking whether or not it can be upgraded to a functor. $\endgroup$ Sep 11, 2016 at 1:37

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Let's say I have some nice mathematical construction $C$ that takes in (say) a group, and outputs (say) a topological space.

Well, groups don't exist in a vacuum! There are lots of homomorphisms between groups, and we learn a lot about a group by learning about the properties of the homomorphisms into and out of it. Similarly for topological spaces - with "homomorphism" replaced by "continuous map."

That is, both $\{$groups$\}$ and $\{$topological space$\}$ are not just sets (fine, classes), but categories. The functoriality of $C$ is the property that $C$ "plays nicely" with this categorial structure: roughly speaking, given a group homomorphism $f: A\rightarrow B$, I should get a continuous map $c_f: C(A)\rightarrow C(B)$ in some reasonable way. Informally, I want to say that $C$ "really is" a functor (although of course this is kind of an abuse of terminology; initially at least, $C$ might only be defined on objects).


Note that you could have a map on objects which can be "upgraded" to a functor in more than one natural way; however, in practice this doesn't tend to happen very often.

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  • $\begingroup$ To sum up, does functoriality mean "the realization of a functor"? $\endgroup$
    – Rubertos
    Sep 11, 2016 at 1:50
  • $\begingroup$ @Rubertos I'm not sure I'd use the word "realization" (that suggests that the thing-which-is-functorial has more (possibly irrelevant) information than the functor - e.g. a topological realization of a simplicial set is more complicated than the simplicial set alone; note that here the opposite is the case) but yes. I would call the thing-which-is-functorial the reduct of a functor, being a logician (although I also quite like Qiaochu's use of upgrade/downgrade). $\endgroup$ Sep 11, 2016 at 3:28

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