1

When we have a category $\mathcal C$, it is usual to define the category $\textrm{Ar}(\mathcal C)$ of morphisms of $\mathcal C$ as the one whose objects are the morphisms of $\mathcal C$ and whose morphisms are commutative squares: If $f:A\to B$ and $g:C\to D$ are elements of $\textrm{Ar}(\mathcal C)$, then a morphism from $f$ to $g$ is a pair of morphisms $(h:A\to C, h':B\to D)$ such that $h'f = gh$.

Therefore, since small categories and functors form a category, it would be expected that the morphisms between functors would be defined this way. But this is not the case, and instead what are used as the morphisms of functors are natural transformations. This is very strange and inelegant: why are morphisms between functors in usual categories defined one way but the ones in the category of categories defined differently?

Note that i'm not asking whether we can define morphisms the way i said, i know we can. I'm also aware of similar questions on this site but none answers where the different treatment of different categories comes from. (And to be honest i don't find any of the anwers in the other related questions satisfying either.)

Carla_
  • 532

1 Answers1

4

A morphism between two morphisms only requires two other maps. Morphisms between functors must do that for every morphism in the source category.

If $F:\mathcal C\to\mathcal D$ and $G:\mathcal C\to\mathcal D$ are two functors then we require a commutative diagram for the four objects $F(A), F(B), G(A), G(B)$ for each $A,B\in\mathcal C$ for which a morphism $f:A\to B$ exists.

$\require{AMScd}$ \begin{CD} F(A) @>F(f)>> F(B)\\ @V \eta_A V V @VV \eta_B V\\ G(A) @>>G(f)> G(B) \end{CD}

A commutative diagram as above is required for every morphism in $\mathcal C$. If we imagine a category as a diagram of nodes and arrows, a functor maps the entire diagram to a diagram in another category. The natural transformation ensures that the entire diagrams of $F(\mathcal C)$ and $G(\mathcal C)$ commute.

John Douma
  • 11,426
  • 2
  • 23
  • 24
  • 1
    "A morphism between two morphisms only requires two other maps." - but functors are morphisms in a category of categories, so why don't they only require two other maps too? – Carla_ Mar 26 '23 at 14:01
  • It is important to remember that category theory was invented to define functors which were invented to define natural transformations. There are a lot of data associated with a functor that must be accounted for in a morphism.

    Try defining morphisms between functors your way. You would end up with two additional functors and four categories. That might define a valid arrow category but it would be too general for our purposes.

     – John Douma Mar 26 '23 at 15:47
  • From "General Theory of Natural Equivalences" by Eilenberg and MacLane

    "It should be observed first that the whole concept of a category is essentially an auxiliary one; our basic concepts are essentially those of a functor and of a natural transformation (the latter is defined in the next chapter). The idea of a category is required only by the precept that every function should have a definite class as domain and a definite class as range, for the categories are provided as the domains and ranges of functors. "

     – John Douma Mar 26 '23 at 16:08
  • 1
    I am aware that category theory was originally invented to define natural transformations. I'm not looking for historical motivation but for a conceptual reason of why natural transformations should be the morphisms between functors. I've got good reasons for why functors should be the morphisms between categories: they preserve the category structure which is morphism composition and identity, just like for other structures in algebra homomorphisms preserve their structure. But what structure does a morphism or a functor have? – Carla_ Mar 26 '23 at 16:16
  • 1
    An abstract morphism only has three things: source, arrow and target. A functor has all of those for each morphism in the source category. Moreover, a functor preserves identities and morphism composition. – John Douma Mar 26 '23 at 16:29
  • Agreed with OP that although this answer is helpful, especially vis a vis the historical context, it doesn't answer the intent of the original question. – hasManyStupidQuestions Feb 24 '24 at 22:17
  • 1
    @hasManyStupidQuestions In a morphism, there is one arrow that goes from one object to another. But categories are composed of many objects and arrows (components) and so any morphism that preserves the structure of a category must operate on diagrams. Those diagrams represent the structure of a category. In fact, the entire category can be viewed as a diagram of objects and arrows. That is why functors are defined hat way. Compare this definition with morphisms in a category whose objects are functions. – John Douma Feb 25 '24 at 01:57
  • I said this in response to another question (https://math.stackexchange.com/questions/4870033/are-commutative-squares-in-some-sense-universal-among-edge-symmetric-double-ca ), but basically I agree with this position while also thinking some caution is advised. It's not a priori clear to me that there exists any obvious unique "best" definition of morphism between diagrams, and because diagrams are equivalent to certain classes of functors between categories, one can not make a choice of morphism between diagrams without also implicitly making a choice of morphism between certain functors. – hasManyStupidQuestions Feb 25 '24 at 17:57
  • At the risk of shamelessly self-promoting, I have made my exact concerns more explicit / concrete in two previous questions: https://math.stackexchange.com/questions/4870033/are-commutative-squares-in-some-sense-universal-among-edge-symmetric-double-ca and https://math.stackexchange.com/questions/4864856/among-morphisms-of-morphisms-what-makes-commutative-squares-special Basically it seems like one can reduce the question of choosing a definition of morphism between diagrams to choosing a definition of morphism between diagrams of the "walking arrow", i.e. of "morphisms-of-morphisms". – hasManyStupidQuestions Feb 25 '24 at 18:02