Composing functors with natural transformations

Question

So I'm doing a project in Category Theory. I fully understand natural transformations and functors, but what does it mean to compose them, for example in the monad axioms where you have something like $GF\eta \circ G\epsilon F = 1$ etc. What exactly is $F\eta$? Thanks

Answer 1

Given functors $$\mathcal{A} \xrightarrow{F} \mathcal{B} \overset{G}{\underset{H}{\rightrightarrows}} \mathcal{C} \xrightarrow{K} \mathcal{D}$$ and a natural transformation $\alpha : G \to H$, we can define $\alpha_F$ (also written $\alpha F$) and $K\alpha$ as follows:

• $\alpha_F : G \circ F \to H \circ F$ is the natural transformation whose components are given by $(\alpha_F)_A = \alpha_{FA}$ for $A \in \mathcal{A}$;
• $K\alpha : K \circ G \to K \circ H$ is the natural transformation whose components are given by $(K\alpha)_B = K(\alpha_B)$ for $B \in \mathcal{B}$.

So here, for example, $F\eta : F \to F \circ G \circ F$ has components $(F\eta)_B = F(\eta_B) : FA \to FGFB$.

Answer 2

$F\eta$ is just short for $(F(\eta_C))_{C\in Ob \mathcal{C}}$, wich defines (in this case) a natural transformation $F\Rightarrow FGF$.

Answer 3

$F\eta$ isn't a composition of a functor and a natural transformation, it's just a natural transformation. If you have a transformation $\alpha:F\to G$, then a little examination will show that there's an induced natural transformation $HF\to HG$ for any $H$ that composes appropriately, often denoted $H\alpha$; and similarly one $FH'\to GH'$ for appropriate $H'$ denoted $\alpha H'$. The component of the former at an object $c$ of $F,G$'s domain is $H(\alpha_c)$; in the latter case, the component at an object $d$ in $H'$'s domain is $\alpha_{H'(d)}$.

Another way to think of it is it's just a bit of an abuse of notation for a horizontal composition of two natural transformations: "$F\eta$" is the horizontal composite $id_F *\eta$, though this may not clarify anything depending on what resources you've learned from.

Answer 4

In my opinion, it becomes clearer if you take the homotopical point of view.

Denote by $\mathcal I$ the interval category: its objects are 0 and 1, and it has a unique non-identity morphism $0\to 1$. Then a natural transformation between $F,G \colon \mathcal C \to \mathcal D$ is just a functor $\alpha \colon \mathcal I \times \mathcal C \to \mathcal D$ making the following commute:

where the vertical arrows are the obvious inclusion at $0$ and $1$.

Then for a functor $H \colon \mathcal D \to \mathcal E$, the functor $H\alpha$ is defined and fit into the diagram:

Similarly, if $J \colon \mathcal B \to \mathcal C$ is a functor, $\alpha J$ is an abuse for the induced functor $\mathcal I \times \mathcal B \to \mathcal D$ fitting in the diagram:

Answer 5

This appears to actually have its own name in category theory: it's called whiskering.

Answer 6

In traditional notation, $F\eta$ is the natural transformation $ F \eta : F \to FGF$ whose components are given by $(F\eta)_x = F(\eta_x)$.

The rest of this answer describes a convenient arithmetic system that, IMO, one really should use for the sort of algebra you're doing. And at the end I use it to compute the answer to your question.

The most convenient algebra is to treat everything like a natural transformation:

• Arithmetically treat functor $F$ as its identity natural transformation $\mathrm{id}_F$
• Arithmetically treat an object $x$ as the functor $\bar{x} : \mathbf{1} \to \mathcal{C}$ whose image is $x$. (which you then treat as a natural transformation $\mathrm{id}_{\bar{x}}$)
• Arithmetically treat an arrow $f : x \to y$ as the corresponding natural transformation $\bar{f} : \bar{x} \to \bar{y}$.

All of the basic operations reduce to horizontal and vertical composition:

• Evaluation of a functor at an object is horizontal composition. (because $\mathrm{id}_{\overline{F(x)}} = \mathrm{id}_F \mathrm{id}_{\bar{x}}$)
• Evaluation of a functor at an arrow is horizontal composition. (because $\overline{F(f)} = \mathrm{id}_F \bar{f}$)
• Composition of arrows is vertical composition. (because $\bar{f} \cdot \bar{g} = \overline{fg}$)
• Composition of functors is horizontal composition. (because $\mathrm{id}_F \mathrm{id}_G = \mathrm{id}_{FG}$)

All of the basic identities relating these things boil down to associativity and the interchange law. E.g. the fact a functor respects composition is $F(f \cdot g) = (F \cdot F)(f \cdot g) = (Ff) \cdot (Fg)$.

So, $F \eta$ denotes the horizontal composite of the functor $F$ with the natural transformation $\eta$. This is the natural transformation where

• The source is determined by composing the sources of the two factors: $F 1_\mathcal{C} = F$
• The target is determined by composing the targets of the two factors: $F GF$
• Its value on objects is determined by the associative law for horizontal composition: $(F\eta)x = F(\eta x)$.

So in the traditional notation, $F\eta$ is the natural transformation $ F \eta : F \to FGF$ whose components are given by $(F\eta)_x = F(\eta_x)$.