Last time I introduced natural transformations, and I think it's important to solve a bunch more puzzles to get a feel for what they're like. First I'll remind you of the basic definitions. I'll go through 'em quickly and informally:
**Definition.** A **category** \\(\mathcal{C}\\) consists of:
1. a collection of **objects** and
2. a set of **morphisms** \\(f : x \to y\\) from any object \\(x\\) to any object \\(y\\),
such that:
a) each pair of morphisms \\(f : x \to y\\) and \\(g: y \to z\\) has a **composite** \\(g \circ f : x \to z \\) and
b) each object \\(x\\) has a morphism \\(1_x : x \to x\\) called its **identity**,
for which
i) the **associative law** holds: \\(h \circ (g \circ f) = (h \circ g) \circ f\\), and
ii) the **left and right unit laws** hold: \\(1_y \circ f = f = f \circ 1_x \\) for any morphism \\(f: x \to y\\).
A category looks like this:
**Definition.** Given categories \\(\mathcal{C}\\) and \\(\mathcal{D}\\), a **functor** \\(F: \mathcal{C} \to \mathcal{D} \\) maps
1. each object \\(x\\) of \\(\mathcal{C}\\) to an object \\(F(x)\\) of \\(\mathcal{D}\\),
2. each morphism \\(f: x \to y\\) in \\(\mathcal{C}\\) to a morphism \\(F(f) : F(x) \to F(y) \\) in \\(\mathcal{D}\\) ,
in such a way that:
a) it preserves composition: \\(F(g \circ f) = F(g) \circ F(f) \\), and
b) it preserves identities: \\(F(1_x) = 1_{F(x)}\\).
A functor looks sort of like this, leaving out some detail:
**Definition.** Given categories \\(\mathcal{C},\mathcal{D}\\) and functors \\(F, G: \mathcal{C} \to \mathcal{D}\\), a **natural transformation** \\(\alpha : F \to G\\) is a choice of morphism
\[ \alpha_x : F(x) \to G(x) \]
for each object \\(x \in \mathcal{C}\\), such that for each morphism \\(f : x \to y\\) in \\(\mathcal{C}\\) we have
\[ G(f) \alpha_x = \alpha_y F(f) ,\]
or in other words, this **naturality square** commutes:
A natural transformation looks sort of like this:
You should also review the [free category on a graph](https://forum.azimuthproject.org/discussion/2204/lecture-36-categories-from-graphs/p1) if you don't remember that.
Okay, now for a bunch of puzzles! If you're good at this stuff, please let beginners do the easy ones.
**Puzzle 129.** Let \\(\mathbf{1}\\) be the free category on the graph with one node and no edges:
Let \\(\mathbf{2}\\) be the free category on the graph with two nodes and one edge from the first node to the second:
How many functors are there from \\(\mathbf{1}\\) to \\(\mathbf{2}\\), and how many natural transformations are there between all these functors? It may help to draw a graph with functors \\(F : \mathbf{1} \to \mathbf{2} \\) as nodes and natural transformations between these as edges.
**Puzzle 130.** Let \\(\mathbf{3}\\) be the free category on this graph:
How many functors are there from \\(\mathbf{1}\\) to \\(\mathbf{3}\\), and how many natural transformations are there between all these functors? Again, it may help to draw a graph showing all these functors and natural transformations.
**Puzzle 131.** How many functors are there from \\(\mathbf{2}\\) to \\(\mathbf{3}\\), and how many natural transformations are there between all these functors? Again, it may help to draw a graph.
**Puzzle 132.** For any category \\(\mathcal{C}\\), what's another name for a functor \\(F: \mathbf{1} \to \mathcal{C}\\)? There's a simple answer using concepts you've already learned in this course.
**Puzzle 133.** For any category \\(\mathcal{C}\\), what's another name for a functor \\(F: \mathbf{2} \to \mathcal{C}\\)? Again, there's a simple answer using concepts you've already learned here.
**Puzzle 134.** For any category \\(\mathcal{C}\\), what's another name for a natural transformation \\(\alpha : F \Rightarrow G\\) between functors \\(F,G: \mathbf{1} \to \mathcal{C}\\)? Yet again there's a simple answer using concepts you've learned here.
**Puzzle 135.** For any category \\(\mathcal{C}\\), what are functors \\(F : \mathcal{C} \to \mathbf{1} \\) like?
**Puzzle 136.** For any natural number \\(n\\), we can define a category \\(\mathbf{n}\\) generalizing the categories \\(\mathbf{1},\mathbf{2}\\) and \\(\mathbf{3}\\) above: it's the free category on a graph with nodes \\(v_1, \dots, v_n\\) and edges \\(f_i : v_i \to v_{i+1}\\) where \\(1 \le i < n\\). How many functors are there from \\(\mathbf{m}\\) to \\(\mathbf{n}\\)?
**Puzzle 137.** How many natural transformations are there between all the functors from \\(\mathbf{m}\\) to \\(\mathbf{n}\\)?
I think Puzzle 137 is the hardest; here are two easy ones to help you recover:
**Puzzle 138.** For any category, what are functors \\(F : \mathbf{0} \to \mathcal{C}\\) like?
**Puzzle 139.** For any category, what are functors \\(F : \mathcal{C} \to \mathbf{0} \\) like?
**[To read other lectures go here.](http://www.azimuthproject.org/azimuth/show/Applied+Category+Theory#Chapter_3)**