Lecture 44 - Categories, Functors and Natural Transformations

Lecture 44 - Categories, Functors and Natural Transformations

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 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?

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