# Seven Sketches

We are currently reading “Seven Sketches in Compositionality: An Invitation to Applied Category Theory”Brendan Fong and David I Spivak, “Seven Sketches in Compositionality: An Invitation to Applied Category Theory,” 2018, http://arxiv.org/abs/1803.05316.

and I’ll write some comments here. (The numbering in the printed book is different, we refer to the arXiv PDF.)

## Ch. 1: Generative Effects: Orders and Adjunctions

### Adjoints of the powerset functor

Example 1.117 speaks about the adjoints of the powerset functor. Given a function between sets, we get a monotone map that is the preimage (you may know this also as ):More about the notation.

The book now presents two adjoints to this:

given by

We can imagine say: lower shriek

as the function that returns all that occur in the image , and as the function that returns all that only occur in the image .

The preimage is called a pullback in the text:

Here is the restriction of to the subset .

### Closure operators

Exercise 1.119:

follows directly from the definition of an adjunction.Remember, left adjoint to and

Theorem: .
Proof: First, we show for all : since the Galois connection yields and is monotonic, we get ; but since , we also get , thus . Now applying on both sides yields the result.

## Ch. 3: Functors, Natural Transformations, and Databases

We review the definition of a natural transformation :

Definition 3.49: Let and be categories, and let be functors. To specify a natural transformation , for each object , one specifies a morphism in , called the -component of .

These components must satisfy the following naturality condition: for every morphism in , the following equation must hold:

Or, as a commutative diagram:

So what happens is that a morphism is mapped to morphisms .