CM:

1. The ‘determination’ (or ‘extension’) problem

Givenfandhas shown, what are allg, if any, for whichh = g ∘ f?

2. The ‘choice’ (or ‘lifting’) problem

Givengandhas shown, what are allg, if any, for whichh = g ∘ f?

These two notions are analogous to division problem.

Definitions: Iff: A => B:

- a
retraction for fis an arrowr: B => Afor whichr ∘ f = 1_{A}- a
section for fis an arrows: B => Afor whichf ∘ s = 1_{B}

Here’s the external diagram for retraction problem:

and one for section problem:

If an arrow

f: A => Bsatisfies the property ‘for anyy: T => Bthere exists anx: T => Asuch thatf ∘ x = y‘, it is often said to be ‘surjective for arrows from T.’

I came up with my own example to think about what surjective means in set theory:

Suppose John and friends are on their way to India, and they are given two choices for their lunch in the flight: chicken wrap or spicy chick peas. Surjective means that given a meal, you can find at least one person who chose the meal. In other words, all elements in codomain are covered.

Now recall that we can generalize the concept of elements by introducing singleton explicitly.

Compare this to the category theory’s definition of surjective: for any *y: T => B* there exists an *x: T => A* such that *f ∘ x = y*. For any arrow going from *1* to *B* (lunch), there is an arrow going from *1* to *A* (person) such that *f ∘ x = y*. In other words, *f* is surjective for arrows from *1*.

Let’s look at this using an external diagram.

This is essentially the same diagram as the choice problem.

Definitions: An arrowfsatisfying the property ‘for any pair of arrowsxand_{1}: T => Ax, if_{2}: T => Af ∘ xthen_{1}= f ∘ x_{2}x‘, it is said to be_{1}= x_{2}injective for arrows from T.If

fis injective for arrows fromTfor everyT, one says thatfisinjective, or is amonomorphism.

Here’s how **injective** would mean in terms of sets:

All elements in codomain are mapped only once. We can imagine a third object *T*, which maps to John, Mary, and Sam. Any of the composition would still land on a unique meal. Here’s the external diagram:

Definition: An arrowfwith this cancellation property ‘iftthen_{1}∘ f = t_{2}∘ ft’ for every T is called an_{1}= t_{2}epimorphism.

Apparently, this is a generalized form of surjective, but the book doesn’t go into detail, so I’ll skip over.

Definition: An endomorphismeis called idempotent ife ∘ e = e.

An arrow, which is both an endomorphism and at the same time an isomorphism, usually called by one word

automorphism.

I think we’ve covered enough ground. Breaking categories apart into internal diagrams really helps getting the hang of it.

learning Scalaz — Determination and choice