HW #1 due Friday Jan 19 -- major correction

1. Prove the assertion from class that "disjoint union with basepoints amalgamated" really is the coproduct in the category of sets-with-basepoints.

2. A directed graph or digraph (V,E,h,t) is a pair of sets V and E (called "vertices" and "edges") plus two functions h,t: E -> V called "head" and "tail".
(In particular, these digraphs may have repeated edges, and h(e)=t(e), whatever.)

a. Given a category, come up with a definition of "the underlying directed graph".

b. Given a directed graph, come up with a (more interesting) definition of "the free category on the directed graph".

You can check that you have the right notion using the following:

c. Given a poset P, define a directed graph G such that the free category on G is the category we associated to P. Oops, this isn't quite possible for infinite P. So assume P finite.
OOPS. Not any poset will do. Assume that there is at most one chain p > q > ... > s connecting any two elements p,s.

d. Let v,w be two vertices of a directed graph G. If the corresponding objects in the free category on G are isomorphic, show v=w.

3. Define a monomorphism f as one such that whenever f o g = f o h, it follows that g = h. (Here f,g,h are various morphisms in C for which o is defined.)

a. Let C = Set. Show f is monic (i.e. a monomorphism) iff f is 1:1.

b. (Harder.) Let C = Grp. Show f is monic iff f is 1:1.

Probably some more as I think of them; Monday at the latest.