HW due Friday Jan 26 -- corrected

1. Define a category Graph whose objects are digraphs, and define a forgetful functor Cat -> Graph.

2. Show that the "free category on a digraph" (from last HW) extends to a functor Graph -> Cat.

3. Show that these two functors (in #1 and #2) are adjoint.

4. Call a category a pre-poset if any two objects A,B have at most one morphism A->B. This defines a subcategory Pre-poset of Cat. Show that the inclusion functor Pre-poset -> Cat has a right left adjoint Cat -> Pre-poset. (In particular, figure out how to assign a pre-poset to a category.)

[This left adjoint is almost a right adjoint... there is a natural transformation from one of the relevant set-valued hom functors to the other; it's just not an isomorphism.]

5. Define the cardinality |C| of a category C as a sum over the isomorphism classes of objects, of 1 / |Aut(X)|. (This definition rarely makes sense -- Aut(X) may be infinite, or the sum may diverge -- but it's cool when it does.)

a. Check that |FinSet| = e, where FinSet is the category of finite sets.

b. Let X be a G-set (both finite), and X/G the category defined in class, whose objects are X and Hom(x,y) = {g : gx = y}. Show that |X/G| = |X|/|G|.

(If you think this stuff is cool, you might check out this chatty paper for lots more about it!)

6. Let C be a category that has products. For every X,Y objects in C, pick a product of X and Y; call this choice P : CxC -> C. Show that P can be extended to a functor.

7. Let Mon be the category of monoids, and Grp -> Mon the inclusion functor. Find left and right adjoints of this functor (two ways of assigning a group to a monoid).