Problem set 1 due Friday, Feb 10

Here are a few problems to test your understanding of the material in lecture 1. (If something isn’t clear about the statements, looking at the notes might help.)

The first two are meant to be done with the first definition of groupoid, as a set with a partially defined multiplication that is associative and has inverses.

1. Show that if in a groupoid both \(ab\) and \(ac\) are defined \(bb^{-1} = cc^{-1}\).

   Corrected! It used to ask to prove \(b^{-1}b = c^{-1}c\), which of course is usually not true.

2. Show that in a groupoid \((a^{-1})^{-1} = a\).

3. Prove that the two definitions we gave are equivalent. The first one is mentioned above, and the second is as a category where all morphisms are invertible.

   This is easy but a little long and was sketched in class. First of all, let’s figure out what we mean by this equivalence: it means that any groupoid according to the first definition can be regarded as the collection of morphisms of a category with invertible morphisms, and that given any category with all morphisms inverible, it’s morphisms (more precisely, the disjointified union of its morphism collections \(\hom_C(X,Y)\)) form a groupoid according to the first definition. The second statement is clear, to prove the first you just need to do the following: given a groupoid according to the first definition, define for it objects, and sources and targets of elements and prove that \(ab\) is defined if and only if \(\text{source}(a) = \text{target}(b)\). (Very briefly explain why this is enough.) We saw two methods for doing this:

   Method 1. Define \(\text{source}(a) = \{ b : ab\ \text{is defined}\}\), and \(\text{target}(a) = \text{source}(a^{-1})\).

   Method 2. Define the objects to be the collection of identities \(\{aa^{-1}\}\) and then \(\text{source}(a) = a^{-1} a\), \(\text{target}(a) = a a^{-1}\).

4. “A partial order is almost category such that for every pair of objects \(X\) and \(Y\) there are either no morphisms \(X \to Y\), or only one.” Explain why the word “almost” appears in the previous sentence and show how to fix the definition of a partial order as a special kind of category.
5. What is the action groupoid of the action of \(G\) on itself by left multiplication?