Problem set 2 due Friday, Feb 17

Here are a few problems about the material in lectures 2 and 3.(If something isn’t clear about the statements, looking at the notes might help.)

Recall the definition of concatenation or composition of unit-time paths: for \(0 \le t \le 1/2\) we set \(\beta \alpha(t) = \alpha(2t)\), and for for \(1/2 \le t \le 1\) we set \(\beta \alpha(t) = \beta(2(t-1/2))\). We claimed that for this definition there are no identities, and in particular that you almost never have \(\alpha\ \text{id}_{\alpha(0)} = \alpha\). For which \(\alpha\) does that equality hold? Be sure to prove the paths you describe are the only examples. [1]
We mentioned two definitions [2] of homotopies (relative to the endpoints) of Moore paths of different domains. Prove they are equivalent.
A space \(X\) is called contractible if there is a function \(C : X \times [0,1] \to X\) and a point \(x_0\) in \(X\), such that for all \(x\) in \(X\) we have \(C(x,0) = x\) and \(C(x,1) = x_0\). You can think of this as giving for every \(x\) a path \(t \mapsto C(x,t)\) between \(x\) and \(x_0\), in such a way that the path depends continuously on \(X\). From the point of view of homotopy theory, a contractible space is equivalent to a single point. Prove that the fundamental groupoid of a contractible space is indiscrete.
We talked about representable functors and mentioned that if two objects in a category represent isomorphic functors they are isomorphic themselves. This exercises guides you to a proof of that, proving the extremely important Yoneda Lemma along the way. The Yoneda Lemma 1) is very very useful, 2) can be pretty confusing the first time you see it, 3) is almost trivial to prove! It’s a good example of something that’s hard to explain understandably to someone else: for some reason it seems to be much easier to learn it by proving it yourself (a couple of times). OK, here we go:

Let \(C\) be a category and let \(X\) be an object of \(C\). Recall that there is a functor \(\hom(X,\_) : C \to \text{Set}\) given on objects by \(Y \mapsto \hom(X,Y) = \) the set of all morphisms \(X \to Y\) in \(C\).
1. Write the definition of \(\hom(X,\_)\) on morphisms.
2. (Yoneda, part 1.) Let \(F : C \to \text{Set}\) be any functor, and let \(x \in F(X)\). Use \(x\) to define a natural transformation \(\hom(X,\_) \to F\) (there is only one reasonable way to do this).
3. (Yoneda, part 2.) Now let \(\eta : \hom(X,\_) \to F\) be any natural transformation at all. Show that it is actually equal to one of the natural transformations you constructed in the previous part. Hint: First figure out which \(x \in F(X)\) this \(\eta\) corresponds to: it must some element of \(F(X)\) for which you can write down a formula using \(\eta\), so there aren’t many options. Now prove that choice of \(x\) does give you \(\eta\), you will (obviously) have to use naturality for \(\eta\).
4. As a corollary of the Yoneda Lemma, we get that there is a bijection between morphisms \(Y \to X\) and natural transformations \(\hom(X,\_) \Rightarrow \hom(Y,\_)\). Prove that if there is a natural isomorphism[\³] \(\hom(X,\_) \Rightarrow \hom(Y,\_)\), then \(X\) and \(Y\) are isomorphic (i.e., in the bijection of the previous sentence, natural isomorphisms correspond to isomorphisms \(Y \to X\)).
Here are two simple functors from the category of groupoids to the category of sets:
1. The object set functor defined on groupoids by \(\text{obj}(G)\) = set of objects of \(G\), and on morphisms of groupoids “in the obvious way” [3], i.e., part of the data in a morphism of groupoids \(f : G \to H\) is precisely a functions from the set of objects of \(G\) into the set of objects of \(H\), so take \(\text{obj}(f)\) to be that function.
2. The morphism set functor, defined on groupoids by \(\text{mor}(G) =\) set of morphisms of \(G\), or, more precisely, the disjoint union of the morphisms sets \(\bigcup_{X,Y} \hom_G(X,Y)\). Again, the definition on morphisms is the obvious one: a morphism of groupoids \(G \to H\) includes a function that sends every morphism in \(G\) to one in \(H\).
Are these two functors representable? If so, what groupoids represent them?

[1] This exercise is to make sure you know the definition of continuous function.

[2] I’ll explain them here again for convenience. Both definitions are in terms of the notion of homotopy (rel. endpoints) of paths with the same domain and the notion of padding a path. Given \(\alpha : [0,t_0] \to X\) and \(t \ge t_0\), we define \(\alpha\) padded to length \(t\) as the function \([0,t] \to X\) which agrees with \(\alpha\) on \([0,t_0]\) and is constant, equal to \(\alpha(t_0)\) on \([t_0,t]\). The two definitions of homotopic rel. endpoints for paths of differing lengths were:

there exists a homotopy after padding the shorter path to the length of the longer path, and
there exists a homotopy after padding both paths to some common length (\(\ge\) the lengths of both).

[3] Category theorists say things like this all the time.