Let's Grothendieck Everything in Sight (Week 4)

last week* I showed you how to use the the fundamental groupoid functor $ \pi_1 : \mathsf{Top} \to \mathsf{Cat}$ to turn the category of topological spaces and continuous functions into the category of pointed topological spaces where the continuous maps are up to a path.

This week I'm going to try to show you how to do something similar with homology. Roughly, this will yield a category where the objects are topological spaces "pointed" with a sum of n-dimensional holes.  The morphisms in this category will be continuous maps which preserve these holes up to another sum of n-dimensional holes.

The first obstacle is that the n-th homology of a space is not a category...it's a mere group. There's a standard way to turn groups into categories.

Theorem:** A group is a one object category where every morphism is invertible.

In his book Quantum Quandaries: a Category-Theoretic Perspective John Baez said the following:

"Every sufficiently good analogy is yearning to be a functor."

The "group-as-a-one-object-category" analogy is no exception! There's a functor

$ C: \mathsf{Grp} \to \mathsf{Cat}$ which

• sends a group $ G$ to the a category $ C(G)$, with one object, a morphism for each element $ g \in G$, and with composition defined by the group multiplication.
• For a group homomorphism $ f: G \to H$ there is a functor $ C(f): C(G) \to C(H)$ which makes the only possible assignment on objects (what is this???) and sends a morphism $ g \in G$ to the morphism $ f(g) \in H$. The fact that $ f$ is a homorphism expresses the fact that $ C(f)$ respects composition and identities.

This functor restricts to a functor $ D: \mathsf{Ab} \to \mathsf{Cat}$ where $ \mathsf{Ab}$ is the category of abelian groups. You can compose $ D$ with the n-th homology functor $ H_n : \mathsf{Top} \to \mathsf{Ab}$ to a get a functor

$ D \circ H_n : \mathsf{Top} \to \mathsf{Cat}$

You know what usually happens next in this blog...but I'm gonna throw a curve. If you take the Grothendieck construction of this you don't a category of topological spaces "pointed with a sum of n-dimensional holes". This is because the categories in the image of $ D \circ H_n$ have only one object so when you take the Grothendieck construction the elements of your homology group are deferred to the morphisms. I'm not saying this is a bad thing; it's also a fun category to think about.

Exercise: Describe the category $ \int D \circ H_n$.

Instead we're going to replace $ D$ with a different functor. Let,

$ E : \mathsf{Ab} \to \mathsf{Cat}$ be the functor which makes the following assignments:

• For an abelian group $ G$, $ E(G)$ is a category where
  • an object $ g$ is an element of $ G$ and,
  • an arrow $ h : g \to g'$ is an element $ h \in G$ with $ h+g = g'$.
  • for arrows $ h: g \to g'$ and $ h': g' \to g''$ their composite is the element $ h'+ h: g \to g''$ satisfying $ h'+h +g = g''$. Note that the identity $ e \in G$ gives an identity morphism for every object in $ E(G)$.

In English; $ E(G)$ is a category where objects are elements of $ G$ and morphisms are ways to add an element with a source element to get to a target element. Note that this category is a connected groupoid so it's equivalent to the category with one object and one morphism. This doesn't mean it's entirely uninteresting though.

• For a homomorphism $ f: G \to H$ we get a functor $ E(f) : E(G) \to E(H)$ which sends
  • objects $ g \in E(G)$ to $ f(g) \in E(H)$ and,
  • morphisms $ h: g \to g'$ to the morphism $ f(h): f(g) \to f(g')$.

I don't know what this construction is called but I'm guessing it already exists and there is a slick description of it. If you know that description please let me know.

Edit: Tim Hosgood found a slick way to describe this: $ E(G)$ is equivalent to the looping of $ D(G)$ (I like the definitions of looping/delooping in this paper).

Exercise: Show that this construction works equally well for non-abelian groups and monoids.

Okay now let's integrate $ E \circ H_n: \mathsf{Top} \to \mathsf{Cat}$ (actually the restriction of $ E$ to $ \mathsf{Ab}$).

$ \int E \circ H_n$ is a category where

• an object is a pair $ (X, K)$ where $ X$ is a topological space and $ K$ is an element of $ H_n (X)$ i.e. a formal sum of n-dimensional holes in $ X$.
• a morphism $ (f, L) : (X,K) \to (Y,K')$ is a continuous function $ f: X \to Y$ such that $ L + H_n (f) (K) = K'$. In words, a morphism is a continuous map $ f$ and an n-dimensional hole $ L$ such that $ L$ adds to the n-dimensional hole $ X$ is equipped with to get the hole that $ Y$ is equipped with.

Okay why is this category interesting?  Before we took the Grothendieck construction of the fundamental groupoid functor in order to soften some of arbitrariness of pointed topological spaces. Now, with homology there is no such arbitrariness because the homology of a space does not depend on a basepoint so instead we're using the Grothendieck construction to glue the homology group of every topological space into one huge category.

Open Ended Question 1: Is there a categorical way to relate $ \int E \circ H_1$ with $ \int \pi_1$ using abelianization?

Open Ended Question 2: Does this category have products and coproducts? What about initial or terminal objects?

*in the land of category theory time moves at a different pace.

** I have no idea who first figured this out. If you know please tell me.