Geometric Homotopy as Chain Homotopy

Question

In Can we think of a chain homotopy as a homotopy, I learned that chain homotopy can be defined in an analogous fashion to homotopy, i.e from the product with an interval object etc. What about the other way around?

Can we think of homotopy as chain homotopy?

The usual definition of chain homotopy via the boundary map probably doesn't encode enough information to define geometric homotopy, but maybe, given additional information, it is possible to recover.

I would be glad to discover homotopy has been categorified, but if that's the case, I ask for a detailed reference or detailed explanation about how to recover geometric homotopy from it. Please don't post two sentence replies using $\infty$-groupoids and things of the sort because I know nothing about them (for now).

Answer 1

I'm not sure I understand your question, but here goes an attempt at placing both examples in the same formalism.

Both what you call geometric homotopy and chain homotopy are instances of the same general idea. In a category $C$ (nothing fancy like $\infty$-groupoid, just a category), suppose that some notion of cylinder object exists. That is, for each object $c$ there exits an object $c\wedge I$ together with canonical morphisms $i_1,i_2\colon c\to c\wedge I$ and a trivializing morphism $j\colon c\wedge I \to c$. These structure morphisms should behave nicely enough, at the very least $j\circ i_j=\rm id$. Now you can define two morphisms $f,g\colon c\to d$ to be homotopic if there exists a morphism $H\colon c\wedge I\to d$, called a homotopy, such that $f=H\circ i_1$ and $g=H\circ i_2$.

This is nothing but a diagramatic way of saying what a homotopy is. In the category of topological spaces, taking $c\wedge I$ to be the usual product of the space $c$ with the interval $I$ is a cylinder object and the notion of homotopy above is the geometric one. In the category of chain complexes, taking $c\wedge I$ to be the tensor of the chain complex $c$ with the interval complex is a cylinder object, and homotopy is chain homotopy.

Now, back to the categorical story. If no further axioms are assumed, the notion of homotopy in $C$ is not very strong and is hardly useful. There are at least three different axiom systems, due to Bauez, Brown, and Quillen, which allow for a significant development of homotopy theory in $C$. By far, Quillen's is the most popular one. I suggest Riehl's "categorical homotopy theory" if you want to get serious (about Quillen model categories). The book does not assume you already know a tremendous amount of category theory, so it's a pretty smooth ride.

Answer 2

Let $X$ and $Y$ be topological spaces, let $\Delta^1$ be the interval, let $h : X \times \Delta^1 \to Y$ be a continuous map, let $f_0 = h (-, 0)$ and let $f_1 = h (-, 1)$. The usual way of turning a continuous map $h$ into a chain homotopy $\alpha : C (f_0) \Rightarrow C (f_1)$ of morphisms $C (X) \to C (Y)$ is a bit difficult to describe explicitly because one has to triangulate the prisms $\Delta^n \times \Delta^1$, but the point is that in degree 0, the map $\alpha_0 : C_0 (X) \to C_1 (Y)$ is defined by sending 0-simplices $x$ in $X$ to the 1-simplices $h (x, -)$ in $Y$.

In particular, yes, there is enough information to recover $h$ from $\alpha$. But not every chain homotopy will come from an actual homotopy.

Answer 3

I suggest for a start the book "Abstract Homotopy and Simple Homotopy Theory" by Kamps and Porter for the notion of a category with a "cylinder object". This book has a section on chain complexes.

Standard algebraic topology is a bit stuck in the simplicial mode, whereas cubical theory has many advantages, one of them being the simplicity of homotopies due to the fact that $$I^m \times I^n \cong I^{m+n}.$$ This is illustrated in Massey's book on "Singular homology theory" (and also the book by Hilton and Wylie).

For chain complexes over a ring $R$ there is an "interval object" $I_R$ which I'll leave you to define so that a homotopy is a map $C \otimes_R I_R \to D$.