A 2-category of chain complexes, chain maps, and chain homotopies?

Question

First-time here... I hope my question isn't silly or anything... anyway...

Consider the category of chain complexes and chain maps. We can also define chain homotopies between chain maps. Does this form a 2-category? I am able to construct vertical and horizontal composition of chain homotopies but am unable to prove that the horizontal composition is associative and that the interchange law holds.

I like to include a lot of details in case the reader isn't familiar with everything, so the following is the problem in detail.

To be a bit more concrete, let's let $(C_n, \partial_n), (C'_n, \partial'_n),$ and $(C''_n, \partial''_n)$ be chain complexes (sorry, my differential is going up in degree) and let $f_n, g_n, h_n: C_n \to C'_n$ and $f'_n, g'_n, h'_n : C'_n \to C''_n$ be chain maps. Suppose that $\sigma : f \Rightarrow g,$ $\tau : g \Rightarrow h,$ $\sigma' : f' \Rightarrow g',$ and $\tau' : g' \Rightarrow h'$ are chain homotopies, where $\sigma_n, \tau_n : C_n \to C'_{n-1}$ and similarly for $\sigma'$ and $\tau'.$ Recall that these definitions say

$f_n \circ \partial_{n-1} = \partial'_{n-1} \circ f_{n-1}$

and similarly for primes, $g$'s, $h$'s, and

$\sigma_{n+1} \circ \partial_{n} + \partial'_{n-1} \circ \sigma_{n} = f_{n} - g_{n},$

$\tau_{n+1} \circ \partial_{n} + \partial'_{n-1} \circ \tau_{n} = g_{n} - h_{n},$

and similarly for primes.

We can define the vertical composition of $\sigma : f \Rightarrow g$ with $\tau : g \Rightarrow h$ by

$(\tau \diamond \sigma)_{n} := \tau_n + \sigma_n$

(I denote vertical composition with a diamond, $\diamond$).

We can also define the horizontal composition of $\sigma : f \Rightarrow g$ with $\sigma' : f' \Rightarrow g'$ by

$(\sigma' \circ \sigma)_n := \partial''_{n-2} \circ \sigma'_{n-1} \circ \sigma_{n} - \sigma'_{n} \circ \sigma_{n+1} \circ \partial_{n} + \sigma'_{n} \circ g_{n} + f'_{n-1} \circ \sigma_{n}.$

Now we just verify that these are indeed chain homotopies. The vertical composition is easy:

$(\tau \diamond \sigma)_{n+1} \circ \partial_{n} + \partial'_{n-1} \circ (\tau \diamond \sigma)_{n} = f_n - g_n + g_n - h_n = f_n - h_n$

and the horizontal composition is a bit more challenging but doable (I won't include the derivation here).

The identity for vertical composition is the zero map and similarly for the horizontal composition (note that for the vertical composition, the zero map is a chain homotopy between any two chain maps provided they are the same while for the horizontal composition, the chain maps are both the identities). The vertical composition is easily seen to be associative. However, the horizontal composition satisfies

$( \sigma'' \circ ( \sigma' \circ \sigma ) )_{n} - ( ( \sigma'' \circ \sigma' ) \circ \sigma )_n$

$= ( f''_{n-1} - g''_{n-1} ) \circ ( f'_{n-1} - g'_{n-1} ) \circ \sigma_{n} - \sigma''_{n} \circ ( f'_{n} - g'_{n} ) \circ ( f_n - g_n ).$

The interchange law doesn't hold either and the difference is given by

$( ( \tau' \diamond \sigma' ) \circ ( \tau \diamond \sigma ) )_{n} - ( ( \tau' \circ \tau ) \diamond ( \sigma' \circ \sigma ) )_{n}$

$= ( f'_{n-1} - g'_{n-1} ) \circ \tau_{n} + (g'_{n-1} - h'_{n-1} ) \circ \sigma_{n} - \tau'_{n} \circ ( f_{n} - g_{n} ) - \sigma'_{n} \circ (g_{n} - h_{n} )$

So I'm not sure if chain complexes, chain maps, and chain homotopies are supposed to form a 2-category, but I would've liked this result to be true. Does anyone know what the correct categorical structure of this category is? I have not yet considered chain homotopies of chain homotopies, so if the answer requires all such higher morphisms, then an answer in that direction would also be acceptable.

Any thoughts? Thanks in advance.

Answer 1

Part of the problem is that you're not using a convenient definition of chain homotopy. I'll use a differential going down in degree. Let $I$ be the interval object in the category of chain complexes; that is, $I$ is $\text{span}(0, 1)$ in degree $0$ and $\text{span}(e)$ in degree $1$ ($k$ the underlying commutative ring), and the differential sends $e$ to $1 - 0$. If $C, D$ are two complexes, I claim that a chain homotopy between two chain maps $f, g : C \to D$ is precisely a chain map $$H : I \to \text{Hom}(C, D)$$

such that the restriction of the map to $0$ is $f$ and the restriction of the map to $1$ is $g$, where $\text{Hom}$ is the hom chain complex. With this definition you can work guided by analogy to the topological situation (there the 2-category is the 2-category of topological spaces, continuous functions, and homotopy classes of homotopies between such functions); all of the maps you need have obvious topological definitions, although I admit I have never worked out the details. I think everything reduces to working with fairly concrete maps between some chain complexes constructed from $I$.

Answer 2

Viewing homotopy via interval objects analogously to topological homotopy as done in Qiaochu Yuan's answer is certainly the best way to consider the problem. Nevertheless it's also possible to get along directly with your definition of homotopy.

In this point of view the problem is that your horizontal composition isn't appropriate: First note that we have $\sigma: f \Rightarrow g,\; \sigma': f' \Rightarrow g'$. Thus (I just write $f'f$ for $f'\circ f$) $$f'f - g'g = f'f - f'g + f'g - g'g = ...=\partial^{''}(f'\sigma+\sigma'g) + (f'\sigma+\sigma'g)\partial. $$

Therefore $\sigma' \circ \sigma:= f'\sigma+\sigma'g: f'f \Rightarrow g'g$ is a homotopy.

Let $\sigma'': f'' \Rightarrow g''$ be another homotopy. By setting in the definition of the former homotopy, one finds that $$(\sigma'' \circ \sigma') \circ \sigma = f''f'\sigma + f'' \sigma' g + \sigma'' g'g = \sigma'' \circ (\sigma' \circ \sigma)$$ is a homotopy $(f'' f')f = f''(f'f) \Rightarrow g''(g'g) = (g''g')g$.

Hence the horizontal composition is associative.

Answer 3

I think the right context for this question is that of monoidal closed categories with a unit interval object, and probably the first place to explore this is the ncat lab.

I mention that we set up a number of monoidal closed categories in our book

R. Brown, P.J. Higgins, R. Sivera, Nonabelian algebraic topology: filtered spaces, crossed complexes, cubical homotopy groupoids, EMS Tracts in Mathematics Vol. 15, 703 pages. (August 2011). http://www.bangor.ac.uk/r.brown/nonab-a-t.html pdf available there.

for example:

crossed complexes, chain complexes with a groupoid of operators,

and explore the relations between them. As explained in the above answer and comments, the cubical approach has some advantages when dealing with homotopies and higher homotopies, basically because of the formula $I^m \otimes I^n \cong I^{m+n}$.

Added as edit: the other point about the cubical formulation is that there is a notion of cubical set with compositions (and also extra structure such as connections) but we do not have a similar notion simplicially, or not so easily. Globular notions have compositions, though multiple compositions are awkward, and tensor products are not so easy.