Chain Complex morphism: Arbitrary maps instead of Homomorphism?

Question

Given two chain complexes $\{ G_i \}$ and $\{ H_i \}$ we usually define a morphism of chain complex as a family of homormophisms $\{ f_i \}$ such that the diagram commutes:

$$ \begin{matrix} \dots &\xrightarrow{\partial} & G_n &\xrightarrow{\partial} &G_{n-1} &\xrightarrow{\partial} &G_{n-2} &\xrightarrow{\partial} & \dots \\ \dots& & f_n \downarrow & &f_{n-1}\downarrow & & f_{n_2} \downarrow & &\dots \\ \dots &\xrightarrow{\partial} & H_n &\xrightarrow{\partial} &H_{n-1} &\xrightarrow{\partial} &H_{n-2} &\xrightarrow{\partial} & \dots \\ \end{matrix} $$

If we weaken the condition that $\{ f_i \}$ are simply maps [That is, they are any function $f_i: G_i \rightarrow H_i$ such that the diagram commutes] what happens? Formally, the maps $\{ f_i \}$ are such that $\partial^H \circ f_n = f_{n-1} \circ \partial^G$. In this case, can we recover a theory of homology?

Clearly, such maps can be arbitrarily bad. For example, consider:

\begin{matrix} \mathbb Z &\xrightarrow{\times 2} &\mathbb Z \\ f \downarrow & & g \downarrow \\ \mathbb Z &\xrightarrow{\times 3} &\mathbb Z \\ \end{matrix}

$$ f(x) \equiv \begin{cases} 1 & x = 1 \\ 0 & \text{otherwise}\end{cases} \quad g(x) \equiv \begin{cases} 3 & \text{x = 2} \\ 0 &\text{otherwise}\end{cases} $$

This diagram commutes by chasing elements:

\begin{bmatrix} 1 & \xrightarrow{\times 2}& 2 \\ \downarrow f& & \downarrow g \\ 1 & \xrightarrow{\times 3} &3 \end{bmatrix}

\begin{bmatrix} \alpha \neq 1 & \xrightarrow{\times 2}& \beta \neq 2 \\ \downarrow f& & \downarrow g \\ 0 & \xrightarrow{\times 3} & 0 \end{bmatrix}

However, can we still salvage the theory and get a notion of a long exact sequence of Homology from arbitrary maps? Or is this the reason why we ask for homorphisms: So that we can build the theory of Snake lemma which we parlay into the long exact sequence of Homology? If it's not possible to salvage, is there some way to prove that this cannot lead to a useful theory?

Answer 1

As requested,

if you want to drop requirements on the maps inducind a morphism of chaincomplexes, you are actually passing into a bigger category while considering the chaincomplexes also just as these more general objects, so something like: Vectorspaces are just special abelian groups.

However, this category will need to have the structure to allow the following:

1. defining chaincomplexes: you need to be able to say $d^2=0$ so you need a $0$ and in particular a pointed cat

2. defining homology: In general homology is the cokernel of the kernel map of the differential, so you will have to consider a more general category that admits at least that.

3. Furthermore, in order to get a long exact sequence you need the snake lemma which you may impose as a condition, but I guess you actually dont want that but actually you just need to consider the category to be abelian.

However all of this structural requirements are very well described in Weibel.

Considering that the embedding of your subcategory should induce something on the derived category, but be aware that sometimes dropping requirements can do more damage than help.

Furthermore, I am sorry if this is not precise enough, but to really work all of that out properly in details I would have to write 2-6 hours.