# Double complexes and mural maps

## Double complexes

Let **double complex** in

in which every row and every column is a chain complex; i.e.,

Here is a situation in which it will be very tempting to write simply

## Mural maps

Suppose we focus on one object in a double complex, say

For notational simplicity, we are going to drop all subscripts and use more human-readable notation, as below:

The morphisms labeled

We have already defined two homology objects at

For notational simplicity, in this context these two objects are sometimes denoted

We now define two additional quotient objects that relate these two homology objects. These new objects are very close to a "diagonal homology" that one could define using diagonal maps above.

First notice that

Similarly, by the commutativity of the bottom-right square both

With the above notation, the **receptor** at

The **donor** at

These four quotient objects (the two homology objects and these two new objects) are closely related:

Continuing the above notation, the identity morphisms (on coset representatives) induce the following commutative diagram in

These morphisms are called the **intramural maps** of

The name "intramural" is meant to indicate that the above morphisms are between objects that all arise from (quotients of subobjects of)

Each morphism **extramural map** associated to

These extramural maps are meant to justify the names "donor" and "receptor". As with the intramural maps, these are not mysterious morphisms. For the sake of concreteness, suppose

Before moving on, we note that these mural maps are closely related to induced maps on homology:

For any horizontal morphism

The analogous statement is true for each vertical morphism.

## Diagram chasing

It will soon be useful to employ a new notation for the induced extramural maps. For a horizontal morphism

For a vertical morphism

This notation makes it visually clear that in every double complex the extramural maps form long diagonal zigzags between donors and receptors:

(Note that I had to adjust the positions of some of the labels in order to make the object labels readable. Also, the q.uiver app I'm using to make these diagrams cannot adjust the arrow angle to match my desired look. Alas.) At the very ends of any such diagonal, we can use the intramural maps to relate the initial donor to the horizontal (or vertical) homology at that donor object, and the final receptor to the horizontal (or vertical) homology at that receptor.

At this first glance this might not look incredibly useful, since the morphisms can't be chained together. However, if there were additional assumptions that guaranteed the mural maps were isomorphisms, then the zig-zag mural maps *could* be chained together to produce a morphism from one homology object to another.