# Extending Reedy dimension to augmented chain complexes of abelian groups

Recall that a normal continuously-graded finite interval is given by a pair $a=([a],f)$ consisting of:

1.) A finite totally ordered set $[a]=[a_0< \dots < a_n]$
2.) A grading function $f:U[a] \to \mathbf{N}$ (where $U[a]$ denotes the underlying set of $[a]$)

satisfying the constraints

i.) Continuity: $f(a_j)=f(a_{j-1})\pm 1$ for $1\leq j \leq n$
ii.) Normality: $f(a_0)=f(a_n)=0$.

For an element $x\in [a]$, define even and odd boundaries: $$\partial^+(x)=\operatorname{min} \{y>x : f(y)=f(x)-1\}$$ and $$\partial^-(x)=\operatorname{max} \{y<x : f(y)=f(x)-1\}$$

We define the Reedy dimension $\operatorname{RDim}$ of $a$ to be the sum of the local maxima of $f$ minus the sum of the local minima of $f$.

From such an object, we may construct an augmented chain complex of free abelian groups:

$$C(a)_m=F_{\operatorname{ab}}(f^{-1}(m))$$

with the boundary map $\partial:C(a)_{m+1}\to C(a)_m$ defined on the generators by the formula $$\partial(x)=\partial^+(x)-\partial^-(x)$$

and with the augmentation $\varepsilon:C(a)_0\to \mathbf{Z}$ defined by sending all generators to $1\in \mathbf{Z}$.

Then the question: Does there exist a "natural way" to extend the Reedy dimension to a dimension function $\operatorname{RDim}:\operatorname{Ob}(\operatorname{AugCh}_{\geq 0})\to \mathbf{N}$ such that

a.) For any normal continuously-graded finite interval $a$, $\operatorname{RDim}(C(a))=\operatorname{RDim}(a)$
b.) It is additive on tensor products: $\operatorname{RDim}(C\otimes D)=\operatorname{RDim}(C)+\operatorname{RDim}(D)$
c.) It satisfies the "dimension condition" on sums: $\operatorname{RDim}(C + D)=\operatorname{RDim}(C)+\operatorname{RDim}(D) - \operatorname{RDim}(C\cap D)$?

That is, can such a dimension map be defined using some kind of homological construction? It seems like it might be related to some kind of additive Euler characteristic, which is the reason why I ask.

-