# Fractal homological algebra

The usual definition of a chain complex requires for the indices to be integer numbers. However, taking inspiration from the theory of Hausdorff dimension, one can think of 'fractal' chain complexes (of $\mathbb{C}$-vector spaces), i.e. $\mathbb{R}$-graded vector spaces $$V=\bigoplus_{\phi \in \mathbb{R}} V_{\phi}$$ equipped with a differential $d: \ V_{\phi} \to V_{\phi +1}$, $d^2=0$.

One can also endow the category of such chain complexes with a monoidal structure, by setting $(V \otimes W)_{\phi} = \bigoplus_{\alpha + \beta=\phi} V_{\alpha} \otimes W_{\beta}$ with differential given by $$d(v \otimes w) = dv \otimes w + e^{\pi \sqrt{-1} \alpha} v \otimes dw$$ when $v \in V_{\alpha}$ (this is somehow a 'fractal' Leibniz rule). We can thus speak of fractal differential graded algebras and categories (i.e. categories enriched over the abovementioned monoidal category) and much more.

The question is: are such constructions present somewhere in literature? Has anyone heard of any attempt to introduce fractal dimensions into homological algebra?

• There are similar constructions in differential calculus with the idea of a "fractional derivative". I would think that fractional is more appropriate than fractal in this context too. – Ali Caglayan Dec 25 '17 at 19:15
• As long as you have a single differential with fixed degree $+1$, these complexes are just direct sums of independent ordered $\mathbb Z$-indexed chain complexes, namely one for each coset of $\mathbb R/\mathbb Z$. A more interesting structure would be something that connects all degree with each other. Maybe a family of differentials $(d^\alpha)_{\alpha\geq 0}$ with $d^0=id$, $d^{\alpha+\beta} = d^\alpha \circ d^\beta$ and $d^c = 0$ for some fixed cutoff $c>0$, but I do not know if such a thing is useful in any way. – Johannes Hahn Dec 25 '17 at 21:06
• Being useful really is the key here. You should figure out at least one example where such an idea can be useful. Is there a geometric/topological/space-ish thing that has 1/2-dimensional "holes" ? Is there anything that feels like it is "between third and fourth cohomology" ? Stuff like that. Find something useful and the right definition may come naturally out of it (that's of course not guaranteed, but it is very much possible) – Johannes Hahn Dec 25 '17 at 21:12
• @darijgrinberg I'm aware of that work. However, it is quite different since the differential is required to satisfy $d^n=0$ for some fixed $n$. – gm01 Dec 25 '17 at 22:33
• @JohannesHahn A fractional complex in this sense is, as you say, just a $\mathbb{R} / \mathbb{Z}$-collection of complexes. However, the monoidal structure is nontrivial: this means that enriched categories (i.e. fractional dg-categories) have interesting structure. I can't think of an example, unfortunately. The fractional De Rham complex idea by Ali could work. – gm01 Dec 25 '17 at 22:38