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?