### Categorification and the Cosmic Cube

#### Posted by David Corfield

I see that Tobias Dyckerhoff’s A categorified Dold-Kan correspondence has just appeared, looking, as its title suggests, to categorify the nLab: Dold-Kan correspondence. As it says there, the latter

interpolates between homological algebra and general simplicial homotopy theory.

So with Dyckerhoff’s paper we seem to be dipping down to the lower layer of the flamboyantly named ‘cosmic cube’, see slide 10 of these notes by John, and discussed at nLab: cosmic cube. Via chain complexes of stable $(\infty, 1)$-categories Dyckerhoff speaks of a ‘categorified homological algebra’, and also through a categorified Eilenberg-Mac Lane spectrum, of a ‘categorified cohomology’.

For old time’s sake, let’s see if anyone is up for the kind of grand vision thing we used to talk about. For one thing we might wonder what plays the role of a categorified homotopy theory, the kind of world where Mike’s suggestions on directed homotopy type theory might find a home.

I see I was raising stratified spaces as relevant back there. In the meantime we now have useful models from A stratified homotopy hypothesis. It turns out that $(\infty, 1)$-categories are equivalent to ‘striation sheaves’, a certain kind of sheaf on ‘conically smooth’ stratified spaces. The relevant fundamental $(\infty, 1)$-category is the exit-path $(\infty, 1)$-category, rather than the entry and exit paths of our older discussions which brought in duals.

In line with Urs’s claim that cohomology concerns mapping spaces in $(\infty, 1)$-toposes (nLab: cohomology), perhaps for categorified cohomology we should be looking for parallels in $(\infty, 2)$-toposes, an important one of which will be that containing all $(\infty, 1)$-categories, or equivalently, all striation sheaves.

Posted at October 26, 2017 10:15 AM UTC
## Re: Categorification and the Cosmic Cube

That particular suggestion on directed homotopy type theory, which has now become this paper, is, I would say, more about embedding it in ordinary homotopy theory. For a “natively” directed version, even the notion of (2,2)-topos is, I think, still poorly understood.