### Categorified Clifford Algebra and weak Lie *n*-Algebras

#### Posted by Urs Schreiber

What is a categorified Grassmann algebra?

What is a categorified Clifford algebra?

What differential algebraic structure are fully weak Lie $n$-algebras equivalent to?

Is there a relation between these questions?

The **principle of least resistance under categorification** says

We understand the true nature of a concept the deeper, the more straightforwardly the definition we use to conceive it lends itself to categorification.

Hence we have a complete understanding of the true meaning of the concept of a Lie *group*. And therefore still a rather good understanding of the concept of Lie algebra.

But do we already, in this sense, understand the true nature of the concepts “Grassmann algebra” and “Clifford algebra”?

Of course I could try to describe a categorified Grassmann algebra as something like an abelian monoidal category equipped with a categorified version of graded-commutativity.

But it turns out that there is something even less resistive:

A Grassmann algebra $\wedge^\bullet V$ over a vector space $V$ is related by Koszul duality to the

abelian Lie algebra on $V$.

(See for instance the beginning of Lie $n$-algebra cohomology for more on how this works.)

But we said Lie algebras are nicely categorified. So we should maybe say

An $n$-Grassmann algebra $\wedge^\bullet V$ over a vector space $V$ is defined to be the Koszul dual to an

abelian semistrict Lie $n$-algebra.

That would imply that an $n$-Grassmann algebra is the graded-commutative algebra

$\wedge^\bullet V$

freely generated over a graded vector space concentrated in degrees $1 \leq d \leq n$.

One generalization of this fact is well known: as we pass from abelian to general Lie $n$-algebras – whose bracket is strictly skew-symmetric but whose Jacobi identity holds only up to coherent equivalence – the Koszul-dual algebraic side generalizes from free graded-commutative algebras to *differential* graded commutative algebras.

In fact, people use precisely this kind of identification to set up their definitions: since on the side of differential graded algebras the generalization to many-objects is obvious, one *defines* a Lie $n$-algebroid to be (dual to) a suitable dg-*manifold*.

This means we are left with two open questions:

- we still need to figure out what happens as we replace Grassmann algebras by Clifford algebras here

- we are still assuming that the skew-symmetry of the bracket functor $[\cdot,\cdot] : S \times S \to S$ holds strictly.

In Detecting Higher Order Necklaces I conjectured that these two items are indeed dual to each other.

If true, this would mean that

An $n$-Clifford algebra over a vector space $V$ is defined to be the Koszul dual to an

abelian fully weak Lie $n$-algebra.

and presumeably that

Fully weak Lie $n$-algebras are Koszul dual to

differential graded Clifford algebras.

Today mankind made one further step towards checking this conjecture: Dmitry Roytenberg has now issued his thoughts on fully weak Lie 2-algebras:

Dmitry Roytenberg
*On weak Lie 2-algebras*

(pdf)

**Exercise:** Give the codifferential coalgebra description of Dmitry Roytenberg’s weak 2-term $L_\infty$-algebras (p. 9). Then dualize to find the corresponding differential algebra. Check if it can be sensibly addressed as a differential graded Clifford algebra.

(Notice that we expect to see “graded” Clifford algebra: the anticommutator of two degree 1 elements is degree 2.)

**Reminder of some of the underlying facts**

John Baez and Alissa Crans essentially defined Lie $n$-algebras to be $(n-1)$-categories $S$ internal to vector spaces, equipped with a product functor

$[\cdot, \cdot] : S \times S \to S$

which is *strictly* skew symmetric and satisfies the Jacobi identiy up to coherent equivalence.

Abstract operad nonsense shows that such a “*semistrict*” Lie $n$-algebra is the same a an $n$-ter, $L_\infty$-algebra, i.e. an $L_\infty$ on a vector space $V$ which is concentrated in degree $1 \leq p \leq n$.

For more details see for instance the beginning of Lie $n$-algebra cohomology and the references given there.

Straightforward computation, in turn, shows that $n$-term $L_\infty$ algebras are the same as free graded commutative coalgebras $S^c V$ equipped with a degree -1 codifferential

$D : S^c V \to S^c V$

of degree -1 such that

$D^2 = 0 \,.$

For $V$ finite, dualizing this statement leads to the statement that Lie $n$-algebras are equivalently encoded in graded Grassmann algebras

$\wedge^\bullet V^*$

equipped with a differential

$d : \wedge^\bullet V^* \to \wedge^\bullet V^*$

of degree +1 and such that

$d^2 = 0 \,.$

## Re: Categorified Clifford Algebra and weak Lie n-Algebras

Any idea what the categorification of the Clifford algebra clock looks like? A categorified clock of categorified Clifford algebras?

Hmm, how to categorify a clock?