The n-Category Café

Weak cokernels and mapping cones

In The Inner Automorphism 3-Group of a Strict 2-Group David Roberts and I considered the mapping cone of the identity morphism on a strict 2-group $G_{(2)}$ as something we started calling a tangent category (pdf).

This is constructed by mapping the “fat point” $2 =\{\bullet \stackrel{\simeq}{\to}\circ\}$ into the one-object 3-groupoid $\Sigma G_{(2)}$ and restricting the morphisms between such maps to fix the left endpoint of the fat point.

As John Baez amplified in Obstructions for $n$-Bundle Lifts, forming the mapping cone of a morphism should correspond to forming its weak cokernel.

Today Enrico Vitale gave a talk on his work

P. Carrasco A. R. Garzó́n and E. M. Vitale
On categorical crossed modules
Theory and Applications of Categories, Vol. 16, 2006, No. 22, pp 585-618

in which he discusses the weak cokernel of morphisms of 2-groups. And, indeed, up to different notation, it comes down to exactly this construction.

Namely, in Obstructions, Tangent Categories and Lie $N$-tegration I had stated the obvious generalization of the mapping cone construction for non-identity morphisms:

Recall

Mapping cones and tangent categories

Recall from the discussion at The Inner Automorphism 3-Group of a Strict 2-Group that the mapping cone of the identity 2-functor $C \to C$, in the world of strict 2-groupoids, amounts to forming what I called the “tangent category” $T C$, which is obtained by mapping the interval $\{\bullet \stackrel{\simeq}{\to} \circ\}$ into $C$ and admitting only those homotopies between such maps which fix the left endpoint.

Now, you won’t be surprised to learn that the mapping cone of an arbitrary injection $t : C_0 \to C_1$ turns out to correspond to maps of the interval into $C_1$, whose homotopies are resticted to fix the left endpoint and to have a right endpoint transverse a 2-path in the image of $t$. But I mention it nevertheless. Since I think it is true and useful.

Indeed, a short inspection shows that this construction is precisely the same as that which Enrico Vitale describes at the beginning of p.11 from a different point of view and using different notation.

But in order to see how his definition amounts to the above one you simply suspend his categorical group to a one-object 2-groupoid and draw the relevant pictures.

Beware, by the way, that Enrico Vitale and his coauthors are also talking about an inner automorphism 2-group. But the concept they mean is the one orthogonal to the one I was talking about: namely the cokernel of the inclusion $G \stackrel{\mathrm{Ad}}{\to} \mathrm{Aut}(G)$ of inner automorphisms into arbitrary autmorphisms. That gives the outer automorphisms!

Compare their example iv) on p. 12.

Weak Lie 2-algebras

A while ago I wrote:

I just had a revelation.

Yesterday I had a followup revelation. (Need to start using a threaded revelation viewer eventually.)

Recall that I conjectured that

general weak Lie $n$-algebroids correspond to NPQ manifolds

Had an extensive discussion of this with Pavol Ševera. We agreed that one ought to be looking for some kind of obvious and/or natural thing in between $A_\infty$ and $L_\infty$. You know, whatever truly weak Lie $n$-algebras are, they must have a description in terms of something we already sort of know.

$A_\infty$ comes from tensor co-algebra with co-differential. $L_\infty$ from (graded) symmetric tensor co-algebra with codifferential.

“Clearly” the strict skew symmetry of the Lie $n$-algebra bracket $n$-functor translates into the graded commutativity of the coalgebra.

What we are looking for is a situation where two things graded commute only up to an additve correction (since composing higher coherence morphisms in this game is just addition).

But this can only mean one thing: we need Clifford algebra instead of exterior algebra.

(Sorry for the boldface. Couldn’t help it.)

Wednesday’s conjecture: Lie $n$-algebras with arbitrary coherently weak skew symmetry and arbitrary coherent weak Jacobi identity correspond to graded differential Clifford algebras.

You see, take a Lie algebra and the corresponding Chevalley Eilenberg algebra $$\wedge^\bullet ( s g^*, d ) \,.$$

Then replace the exterior algebra (of left invariant differential forms on $G$, really), with the Clifford algebra coming from some bilinear form on $g$, but such that the $\mathbb{Z}$-grading is respected.

So we throw in one single degree 2-generator $b$ and demand that for $t^a$ and $t^b$ any two elements in $s g^*$, instead of

$$t^a t^b = - t^b t^a$$

we have

$$k_{ab} t^a t^b = b \,,$$

where $k_{ab}$ are the components of the symmetric bilinear form.

For this to be compatible with the differential, we find that $k$ has to be invariant. Just as if $k_{ab}$ were the components of a symplectic 2-form on the dg-manifold.

Just imagine this turns out to hold water: we’d be able to say

An ordinary Clifford algebra on the vector space $V$ with bilinear form $\langle \cdot, \cdot\rangle$ is an abelian weak Lie 2-algebra with space of objects being $V$ and space of morphisms being $V \oplus \mathbb{R}$ where everything is strict, except that the skew symmetry of the bracket functor holds only up to the natural isomorphism $$[x,y] \stackrel{\langle x,y \rangle}{\to} [y,x] \,.$$

It’d be very nice if this indeed holds water. It would provide a bunch of missing puzzle pieces. Like explaining what Clifford algebra really is, how it categorifies, how the $\mathbb{Z}$-graded structure introduced by the presence of Lie $n$-algebras gives also rise to the $\mathbb{Z}_2$-grading found in supersymmetry, in fact supersymmetry itself would start entering the big picture here.

Pavol’s very thoughtful remark on this last remark of mine:

I knew you $n$-category guys are crazy.

When I find the time I should report on Pavol’s cool talk on his work on differentiating Kan simplicial complexes to Lie $n$-algebras, as described in

Pavol Ševera
$L_\infty$ algebras as 1-jets of simplicial manifolds (and a bit beyond)
math/0612349