Infinitesimal 3-braidings

The notion of an infinitesimal 2-braiding was introduced by Lucio Simone Cirio and João Goncalves Faria Martins in 2013 as a means to go one level higher in category theory than that provided by Pierre Cartier's notion of an infinitesimal braiding (defined in 1993). In particular, they defined infinitesimal 2-braidings as pseudonatural transformations instead of just 2-natural transformations to make room for the examples coming from semiclassical structures on higher Lie algebras (i.e. 'quasi-invariant symmetric tensors' in differential crossed modules) and the concrete cases thereof (e.g. Wagemann's string Lie 2-algebra construction applied to the special linear Lie algebra of order 2). 

The paper Infinitesimal 2-braidings from 2-shifted Poisson structures reproduced this pseudonaturality feature within the following context. Consider the category of cochain complexes concentrated in degrees {−1, 0}; this is a symmetric monoidal category (with the monoidal product given by a truncated tensor product) thus it can be used as a base for enrichment. Define a cochain 2-category as a category enriched over the category of cochain complexes concentrated in degrees {−1, 0}; this is indeed a 2-category under the obvious assignments and, likewise, a functor which respects this enrichment is a special type of 2-functor hence we call it a cochain 2-functor.  Actually, cochain 2-categories are special types of (2,1)-categories hence every lax natural transformation between cochain 2-functors is a pseudonatural transformation and every modification between pseudonatural transformations is invertible with respect to the highest composition. Given a dg-category, one can construct from it a cochain 2-category by restricting the degree 0 morphisms to the 0-cocycles and quotienting the degree −1 morphisms by the (−1)-coboundaries; better yet, this construction preserves symmetric monoidal structures.  Applied to the dg-category of semi-free dg-modules over a semi-free CDGA, one can explicitly construct an infinitesimal 2-braiding given a 2-shifted Poisson structure on the underlying CDGA.  

Back to the bigger picture, the compositional calculus of pseudofunctors, pseudonatural transformations and modifications has been well understood for quite some time; see Niles Johnson and Donald Yau's book 2-Dimensional Categories for an excellent exposition of the concept of a tricategory of bicategories, pseudofunctors, pseudonatural transformations and modifications. Furthermore, this tricategory admits a symmetric monoidal structure which provides the context needed to define a braided monoidal bicategory; see Appendix C of Christopher J Schommer-Pries' thesis The Classification of Two-Dimensional Extended Topological Field Theories for a clear description of all the data and coherence conditions entailed therein. Again, our context speaks of cochain 2-categories, cochain 2-functors, pseudonatural transformations and (invertible) modifications; these assemble into a weak (3,2)-category where the only weakness is in the fact that the horizontal and vertical composition of pseudonatural transformations does not satisfy the exchange law but instead this deficit is measured by a modification. 

Going one level higher again, we must lift the truncation of the hom-complexes and consider cochain 3-categories, i.e. categories enriched over the category of cochain complexes concentrated in degrees {−2,−1, 0} . This time we have a bit of a misnomer because we don't recover 3-categories under the obvious assignments; we actually recover weak (3,1)-categories because the horizontal and vertical composition of homotopies doesn't satisfy the exchange law but only up to a 2-homotopy (which was previously killed by the truncation to degrees {−1, 0}). The goal is to find a sensible definition of a pseudonatural transformation between cochain 3-functors where a "sensible definition" means one such that the two compositions are closed, associative, unital, etc. Just as modifications are needed to remedy the lack of exchange between vertical and horizontal composition of pseudonatural transformations in the 2-categorical world, so too one must find a sensible definition of a 'pseudomodification' in this 3-categorical world. Actually, we would need to go one step even further and also define the notion of a 'mutation between pseudomodifications'. Todd H. Trimble introduced the definition of a tetracategory in 2006; for our purposes we would like to show that there exists a tetracategory of cochain 3-categories, cochain 3-functors, pseudonatural transformations, pseudomodifications and mutations. We expect this to be a weak (4,2)-category where, again, the only weakness comes from the lack of exchange. 

As explicated by Alexander E. Hoffnung in 2011, a monoidal tricategory is simply a one-object tetracategory in the sense of Trimble. In order to approach 'infinitesimal 3-braidings' we would first need to find a way to define a braided monoidal tricategory. The axioms of such a structure must be coherence conditions as was historically the case for the founding of braided monoidal categories and braided monoidal bicategories; the  general picture being provided by the Stasheff polytopes, Kapranov-Voevodsky shuffle polytopes and permutohedra. Actually, such a general definition would be overkill; we would only need to define a 'braided strictly-unital monoidal cochain 3-category' akin to how Syllepses from 3-shifted Poisson structures and second-order integration of infinitesimal 2-braidings showed that only "braided strictly-unital monoidal cochain 2-categories" are relevant for our purposes rather than utmost general braided monoidal bicategories. 

Homotopy Lie algebra representations and infinitesimal 2-braidings

Cartier integration of infinitesimal 2-braidings via 2-holonomy of the CMKZ 2-connection, II: The Pentagonator and its axioms