Categorical bialgebras arising from 2-Segal spaces

Topological field theory (TFT) is the study of representations of the cobordism category of manifolds. Such representations are often constructed from purely algebraic inputs. For example, 2d TFTs are constructed from commutative Frobenius algebras. Categories of representations of finite dimensional involutory Hopf algebras and quantum groups at roots of unity define 3d TFTs. In general, the algebraic inputs for 4d TFTs are much less well understood. Crane-Frenkel proposed a construction of a 4d TFT from a bimonoidal category, that is, a category equipped with both a monoidal and comonoidal structure such that the data defining the comonoidal structure is itself monoidal. Said another way, a bimonoidal category is a categorification of a bialgebra. This proposal has been hugely influential, essentially kick-starting the whole field of categorification. Despite this, it was not until the work of Gal-Gal almost 20 years later that the first non-trivial examples of bimonoidal categories appeared.
A rich source of bialgebraic structures, including the (positive part of) quantum groups, are the Hall algebras of finitary abelian categories. A new perspective, introduced independently by Dyckerhoff-Kapranov and Gálvez-Carrillo-Kock-Tonks, separates the construction of the Hall algebra of an abelian category A into three steps. In the first step one associates to A its Waldhausen S-construction: a simplicial groupoid SY(A) with Sn(A) the groupoid of length n flags in A. The second stop associates to the simplicial groupoid S·(A) its universal Hall algebra, an algebra in the category of spans of groupoids whose underlying groupoid is the groupoid of objects of A. In the final step this algebra is linearised by passing to the finitely supported functions on the isomorphism classes of objects of A and implementing the product via a pull-push construction.
This new perspective on Hall algebras opens up the possibility of vastly generalising the construction, as was realised by both sets of authors. The associativity of the product on the universal Hall algebra of A was a direct consequence of S·(A) being a particular example of a 2-Segal groupoid, a simplicial groupoid satisfying a 2-dimensional analogue of the Segal conditions. Both sets of authors define the universal Hall algebra for a general 2-Segal space. A second, independent direction of generalisation is the freedom of choosing alternate linearisation procedures.
We exploit these possibilities to construct new examples of bimonoidal categories. The main contribution of this thesis is the definition of the Hall bimonoidal category, ℌ(A), of a bifinitary abelian category A. On a purely formal level, the universal Hall algebra can be dualised to yield the universal Hall coalgebra, a coassociative coproduct on the groupoid of objects of A. Our first step in the definition of the Hall bimonoidal category of A is to extend the universal Hall algebra and coalgebra of A to a universal Hall bialgebra. We show that the product and coproduct are laxly compatible, endowing the groupoid of objects of A with the structure of a lax bialgebra as an object in the 2-category of iterated spans of groupoids. In fact, we construct the universal Hall bialgebra on the level of generality of so-called double 2-Segal spaces, which are bisimplicial spaces &Xscr;·,· such that the simplicial spaces &Xscr;·;1 and &Xscr;1;· are 2-Segal spaces.
The Hall bimonoidal category of A is obtained by applying a categorical linearisation, due to Morton, to the universal Hall bialgebra. As a result, the monoidal and comonoidal structures on the universal Hall bimonoidal category ℌ(A) are, in general, only laxly compatible. We prove that ℌ(A) is a bimonoidal category if and only if A is semisimple. In particular, taking A = Vect&Fopf;q we obtain a bimonoidal structure on the category of representations of the general linear groupoid extending the monoidal structure introduced by Joyal-Street. Furthermore, the bimonoidal category ℌ(Vect&Fopf;q ) is a categorification of one of the primary examples of Zelevinsky's positive, self-adjoint Hopf algebras.

Advisor: Douglas, C