We use a special kind of 2-dimensional extended Topological Quantum Field Theories (TQFTs), so-called open-closed TQFTs, in order to extend Khovanov homology from links to arbitrary tangles, not necessarily even. For every plane diagram of an oriented tangle, we construct a chain complex whose homology is invariant under Reidemeister moves. The terms of this chain complex are modules of a suitable algebra A such that there is one action of A or A op for every boundary point of the tangle. We give examples of such algebras A for which our tangle homology theory reduces to the link homology theories of Khovanov, Lee, and Bar-Natan if it is evaluated for links. As a consequence of the Cardy condition, Khovanov’s graded theory can only be extended to tangles if the underlying field has finite characteristic. In all cases in which the algebra A is strongly separable, i.e. for Bar-Natan’s theory in any characteristic and for Lee’s theory in characteristic other than 2, we also provide the required algebraic operation for the composition of oriented tangles. Just as Khovanov’s theory for links can be recovered from Lee’s or Bar-Natan’s by a suitable spectral sequence, we provide a spectral sequence in order to compute our tangle extension of Khovanov’s theory from that of Bar-Natan’s or Lee’s theory. Thus, we provide a tangle homology theory that is locally computable and still strong enough to recover characteristic p Khovanov homology for links.

in a somewhat unconventional manner. Our main focus will be on monoidal categories, mainly symmetric ones, for which we propose a physical interpretation. Special attention is given to the category of sets and relations, posetal categories, diagrammatic calculi, strictification, compact categories, biproduct categories and abstract matrix calculi, internal structures, and topological quantum field theories. In our attempt to complement the existing literature we (on purpose) omitted some very basic topics for which we point to other available sources. 0 Prologue: cooking with vegetables Consider a raw potato. Conveniently, we refer to it as A. Raw potato A admits several states e.g. ‘dirty’, ‘clean’, ‘skinned’,... We usually don’t eat raw potatoes so we need to process A such that it becomes eatable. We refer to this cooked version of A as B. Also B admits several states e.g. ‘boiled’, ‘fried’, ‘baked with skin’, ‘baked without skin’,... Correspondingly, there are several ways to turn raw potato A into cooked potato B e.g. ‘boiling’, ‘frying’, ‘baking’, respectively referred to as f, f ′ and f ′ ′. We make the fact that these cooking processes apply to raw potato A and produce cooked potato B explicit by labelled arrows: A f ✲ B A f ′

Abstract. Mednykh [Me78] proved that for any finite group G and any orientable surface S, there is a formula for #Hom(π1(S), G) in terms of the Euler characteristic of S and the dimensions of the irreducible representations of G. A similar formula in the nonorientable case was proved by Frobenius and Schur [FS06]. Both of these proofs use character theory and an explicit presentation for π1. These results have been reproven using quantum field theory ([FQ93], [MY05], and others). Here we present a greatly simplified proof of these results which uses only elementary topology and combinatorics. The main tool is an elementary invariant of surfaces attached to a semisimple algebra called a lattice topological quantum field theory. 1.

We study a special sort of 2-dimensional extended Topological Quantum Field Theories (TQFTs) which we call open-closed TQFTs. These are defined on open-closed cobordisms by which we mean smooth compact oriented 2-manifolds with corners that have a particular global structure in order to model the smooth topology of open and closed string worldsheets.

We study the sewing constraints for rational two-dimensional conformal field theory on oriented surfaces with possibly non-empty boundary. The boundary condition is taken to be the same on all segments of the boundary. The following uniqueness result is established: For a solution to the sewing constraints with nondegenerate closed state vacuum and nondegenerate two-point correlators of boundary fields on the disk and of bulk fields on the sphere, up to equivalence all correlators are uniquely determined by the one-, two,- and three-point correlators on the disk. Thus for any such theory every consistent collection of correlators can be obtained by the TFT approach of [1, 2]. As morphisms of the category of world sheets we include not only homeomorphisms, but also sewings; interpreting the correlators as a natural transformation then encodes covariance both under homeomorphisms and under sewings of world sheets. Contents

We study the sewing constraints for rational two-dimensional conformal field theory on oriented surfaces with possibly non-empty boundary. The boundary condition is taken to be the same on all segments of the boundary. The following uniqueness result is established: For a solution to the sewing constraints with nondegenerate closed state vacuum and nondegenerate two-point correlators of boundary fields on the disk and of bulk fields on the sphere, up to equivalence all correlators are uniquely determined by the one-, two,- and three-point correlators on the disk. Thus for any such theory every consistent collection of correlators can be obtained by the TFT approach of [1, 2]. As morphisms of the category of world sheets we include not only homeomorphisms, but also sewings; interpreting the correlators as a natural transformation then encodes covariance both under homeomorphisms and under sewings of world sheets. Contents