Universal construction [C. Blanchet et al., Topology 34, No. 4, 883–927 (1995; Zbl 0887.57009); M. Khovanov, “Decorated one-dimensional cobordisms and tensor envelopes of noncommutative recognizable power series”, Preprint, arXiv:2010.05730] starts with an evaluation function for closed \(n\)-manifolds to produce state space for closed \(\left( n-1\right) \)-manifolds and maps between these spacces associated to \(n\)-cobordisms, which results in a functor from the category of \(n\)-dimensional cobordisms to the category of vector spaces usually failing to be a TQFT with the tensor product of states for two \(\left( n-1\right) \)-manifolds \(N_{1},N_{2}\)properly embedded into the state space for their union \[ A\left( N_{1}\right) \otimes A\left( N_{2}\right) \hookrightarrow A\left( N_{1}\sqcup N_{2}\right) \]
The universal construction turns out to be riveting already in low-dimensions, including in dimensions two [M. Khovanov, “Universal construction of topological theories in two dimensions”, Preprint, arXiv:2007.03361; M. Khovanov et al., Sel. Math., New Ser. 28, No. 4, Paper No. 71, 68 p. (2022; Zbl 1496.18018); M. Khovanov et al., Commun. Math. Phys. 385, No. 3, 1835–1870 (2021; Zbl 1490.57039); M. Khovanov and R. Sazdanovic, J. Pure Appl. Algebra 225, No. 6, Article ID 106592, 24 p. (2021; Zbl 1480.16037)] and one [P. Gustafson et al., Lett. Math. Phys. 113, No. 5, Paper No. 93, 38 p. (2023; Zbl 07743387); M. S. Im and M. Khovanov, “Topological theories and automata”, Preprint, arXiv:2202.13398; M. S. Im et al., “Universal construction in monoidal and non-monoidal settings, the Brauer envelope, and pseudocharacters”, Preprint, arXiv:2303.02696; M. S. Im and P. Zimmer, Involve 15, No. 2, 319–331 (2022; Zbl 1499.18039)]. In the latter case, one needs to add zero-dimensional defects with labels in a set \(\Sigma\). An oriented interval with a collection of \(\Sigma\)-labelled defects encodes a word \(\omega\), that is an element of the free monoid \(\Sigma^{\ast}\)on the set \(\Sigma\). An oriented circle with labels in \(\Sigma\)encodes a word up to cyclic eigenvalues. Given an evaluation of each word and a separate evaluation of words up to cyclic equivalence, there is an associated rigid linear monoidal category [M. Khovanov, “Decorated one-dimensional cobordisms and tensor envelopes of noncommutative recognizable power series”, Preprint, arXiv:2010.05730].
This paper studies this category for a rational evaluation \(\alpha\). The Karoubi closure of the resulting category can be reduced to the Karoubi closure of a category built from a symmetric Frobenius algebra \(\mathcal{K}\) that can be extracted from \(\alpha\) (§2.4). §§2.1–2.3 deal with the setup, basic theory and various examples. §3 reviews thin flat surface 2D TQFTs associated to symmetric Frobenius algebras, explaining how to enhance these TQFTsby 0-dimensional defects floating along the boundary that carry elements of the algebra. Comparisons between one-dimensional theories with defects and two-dimensional theories without defects are addressed throughout the paper. The Boolean analogues of these categories and their relation to automata and regular languages were investigated in [M. S. Im and M. Khovanov, “Topological theories and automata”, Preprint, arXiv:2202.13398].
For the entire collection see [Zbl 1531.17003].