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A category equivalent to the category of 3-dimensional cobordisms is defined in terms of planar diagrams. The operation of composition in this category is completely described via these diagrams. [ABSTRACT FROM AUTHOR]

This paper explains how any nondeterministic automaton for a regular language L gives rise to a one-dimensional oriented topological quantum field theory (TQFT) with inner endpoints and zero-dimensional defects labeled by letters of the alphabet for L. The TQFT is defined over the Boolean semiring B . Different automata for a fixed language L produce TQFTs that differ by their values on decorated circles, while the values on decorated intervals are described by the language L. The language L and the TQFT associated with an automaton can be given a path integral interpretation. In this TQFT, the state space of a one-point 0-manifold is a free module over B with the basis of states of the automaton. Replacing a free module by a finite projective B -module P allows to generalize automata and this type of TQFT to a structure where defects act on open subsets of a finite topological space. Intersection of open subsets induces a multiplication on P allowing to extend the TQFT to a TQFT for one-dimensional foams (oriented graphs with defects modulo a suitable equivalence relation). A linear version of these constructions is also explained, with the Boolean semiring replaced by a commutative ring. [ABSTRACT FROM AUTHOR]

A bicommutant category is a higher categorical analog of a von Neumann algebra. We study the bicommutant categories which arise as the commutant C ′ of a fully faithful representation C → Bim (R) of a unitary fusion category C . Using results of Izumi, Popa, and Tomatsu about existence and uniqueness of representations of unitary (multi)fusion categories, we prove that if C and D are Morita equivalent unitary fusion categories, then their commutant categories C ′ and D ′ are equivalent as bicommutant categories. In particular, they are equivalent as tensor categories: ( C ≃ Morita D ) ⟹ ( C ′ ≃ tensor D ′ ). This categorifies the well-known result according to which the commutants (in some representations) of Morita equivalent finite dimensional C ∗ -algebras are isomorphic von Neumann algebras, provided the representations are 'big enough'. We also introduce a notion of positivity for bi-involutive tensor categories. For dagger categories, positivity is a property (the property of being a C ∗ -category). But for bi-involutive tensor categories, positivity is extra structure. We show that unitary fusion categories and Bim (R) admit distinguished positive structures, and that fully faithful representations C → Bim (R) automatically respect these positive structures. This is the published version of arXiv:2004.08271. [ABSTRACT FROM AUTHOR]

Categorical probability has recently seen significant advances through the formalism of Markov categories, within which several classical theorems have been proven in entirely abstract categorical terms. Closely related to Markov categories are gs-monoidal categories, also known as CD categories. These omit a condition that implements the normalization of probability. Extending work of Corradini and Gadducci, we construct free gs-monoidal and free Markov categories generated by a collection of morphisms of arbitrary arity and coarity. For free gs-monoidal categories, this comes in the form of an explicit combinatorial description of their morphisms as structured cospans of labeled hypergraphs. These can be thought of as a formalization of gs-monoidal string diagrams (= term graphs) as a combinatorial data structure. We formulate the appropriate 2-categorical universal property based on ideas of Walters and prove that our categories satisfy it. We expect our free categories to be relevant for computer implementations and we also argue that they can be used as statistical causal models generalizing Bayesian networks. [ABSTRACT FROM AUTHOR]

To any Frobenius superalgebra A we associate an oriented Frobenius Brauer category and an affine oriented Frobenius Brauer category. We define natural actions of these categories on categories of supermodules for general linear Lie superalgebras g l m | n (A) with entries in A. These actions generalize those on module categories for general linear Lie superalgebras and queer Lie superalgebras, which correspond to the cases where A is the ground field and the two-dimensional Clifford superalgebra, respectively. [ABSTRACT FROM AUTHOR]