Pseudo-commutative Monads

AbstractWe introduce the notion of pseudo-commutative monad together with that of pseudo-closed 2-category, the leading example being given by the 2-monad on Cat whose 2-category of algebras is the 2-category of small symmetric monoidal categories. We prove that for any pseudo-commutative 2-monad on Cat, its 2-category of algebras is pseudo-closed. We also introduce supplementary definitions and results, and we illustrate this analysis with further examples such as those of small categories with finite products, and examples arising from wiring, interaction, contexts, and the logic of Bunched Implication

Conservative descent for semi-orthogonal decompositions

Motivated by the local flavor of several well-known semi-orthogonal decompositions in algebraic geometry, we introduce a technique called conservative descent, which shows that it is enough to establish these decompositions locally. The decompositions we have in mind are those for projectivized vector bundles and blow-ups, due to Orlov, and root stacks, due to Ishii and Ueda. Our technique simplifies the proofs of these decompositions and establishes them in greater generality for arbitrary algebraic stacks.Comment: Final versio

The weak theory of monads

We construct a `weak' version EM^w(K) of Lack & Street's 2-category of monads in a 2-category K, by replacing their compatibility constraint of 1-cells with the units of monads by an additional condition on the 2-cells. A relation between monads in EM^w(K) and composite pre-monads in K is discussed. If K admits Eilenberg-Moore constructions for monads, we define two symmetrical notions of `weak liftings' for monads in K. If moreover idempotent 2-cells in K split, we describe both kinds of a weak lifting via an appropriate pseudo-functor EM^w(K) --> K. Weak entwining structures and partial entwining structures are shown to realize weak liftings of a comonad for a monad in these respective senses. Weak bialgebras are characterized as algebras and coalgebras, such that the corresponding monads weakly lift for the corresponding comonads and also the comonads weakly lift for the monads.Comment: 30 page

Exact sequences of tensor categories

We introduce the notions of normal tensor functor and exact sequence of tensor categories. We show that exact sequences of tensor categories generalize strictly exact sequences of Hopf algebras as defined by Schneider, and in particular, exact sequences of (finite) groups. We classify exact sequences of tensor categories C' -> C -> C'' (such that C' is finite) in terms of normal faithful Hopf monads on C'' and also, in terms of self-trivializing commutative algebras in the center of C. More generally, we show that, given any dominant tensor functor C -> D admitting an exact (right or left) adjoint there exists a canonical commutative algebra A in the center of C such that F is tensor equivalent to the free module functor C -> mod_C A, where mod_C A denotes the category of A-modules in C endowed with a monoidal structure defined using the half-braiding of A. We re-interpret equivariantization under a finite group action on a tensor category and, in particular, the modularization construction, in terms of exact sequences, Hopf monads and commutative central algebras. As an application, we prove that a braided fusion category whose dimension is odd and square-free is equivalent, as a fusion category, to the representation category of a group.Comment: 39 page

Monads with arities and their associated theories

After a review of the concept of "monad with arities" we show that the category of algebras for such a monad has a canonical dense generator. This is used to extend the correspondence between finitary monads on sets and Lawvere's algebraic theories to a general correspondence between monads and theories for a given category with arities. As application we determine arities for the free groupoid monad on involutive graphs and recover the symmetric simplicial nerve characterisation of groupoids.Comment: New introduction; Section 1 shortened and redispatched with Section 2; Subsections on symmetric operads (3.14) and symmetric simplicial sets (4.17) added; Bibliography complete

Type-Decomposition of a Pseudo-Effect Algebra

The theory of direct decomposition of a centrally orthocomplete effect algebra into direct summands of various types utilizes the notion of a type-determining (TD) set. A pseudo-effect algebra (PEA) is a (possibly) noncommutative version of an effect algebra. In this article we develop the basic theory of centrally orthocomplete PEAs, generalize the notion of a TD set to PEAs, and show that TD sets induce decompositions of centrally orthocomplete PEAs into direct summands.Comment: 18 page