We discuss a general way of defining contexts in linear logic, based on the observation that linear universal algebra can be symmetrized by assigning an additional variable to represent the output of a term. We give two approaches to this, a syntactical one based on a new, reversible notion of term, and an algebraic one based on a simple generalization of typed operads. We relate these to each other and to known examples of logical systems, and show new examples, in particular discussing the relationship between intuitionistic and classical systems. We then present a general framework for extracting deductive systems from a given theory of contexts, and give a generic proof that all these systems have cut-elimination.

Abstract. We show that any free ∗-autonomous category is equivalent (in a strict sense) to a free ∗-autonomous category in which the double-involution (−) ∗∗ is the identity functor and the canonical isomorphism A ≃ A∗ ∗ is an identity arrow for all A. 1.

We investigate the reasons for which the existence of certain right adjoints implies the existence of some nal coalgebras, and vice-versa. In particular we prove and discuss the following theorem which has been partially available in the literature: let F a G be a pair of adjoint functors, and suppose that an initial algebra F (X) of the functor H(Y ) = X + F (Y ) exists; then a right adjoint G(X) to F (X) exists if and only if a nal coalgebra G(X) of the functor K(Y ) = X G(Y ) exists. Motivated by the problem of understanding the structures that arise from initial algebras, we show the following: if F is a left adjoint with a certain commutativity property, then an initial algebra of H(Y ) = X + F (Y ) generates a subcategory of functors with inductive types where the functorial composition is constrained to be a Cartesian product.

. Starting from symmetric monoidal closed (= autonomous) categories, Po-Hsiang Chu showed how to construct new - autonomous categories, i.e., autonomous categories that are self-dual by virtue of having a dualizing object. Recently, Michael Barr extended this to the non-symmetric, but closed, case, utilizing monads and modules between them. Since these notions are well-understood for bicategories, we introduce a notion of cyclic - autonomy for these that implies closedness and, moreover, is inherited when forming bicategories of monads and of interpolads. Since the initial step of Barr's construction also carries over to the bicategorical setting, we recover his main result as an easy corollary. Furthermore, the Chu-construction at this level may be viewed as a procedure for turning the endo1 -cells of a closed bicategory into the objects of a new closed bicategory, and hence conceptually is similar to constructing bicategories of monads and of interpolads. Keywords: closed bicate...

Linear bicategories are a generalization of ordinary bicategories in which there are two horizontal (1-cell) compositions corresponding to the "tensor" and "par" of linear logic. Benabou's notion of a morphism (lax 2-functor) of bicategories may be generalized to linear bicategories, where they are called linear functors. Unfortunately, as for the bicategorical case, it is not obvious how to organize linear functors smoothly into a higher dimensional structure. Not only do linear functors seem to lack the two compositions expected for a linear bicategory but, even worse, they inherit from the bicategorical level the failure to combine well with the obvious notion of transformation.

The notion of Frobenius algebra originally arose in ring theory, but it is a fairly easy observation that this notion can be extended to arbitrary monoidal categories. But, is this really the correct level of generalisation?

Poly-categories form a rather natural generalization of multi-categories. Besides the domains also the codomains of morphisms are allowed to be strings of objects. Multi-categories are known to have an elegant global characterization as monads in a suitable bicategory of special spans with free monoid as domains. To describe poly-categories in similar terms, we investigate distributive laws in the sense of Beck between cartesian monads as tools for constructing new bicategories of modi ed spans. Three very simple such laws produce a bicategory in which the monads are precisely the planar poly-categories (where composition only is de ned if the corresponding circuit diagram is planar). General poly-categories, which only satisfy a local planarity condition, require a slightly more complicated construction.

The cyclic Chu-construction for closed bicategories, generalizing the original Chu-construction for symmetric monoidal closed categories, turns out to have a non-cyclic counterpart. Both constructions are based on so-called Chu-cells and can be generalized to chains of composable 1-cells. This leads to two hierarchies of closed bicategories for any closed bicategory with local pullbacks. Chu-cells in rel correspond to bipartite state transition systems. Even though their vertical composition may fail here due to the lack of pullbacks, basic game-theoretic constructions can be performed on cyclic Chu-cells. These generalize to all symmetric monoidal closed categories. If finite limits exist, the cyclic Chu-cells form the objects of a *-autonomous category.

Abstract. It is well known that strong monoidal functors preserve duals. In this short note we show that a slightly weaker version of functor, which we call “Frobenius monoidal”, is sufficient. The idea of this note became apparent from Prop. 2.8 in the paper of R. Rosebrugh, N. Sabadini, and R.F.C. Walters [4]. Throughout suppose that A and B are strict 1 monoidal categories. Definition 1. A Frobenius monoidal functor is a functor F: A � � B which is monoidal (F, r, r0) and comonoidal (F, i, i0), and satisfies the compatibility conditions for all A, B, C ∈ A. ir = (1 ⊗ r)(i ⊗ 1) : F(A ⊗ B) ⊗ FC ir = (r ⊗ 1)(1 ⊗ i) : FA ⊗ F(B ⊗ C) � � FA ⊗ F(B ⊗ C) � � F(A ⊗ B) ⊗ FC, The compact case ( ⊗ = ⊕) of Cockett and Seely’s linearly distributive functors [2] are precisely Frobenius monoidal functors, and Frobenius monoidal functors with ri = 1 have been called split monoidal by Szlachányi in [5]. A dual situation in A is a tuple (A, B, e, n), where A and B are objects of A and e: A ⊗ B � � I n: I � � B ⊗ A are morphisms in A, called evaluation and coevaluation respectively, satisfying the “triangle identities”: 1⊗n n⊗1

2.2. Soergel bimodules 6