Initial Semantics for Strengthened Signatures

We give a new general definition of arity, yielding the companion notions of signature and associated syntax. This setting is modular in the sense requested by Ghani and Uustalu: merging two extensions of syntax corresponds to building an amalgamated sum. These signatures are too general in the sense that we are not able to prove the existence of an associated syntax in this general context. So we have to select arities and signatures for which there exists the desired initial monad. For this, we follow a track opened by Matthes and Uustalu: we introduce a notion of strengthened arity and prove that the corresponding signatures have initial semantics (i.e. associated syntax). Our strengthened arities admit colimits, which allows the treatment of the \lambda-calculus with explicit substitution.Comment: In Proceedings FICS 2012, arXiv:1202.317

High-level signatures and initial semantics

We present a device for specifying and reasoning about syntax for datatypes, programming languages, and logic calculi. More precisely, we study a notion of signature for specifying syntactic constructions. In the spirit of Initial Semantics, we define the syntax generated by a signature to be the initial object---if it exists---in a suitable category of models. In our framework, the existence of an associated syntax to a signature is not automatically guaranteed. We identify, via the notion of presentation of a signature, a large class of signatures that do generate a syntax. Our (presentable) signatures subsume classical algebraic signatures (i.e., signatures for languages with variable binding, such as the pure lambda calculus) and extend them to include several other significant examples of syntactic constructions. One key feature of our notions of signature, syntax, and presentation is that they are highly compositional, in the sense that complex examples can be obtained by assembling simpler ones. Moreover, through the Initial Semantics approach, our framework provides, beyond the desired algebra of terms, a well-behaved substitution and the induction and recursion principles associated to the syntax. This paper builds upon ideas from a previous attempt by Hirschowitz-Maggesi, which, in turn, was directly inspired by some earlier work of Ghani-Uustalu-Hamana and Matthes-Uustalu. The main results presented in the paper are computer-checked within the UniMath system.Comment: v2: extended version of the article as published in CSL 2018 (http://dx.doi.org/10.4230/LIPIcs.CSL.2018.4); list of changes given in Section 1.5 of the paper; v3: small corrections throughout the paper, no major change

Relating Operator Spaces via Adjunctions

This chapter uses categorical techniques to describe relations between various sets of operators on a Hilbert space, such as self-adjoint, positive, density, effect and projection operators. These relations, including various Hilbert-Schmidt isomorphisms of the form tr(A-), are expressed in terms of dual adjunctions, and maps between them. Of particular interest is the connection with quantum structures, via a dual adjunction between convex sets and effect modules. The approach systematically uses categories of modules, via their description as Eilenberg-Moore algebras of a monad

Monads and extensive quantities

If T is a commutative monad on a cartesian closed category, then there exists a natural T-bilinear pairing from T(X) times the space of T(1)-valued functions on X ("integration"), as well as a natural T-bilinear action on T(X) by the space of these functions. These data together make the endofunctors T and "functions into T(1)" into a system of extensive/intensive quantities, in the sense of Lawvere. A natural monad map from T to a certain monad of distributions (in the sense of functional analysis (Schwartz)) arises from this integration

Hopf monads on monoidal categories

We define Hopf monads on an arbitrary monoidal category, extending the definition given previously for monoidal categories with duals. A Hopf monad is a bimonad (or opmonoidal monad) whose fusion operators are invertible. This definition can be formulated in terms of Hopf adjunctions, which are comonoidal adjunctions with an invertibility condition. On a monoidal category with internal Homs, a Hopf monad is a bimonad admitting a left and a right antipode. Hopf monads generalize Hopf algebras to the non-braided setting. They also generalize Hopf algebroids (which are linear Hopf monads on a category of bimodules admitting a right adjoint). We show that any finite tensor category is the category of finite-dimensional modules over a Hopf algebroid. Any Hopf algebra in the center of a monoidal category C gives rise to a Hopf monad on C. The Hopf monads so obtained are exactly the augmented Hopf monads. More generally if a Hopf monad T is a retract of a Hopf monad P, then P is a cross product of T by a Hopf algebra of the center of the category of T-modules (generalizing the Radford-Majid bosonization of Hopf algebras). We show that the comonoidal comonad of a Hopf adjunction is canonically represented by a cocommutative central coalgebra. As a corollary, we obtain an extension of Sweedler's Hopf module decomposition theorem to Hopf monads (in fact to the weaker notion of pre-Hopf monad).Comment: 45 page