The objective of this thesis is to develop a semantics for higher-order quantum information. Following the work done in the author’s M.Sc. thesis, we study a lambda calculus for quantum computation with classical control. The language features two important properties. The first one, arising from the so-called no-cloning theorem of quantum computation, is the need for a distinction between dupl...

For a Hopf algebra H over a commutative ring k, the category MH of right Hopf modules is equivalent to the category Mk of k-modules, that is, the comparison functor −⊗k H : Mk → MH is an equivalence (Fundamental theorem of Hopf modules). This was proved by Larson and Sweedler via the notion of coinvariants McoH for any M ∈ MH . The coinvariants functor (−) coH : MH → Mk is right adjoint to the ...

Speaker: Gregory Arone (Virginia) Title: Part 1: operads, modules and the chain rule Abstract: Let F be a homotopy functor between the categories of pointed topological spaces or spectra. By the work of Goodwillie, the derivatives of F form a symmetric sequence of spectra ∂∗F . This symmetric sequence determines the homogeneous layers in the Taylor tower of F , but not the extensions in the tow...

For any total category K , with defining adjunction ∨ a Y : K // set op , the expression W (A)(K) = set K op (K (A, ∨ −), [K,−]), where [K,−] is evaluation at K, provides a well-defined functor W : K // K̂ = set op . Also, there are natural transformations β : W ∨ // 1 K̂ and γ : ∨ W // 1K satisfying ∨ β = γ ∨ and βW = Wγ. A total K is totally distributive if ∨ has a left adjoint. We show that K ...

The purpose of this paper is to develop a theory of bimonads and Hopf monads on arbitrary categories thus providing the possibility to transfer the essentials of the theory of Hopf algebras in vector spaces to more general settings. There are several extensions of this theory to monoidal categories which in a certain sense follow the classical trace. Here we do not pose any conditions on our ba...

This paper revisits the authors’ notion of a differential category from a different perspective. A differential category is an additive symmetric monoidal category with a comonad (a “coalgebra modality”) and a differential combinator. The morphisms of a differential category should be thought of as the linear maps; the differentiable or smooth maps would then be morphisms of the coKleisli categ...

In [3], Christine Tasson introduces an algebraic notion of totality for a denotational model of linear logic. The notion of total boolean function is, in a way, quite intuitive. This note provides a positive answer to the question of completeness of the " boolean centroidal calculus " w.r.t. total boolean functions. 0. Introduction. Even though the question answered in this note has its roots i...

We give a class of proof nets for Intuitionistic Linear Logic with the connectives (;!, prove a correctness criterion for them and show that a games semantics can be directly derived from these nets, along with a full completeness theorem. It is well-known that games semantics is intimately connected to linear logic, but there is an important example of games semantics where the connection is f...

Speaker: Gregory Arone (Virginia) Title: Part 1: operads, modules and the chain rule Abstract: Let F be a homotopy functor between the categories of pointed topological spaces or spectra. By the work of Goodwillie, the derivatives of F form a symmetric sequence of spectra ∂∗F . This symmetric sequence determines the homogeneous layers in the Taylor tower of F , but not the extensions in the tow...

Many concepts from category theory have proven useful tools for program abstraction, particularly in functional programming. For example, many parametric data types admit operations which are analogous to a functor and a monad. However, some parametric data types whose operations are restricted in their parametricity are not amenable to traditional category-theoretic abstractions in Haskell, de...