A type theory for synthetic ∞\infty-categories

We propose foundations for a synthetic theory of $(\infty,1)$-categories within homotopy type theory. We axiomatize a directed interval type, then define higher simplices from it and use them to probe the internal categorical structures of arbitrary types. We define Segal types, in which binary composites exist uniquely up to homotopy; this automatically ensures composition is coherently associative and unital at all dimensions. We define Rezk types, in which the categorical isomorphisms are additionally equivalent to the type-theoretic identities - a "local univalence" condition. And we define covariant fibrations, which are type families varying functorially over a Segal type, and prove a "dependent Yoneda lemma" that can be viewed as a directed form of the usual elimination rule for identity types. We conclude by studying homotopically correct adjunctions between Segal types, and showing that for a functor between Rezk types to have an adjoint is a mere proposition. To make the bookkeeping in such proofs manageable, we use a three-layered type theory with shapes, whose contexts are extended by polytopes within directed cubes, which can be abstracted over using "extension types" that generalize the path-types of cubical type theory. In an appendix, we describe the motivating semantics in the Reedy model structure on bisimplicial sets, in which our Segal and Rezk types correspond to Segal spaces and complete Segal spaces.Comment: 78 pages; v2 has minor updates inspired by discussions at the Mathematics Research Community on Homotopy Type Theory; v3 incorporates many expository improvements suggested by the referee; v4 is the final journal version to appear in Higher Structures with a more precise syntax for our type theory with shape

How to Derive the Hilbert-Space Formulation of Quantum Mechanics From Purely Operational Axioms

In the present paper I show how it is possible to derive the Hilbert space formulation of Quantum Mechanics from a comprehensive definition of "physical experiment" and assuming "experimental accessibility and simplicity" as specified by five simple Postulates. This accomplishes the program presented in form of conjectures in the previous paper quant-ph/0506034. Pivotal roles are played by the "local observability principle", which reconciles the holism of nonlocality with the reductionism of local observation, and by the postulated existence of "informationally complete observables" and of a "symmetric faithful state". This last notion allows one to introduce an operational definition for the real version of the "adjoint"--i. e. the transposition--from which one can derive a real Hilbert-space structure via either the Mackey-Kakutani or the Gelfand-Naimark-Segal constructions. Here I analyze in detail only the Gelfand-Naimark-Segal construction, which leads to a real Hilbert space structure analogous to that of (classes of generally unbounded) selfadjoint operators in Quantum Mechanics. For finite dimensions, general dimensionality theorems that can be derived from a local observability principle, allow us to represent the elements of the real Hilbert space as operators over an underlying complex Hilbert space (see, however, a still open problem at the end of the paper). The route for the present operational axiomatization was suggested by novel ideas originated from Quantum Tomography.Comment: Work presented at the conference "On the Present Status of Quantum Mechanics" held on 7-9 September 2005, Mali Losinj, Croatia, in celebration of the 70th birthday of Gian Carlo Ghirardi. The dimensionality theorems along with all results already published in quant-ph/0506034 have been presented at the conference, whereas the operational definition of the real adjoint and the Hilbert spaces derivations have been presented as work in progress, and were completed two months late

Iterated spans and classical topological field theories

We construct higher categories of iterated spans, possibly equipped with extra structure in the form of "local systems", and classify their fully dualizable objects. By the Cobordism Hypothesis, these give rise to framed topological quantum field theories, which are the framed versions of the "classical" TQFTs considered in the quantization programme of Freed-Hopkins-Lurie-Teleman. Using this machinery, we also construct an infinity-category of Lagrangian correspondences between symplectic derived algebraic stacks and show that all its objects are fully dualizable.Comment: Accepted version, plus corrections to Remarks 10.5 and 10.7. 64 page

(Op)lax natural transformations, twisted quantum field theories, and "even higher" Morita categories

Motivated by the challenge of defining twisted quantum field theories in the context of higher categories, we develop a general framework for lax and oplax transformations and their higher analogs between strong (∞,n)-functors. We construct a double (∞,n)-category built out of the target (∞,n)-category governing the desired diagrammatics. We define (op)lax transformations as functors into parts thereof, and an (op)lax twisted field theory to be a symmetric monoidal (op)lax natural transformation between field theories. We verify that lax trivially-twisted relative field theories are the same as absolute field theories. As a second application, we extend the higher Morita category of E d -algebras in a symmetric monoidal (∞,n)-category C to an (∞,n+d)-category using the higher morphisms in C