Claudio Hermida

On weak higher dimensional categories I: Part 1

Inspired by the concept of opetopic set introduced in a recent paper by John C. Baez and James Dolan, we give a modified notion called multitopic set. The name reflects the fact that, whereas the Baez/Dolan concept is based on operads, the one in this paper is based on multicategories. The concept of multicategory used here is a mild generalization of the same-named notion introduced by Joachim Lambek in 1969. Opetopic sets and multitopic sets are both intended as vehicles for concepts of weak higher dimensional category. Baez and Dolan define weak n-categories as (n+1)-dimensional opetopic sets satisfying certain properties. The version intended here, multitopic n-category, is similarly related to multitopic sets. Multitopic n-categories are not described in the present paper; they are to follow in a sequel. The present paper gives complete details of the definitions and basic properties of the concepts involved with multitopic sets. The category of multitopes, analogs of opetopes of Baez and Dolan, is presented in full, and it is shown that the category of multitopic sets is equivalent to the category of set-valued functors on the category of multitopes.

Some properties of Fib as a fibred 2-category

Abstract We consider some basic properties of the 2-category Fib of fibrations over arbitrary bases, exploiting the fact that it is fibred over Cat. We show a factorisation property for adjunctions in Fib, which has direct consequences for fibrations, e.g. a characterisation of limits and colimits for them. We also consider oplax colimits in Fib, with the construction of Kleisli objects as a particular example. All our constructions are based on an elementary characterisation of Fib as a 2-fibration.

From coherent structures to universal properties

Abstract Given a 2-category K admitting a calculus of bimodules, and a 2-monad T on it compatible with such calculus, we construct a 2-category L with a 2-monad S on it such that • S has the adjoint-pseudo-algebra property. • The 2-categories of pseudo-algebras of S and T are equivalent. Thus, coherent structures (pseudo- T -algebras) are transformed into universally characterised ones (adjoint-pseudo- S -algebras). The 2-category L consists of lax algebras for the pseudo-monad induced by T on the bicategory of bimodules of K . We give an intrinsic characterisation of pseudo- S -algebras in terms of representability. Two major consequences of the above transformation are the classifications of lax and strong morphisms, with the attendant coherence result for pseudo-algebras. We apply the theory in the context of internal categories and examine monoidal and monoidal globular categories (including their monoid classifiers) as well as pseudo-functors into C at .

A categorical outlook on relational modalities and simulations

We characterise bicategories of spans, relations and partial maps universally in terms of factorisations involving maps. We apply this characterisation to show that the standard modalities @? and @? arise canonically as extensions of a predicate logic from functions to (abstract) relations. When relations and partial maps are representable, we exhibit logical predicates for the power-object and partial-map-classifier monads. We also show that the @? modality gives the relevant pullbacks of subobjects in the internal logic of categories of partial maps. Organising modal formulae fibrationally, we exhibit an intrinsic relationship between their satisfaction relative to transition systems and the notion of simulation. In this setting, we use the biclosed structure of the bicategory of relations to give a new proof of the standard fact that observational similarity implies similarity.

On weak higher-dimensional categories. I.2

The present second part of a three-part paper gives the detailed treatment of the new notion of multicategory, and that of the construction of the particular multicategory of function replacement. For the overall purpose of the whole paper, see the abstract in Part 1.

Higher dimensional multigraphs

We introduce the notion of higher dimensional multigraph. This notion extends that of multigraph, which underlies multicategories and is essentially equivalent to the notion of context-free grammar. We develop the definition and explain how it gives a semantically coherent category theoretic approach to the notion of higher order context-free grammar. It also gives a conceptual framework in which one can study rewrites, and rewrites of rewrites, etcetera, for proofs of sequent calculus. The definition involves a subtle interaction between geometry and linearly defined syntax; we explore the latter here, outlining the geometric intuition.

Paracategories I: internal paracategories and saturated partial algebras

Based on the monoid classifier Δ, we give an alternative axiomatization of Freyds paracategories, which can be interpreted in any bicategory of partial maps. Assuming furthermore a free-monoid monad T in our ambient category, and coequalisers satisfying some exactness conditions, we give an abstract envelope construction, putting paramonoids (and paracategories) in the more general context of partial algebras. We introduce for the latter the crucial notion of saturation, which characterises those partial algebras which are isomorphic to the ones obtained from their enveloping algebras. We also set up a factorisation system for partial algebras, via epimorphisms and (monic) Kleene morphisms and relate the latter to saturation.

Descent on 2-Fibrations and Strongly 2-Regular 2-Categories

We consider pseudo-descent in the context of 2-fibrations. A 2-category of descent data is associated to a 3-truncated simplicial object in the base 2-category. A morphism q in the base induces (via comma-objects and pullbacks) an internal category whose truncated simplicial nerve induces in turn the 2-category of descent data for q. When the 2-fibration admits direct images, we provide the analogous of the Beck–Bénabou–Roubaud theorem, identifying the 2-category of descent data with that of pseudo-algebras for the pseudo-monad q*Σq. We introduce a notion of strong 2-regularity for a 2-category R, so that its basic 2-fibration of internal fibrations cod:Fib(R)→R admits direct images. In this context, we show that essentially-surjective-on-objects morphisms, defined by a certain lax colimit, are of effective descent by means of a Beck-style pseudo-monadicity theorem.

Paracategories II: adjunctions, fibrations and examples from probabilistic automata theory

In this sequel to Hermida and Mateus (Paracategories I: internal paracategories and saturated partial algebras, Theoret. Comput. Sci., in press), we explore some of the global aspects of the category of paracategories. We establish its (co)completeness and cartesian closure. From the closed structure we derive the relevant notion of transformation for paracategories. We set up the relevant notion of adjunction between paracategories and apply it to define (co)completeness and cartesian closure, exemplified by the paracategory of bivariant functors and dinatural transformations. We introduce partial multicategories to account for partial tensor products. We also consider fibrations for paracategories and their indexed-paracategory version. Finally, we instantiate all these concepts in the context of probabilistic automata.