**Publications** | Teaching

- roadmap.ps
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**A roadmap to the unification of weak categorical structures: transformations and equivalences among the various notions of pseudo-algebra**(*June 2004*)

A quick tour of the constructions/equivalences*monoidal category :: representable multicategory :: covariantly fibrant multicategory :: pseudo-monoid in*, and their general pseudo-algebra versions.**Cat** - paracategories-2.ps
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**Paracategories II: Adjunctions, fibrations and examples from probabilistic automata theory**by C. Hermida and P. Mateus (*revised version in*).**Theoretical Computer Science**, 311, 71-103 2004In this sequel to

Abstract:**Paracategories I**, 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. - paracategories-1.ps
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**Paracategories I: Internal Paracategories and Saturated Partial Algebras**by C. Hermida and P. Mateus (*revised version in*).**Theoretical Computer Science**, 309, 125-156 2003Based on the monoid classif{i}er, we give an alternative axiomatization of Freyd's paracategories, which can be interpreted in any

Abstract:*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. - 2-descent.pdf
**Descent on 2-fibrations and 2-regular 2-categories**(*revised version in*).**Applied Categorical Structures**, 12(5-6), 427--459, 2004We 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

Abstract:**$q$**in the base induces (via*comma-objects*and pullbacks) an internal category whose truncated nerve allows the definition of the 2-category of descent data for**$q$**. When the 2-fibration admits direct images, we provide the analogous of the Beck-Benabou-Roubaud theorem, identifying the 2-category of descent data with that of pseudo-algebras for the pseudo-monad**$q*\Sigma $**. We introduce a notion of_{q}*2-regularity*for a 2-category, so that its basic 2-fibration of internal fibrations $$**R***cod*:**Fib (**→*R*)admits direct images. In this context, we show that**R***essentially-surjective-on-objects*morphisms, defined by a certain lax colimit, are of effective descent by means of a Beck-style pseudo-monadicity theorem. - coh-univ.ps
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**From coherent structures to universal properties**(*version March 2000; final version in*).**Journal of Pure and Applied Algebra**165 (1),7-61, 2001

*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: i) S has the adjoint-pseudo-algebra property. ii) 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 Cat. - rep-mult.ps
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**Representable multicategories**(*version June 8, 1999 , revised in*).**Advances in Mathematics**151, 164-225, 2000

*Abstract:*We introduce the notion of*representable multicategory*, which stands in the same relation to that of monoidal category as fibration does to contravariant pseudofunctor (into**Cat**). We give an abstract reformulation of multicategories as monads in a suitable Kleisli bicategory of spans. We describe representability in elementary terms via*universal arrows*. We also give a doctrinal characterisation of representability based on a fundamental monadic adjunction between the 2-category of multicategories and that of strict monoidal categories. The first main result is the coherence theorem for representable multicategories, asserting their equivalence to strict ones, which we establish via a new technique based on the above doctrinal characterisation. The other main result is a 2-equivalence between the 2-category of representable multicategories and that of monoidal categories and strong monoidal functors. This correspondence extends smoothly to one between bicategories and a localised version of representable multicategories. - 2-fib.ps
2-fib.pdf
**Some Properties of Fib as a fibred 2-category**(*revised version in*).**Journal of Pure and Applied Algebra**134 (1), 83-109, 1999

*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, eg. 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. - induction.ps
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**Structural Induction and Coinduction in a fibrational setting**by C. Hermida and B. Jacobs (*revised version in*).**Information and Computation**145 (2), 107-152, 1998

*Abstract:*We present a categorical logic formulation of induction and coinduction principles for reasoning about inductively and coinductively defined types. Our main results provide sufficient criteria for the validity of such principles: in the presence of comprehension, the induction principle for initial algebras is admissible, and dually, in the presence of quotient types, the coinduction principle for terminal coalgebras is admissible. After giving an alternative formulation of induction in terms of binary relations, we combine both principles and obtain a mixed induction/coinduction principle which allows us to reason about minimal solutions $X\cong \sigma (X)$ where $X$ may occur both positively and negatively in the type constructor $\sigma $ . We further strengthen these logical principles to deal with contexts and prove that such strengthening is valid when the (abstract) logic we consider is contextually/functionally complete. All the main results follow from a basic result about adjunctions between `categories of algebras' (inserters). - comp-fact.ps
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**A note on comprehensive factorisation**. - sat-sim.ps
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(DRAFT)
**A categorical outlook on relational modalities and simulations***presented at IMLA'02.*We characterise bicategories of spans, relations and partial maps universally in terms of factorisations involving

Abstract:*maps*. We apply this characterisation to show that the standard modalities arise canonically as the extension of a predicate logic from*functions*to (abstract)*relations*. With the resulting fibrational interpretation of modalities, we show how to deal with*representability*, thereby deriving*logical predicates*for the power-object and partial-map-classifier monads. In the second part of the article, we exhibit an intrinsic relationship between satisfaction of modal formuale (in a transition system) and simulations, and apply it to exhibit the role of the biclosed nature of the bicategory of relations in proving that*observational similarity*implies*similarity*. - col-dec.ps/pdf
(DRAFT)
**Colimimit decomposition for diagrams indexed by a cofibred category**We characterise small cocompleteness for a category as a 2-adjoint, and obtain from this characterisation a colimit decomposition for diagrams indexed by a cofibration.

Abstract: - FibAbstractMulticat-fields.ps
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**Fibrations for abstract multicategories**(*revised version to appear in Proceedings of*). ( Slides and audio).**Workshop on Categorical Structures for Descent and Galois Theory, Hopf Algebras and Semiabelian Categories**, Fields Institute, Toronto, September 23-28, 2002Building upon the theory of 2-dimensional fibrations and that of (abstract) multicategories, we present the basics of a theory of

Abstract:*fibred multicategories*. We show their intrinsic role in the general theory: a multicategory is representable precisely when it is covariantly fibrant over the terminal such. Futhermore, such fibred structures allow for a treatment of*algebras for operads*in the internal category setting. We obtain thus a conceptual proof of the `slices of categories of algebras are categories of algebras' property, which is instrumental in setting up Baez-Dolan's opetopes.

**Publications** | Teaching