4,548 research outputs found
Tensor products of finitely cocomplete and abelian categories
The purpose of this article is to study the existence of Deligne's tensor
product of abelian categories by comparing it with the well-known ten- sor
product of finitely cocomplete categories. The main result states that the
former exists precisely when the latter is an abelian category, and moreover in
this case both tensor products coincide. An example of two abelian categories
whose Deligne tensor product does not exist is given.Comment: 14 page
Lax orthogonal factorisation systems
This paper introduces lax orthogonal algebraic weak factorisation systems on
2-categories and describes a method of constructing them. This method rests in
the notion of simple 2-monad, that is a generalisation of the simple
reflections studied by Cassidy, H\'ebert and Kelly. Each simple 2-monad on a
finitely complete 2-category gives rise to a lax orthogonal algebraic weak
factorisation system, and an example of a simple 2-monad is given by completion
under a class of colimits. The notions of KZ lifting operation, lax natural
lifting operation and lax orthogonality between morphisms are studied.Comment: 59 page
Consciosusness in Cognitive Architectures. A Principled Analysis of RCS, Soar and ACT-R
This report analyses the aplicability of the principles of consciousness developed in the ASys project to three of the most relevant cognitive architectures. This is done in relation to their aplicability to build integrated control systems and studying their support for general mechanisms of real-time consciousness.\ud
To analyse these architectures the ASys Framework is employed. This is a conceptual framework based on an extension for cognitive autonomous systems of the General Systems Theory (GST).\ud
A general qualitative evaluation criteria for cognitive architectures is established based upon: a) requirements for a cognitive architecture, b) the theoretical framework based on the GST and c) core design principles for integrated cognitive conscious control systems
Hopf measuring comonoids and enrichment
We study the existence of universal measuring comonoids for a pair
of monoids , in a braided monoidal closed category, and the associated
enrichment of a category of monoids over the monoidal category of comonoids. In
symmetric categories, we show that if is a bimonoid and is a
commutative monoid, then is a bimonoid; in addition, if is a
cocommutative Hopf monoid then always is Hopf. If is a Hopf
monoid, not necessarily cocommutative, then is Hopf if the fundamental
theorem of comodules holds; to prove this we give an alternative description of
the dualizable -comodules and use the theory of Hopf (co)monads. We
explore the examples of universal measuring comonoids in vector spaces and
graded spaces.Comment: 30 pages. Version 2: re-arrangement of material; expansion of
previous section 6, splitting into current sections 6,7,8; fix of graded
algebras example, section 11; appendix removed; other minor fixes and edit
Principles for Consciousness in Integrated Cognitive Control
In this article we will argue that given certain conditions for the evolution of bi- \ud
ological controllers, these will necessarily evolve in the direction of incorporating \ud
consciousness capabilities. We will also see what are the necessary mechanics for \ud
the provision of these capabilities and extrapolate this vision to the world of artifi- \ud
cial systems postulating seven design principles for conscious systems. This article \ud
was published in the journal Neural Networks special issue on brain and conscious- \ud
ness
Autonomous Pseudomonoids
In this dissertation we generalise the basic theory of Hopf algebras to the context of autonomous pseudomonoids in monoidal bicategories.
Autonomous pseudomonoids were introduced in [13] as generalisations of both autonomous monoidal categories and Hopf algebras. Much of the theory of autonomous pseudomonoids developed in [13] was inspired by the example of autonomous (pro)monoidal enriched categories. The present thesis aims to further develop the theory with results inspired by Hopf algebra theory instead. We study three important results in Hopf algebra theory: the so-called "fundamental theorem of Hopf modules", the "Drinfel'd quantum double" and its relation with the centre of monoidal categories, and " Radford's formula".
The basic result of this work is a general fundamental theorem of Hopf modules that establishes conditions equivalent to the existence of a left dualization. With this result as a base, we are able to construct the centre (defined in [83]) and the lax centre of an autonomous pseudomonoid as an Eilenberg-Moore construction for certain monad. As an application we show that the Drinfel'd double of a finite-dimensional Hopf algebra is equivalent to the centre of the associated pseudomonoid.
The next piece of theory we develop is a general Radford's formula for autonomous map pseudomonoids formula in the case of a (coquasi) Hopf algebra. We also introduce "unimodular" autonomous pseudomonoids.
In the last part of the dissertation we apply the general theory to enriched categories with a (chosen) class of (co)limits, with emphasis in the case of finite (co)limits. We construct tensor products of such categories by means of pseudo-commutative enriched monads (a slight generalisation of the pseudo-commutative 2-monads of [37], and showing that lax-idempotent 2-monads are pseudo-commutative. Finally we apply the general theory developed for pseudomonoids to deduce the main results of [27].This work was supported by an Internal Graduate Studentship, Trinity College, Cambridge
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