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Tag higher-dimensional-algebra [12 articles]

 
Recent papers classified by the tag higher-dimensional-algebra.
 

Hypernetworks for Reconstructing the Dynamics of Multilevel Systems

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In Proc. European Conference on Complex Systems (25-29 September 2006)
by J. Johnson
edited by University, Oxford
posted to chu-space chu-spaces complexes dynamics graph graph-theory graphs groupoid groupoids hierarchy-of-lattices higher-dimensional-algebra higher-dimensional-graphs higher-order-graphs hypergraphs hypernetworks hypersurfaces interaction interactions interactivity interconnectivity interference mathematical-structures merging modeling multi-level-systems multidimensional multilevel multiscale network network-theory networks neural-network nodal-domains polyhedral-dynamics polyhedralization polytopes q-analysis reconstructing relational-structures representation simplexes superlattice superposition tetrahedralization topology by thuraux on 2012-06-28 11:05:35 **
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✔ Type theory and homotopy

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(9 Oct 2010)
by Steve Awodey
posted to algebraic_topology category_theory computational-homotopy formal-physics fundamental-groupoids groupoids higher-dimensional higher-dimensional-algebra homotopies homotopy homotopy-invariance homotopy-theory intensional-type-theory intensionality logics model-categories propositional semigroups topos-theory type-theory type_theory by Scis0000002 on 2010-12-23 23:41:52 ** along with 2 people jrw pedagand
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Abstract

The purpose of this survey article is to introduce the reader to a connection between Logic, Geometry, and Algebra which has recently come to light in the form of an interpretation of the constructive type theory of Martin-Löf into homotopy theory, resulting in new examples of higher-dimensional categories. ...
 

✔ Higher-Dimensional Algebra V: 2-Groups

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(27 Oct 2004)
by John C. Baez, Aaron D. Lauda
posted to 2-algebras 2-groups higher-dimensional-algebra n-algebras by Scis0000002 on 2010-03-14 18:16:44 ** along with 1 person mstone
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Abstract

A 2-group is a "categorified" version of a group, in which the underlying set G has been replaced by a category and the multiplication map has been replaced by a functor. Various versions of this notion have already been explored; our goal here is to provide a detailed introduction to two, which we call "weak" and "coherent" 2-groups. A weak 2-group is a weak monoidal category in which every morphism has an inverse and every object x has a "weak inverse": an object y such that x ...
 

✔ Higher-Dimensional Algebra VI: Lie 2-Algebras

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(25 Jan 2010)
by John C. Baez, Alissa S. Crans
posted to 2-algebras higher-dimensional-algebra lie-algebras by Scis0000002 on 2010-02-20 18:59:38 ** along with 1 person jrw
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Abstract

The theory of Lie algebras can be categorified starting from a new notion of "2-vector space", which we define as an internal category in Vect. There is a 2-category 2Vect having these 2-vector spaces as objects, "linear functors" as morphisms and "linear natural transformations" as 2-morphisms. We define a "semistrict Lie 2-algebra" to be a 2-vector space L equipped with a skew-symmetric bilinear functor satisfying the Jacobi identity up to a completely antisymmetric trilinear natural transformation called the "Jacobiator", which in turn must satisfy a certain law of ...
 

✔ The Maslov dequantization, idempotent and tropical mathematics: A brief introduction

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ArXiv Mathematics e-prints (1 Jul 2005)
by G. L. Litvinov
posted to algebraic-geometry classicality concept-lattice correspondence extremal-mathematics extremal-physics extremality extremum higher-dimensional-algebra idempotence idempotent-mathematics optimality optimization-theory parallelization probability-theory variational-physics by Scis0000002 on 2007-08-24 13:16:58 **
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This paper is a brief introduction to idempotent and tropical mathematics. Tropical mathematics can be treated as a result of the so-called Maslov dequantization of the traditional mathematics over numerical fields as the Planck constant $\hbar$ tends to zero taking imaginary values. ...
 

✔ Complex Non-linear Biodynamics in Categories, Higher Dimensional Algebra and Łukasiewicz–Moisil Topos: Transformations of Neuronal, Genetic and Neoplastic Networks

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Axiomathes, Vol. 16, No. 1. (15 March 2006), pp. 65-122, doi:10.1007/s10516-005-3973-8
by I. Baianu, R. Brown, G. Georgescu, J. Glazebrook
posted to algebraicity biodynamics biological-dynamics biological-models biological-structures biological-theories category-of-quantum-automata dynamics epigenetics formal-biology groupoids hd higher-dimensional-algebra higher-dimensional-category-theory higher-dimensional-topos-theory hyperalgebras hypergroups hyperstructures lukasiewicz-moisil-algebras mathematical-biology mathematical-structures metalogics neurodynamics nonlinearity omega-algebras omega-categories omegaalgebras organismal-structures quantum-genetics relational-biology superalgebras supercategories topos-theory transforming by Scis0000002 on 2007-07-11 14:59:22 ** along with 2 people Constructivity CurAbility
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Abstract

Abstract  A categorical, higher dimensional algebra and generalized topos framework for Łukasiewicz–Moisil Algebraic–Logic models of non-linear dynamics in complex functional genomes and cell interactomes is proposed. Łukasiewicz–Moisil Algebraic–Logic models of neural, genetic and neoplastic cell networks, as well as signaling pathways in cells are formulated in terms of non-linear dynamic systems with n-state components that allow for the generalization of previous logical models of both genetic activities and neural networks. An algebraic formulation of variable ‘next-state functions’ is extended to a Łukasiewicz–Moisil ...
 

✔ A common generalization for MV-algebras and LukasiewiczMoisil algebras

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Archive for Mathematical Logic, Vol. 45, No. 8. (November 2006), pp. 947-981, doi:10.1007/s00153-006-0020-4
by Georgescu, George, Popescu, Andrei
posted to heyting-algebra higher-dimensional-algebra hyperalgebras lukasiewicz-moisil-algebras metalogics mv-algebras by Scis0000002 on 2007-07-11 14:50:23 **
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✔ Category Theory and Higher Dimensional Algebra: potential descriptive tools in neuroscience

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ArXiv Mathematics e-prints (June 2003)
by R. Brown, T. Porter
posted to biophysics category-theory higher-dimensional-algebra higher-dimensional-category-theory memory-evolutive-system neuroscience by scis0000001 on 2007-04-12 07:51:50 ** along with 1 person jrw
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Abstract

We explain the notion of colimit in category theory as a potential tool for describing structures and their communication, and the notion of higher dimensional algebra as a potential yoga for dealing with processes and processes of processes. ...
 

✔ Complex Systems Analysis of Cell Cycling Models in Carcinogenesis

  [CiTO]
eprint arXiv:q-bio/0406045 (June 2004)
by I. C. Baianu
posted to biology carcinogenesis category-theory higher-dimensional-algebra relation relational-biology topos-theory by scis0000001 on 2007-04-12 07:48:36 **
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Abstract

Carcinogenesis is a complex process that involves dynamically inter-connected modular sub-networks that evolve under the influence of micro-environmentally induced perturbations, in non-random, pseudo-Markov chain processes. An appropriate n-stage model of carcinogenesis involves therefore n-valued Logic treatments of nonlinear dynamic transformations of complex functional genomes and cell interactomes. Lukasiewicz Algebraic Logic models of genetic networks and signaling pathways in cells are formulated in terms of nonlinear dynamic systems with n-state components that allow for the generalization of previous, Boolean or "fuzzy", logic models of genetic activities in vivo. Such models are then ...
 

✔ Higher-Dimensional Algebra and Planck-Scale Physics

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(4 Feb 1999)
by John C. Baez
posted to higher-dimensional-algebra planck-scale scale by scis0000001 on 2007-01-16 10:18:35 ** along with 1 person jrw
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Abstract

This is a nontechnical introduction to recent work on quantum gravity using ideas from higher-dimensional algebra. We argue that reconciling general relativity with the Standard Model requires a `background-free quantum theory with local degrees of freedom propagating causally'. We describe the insights provided by work on topological quantum field theories such as quantum gravity in 3-dimensional spacetime. These are background-free quantum theories lacking local degrees of freedom, so they only display some of the features we seek. However, they suggest a deep link between the concepts of `space' ...
 

✔ Category Theory and Higher Dimensional Algebra: potential descriptive tools in neuroscience

  [CiTO]
(13 Jun 2003)
by R. Brown, T. Porter
posted to hd higher-dimensional-algebra neuroscience by scis0000001 ✚ on 2007-01-16 10:02:17 ** along with 5 people and 1 group ansobol jrw mstone rgb tmiya pileWorks
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Abstract

We explain the notion of colimit in category theory as a potential tool for describing structures and their communication, and the notion of higher dimensional algebra as a potential yoga for dealing with processes and processes of processes. ...
 

✔ Higher-Dimensional Algebra III: n-Categories and the Algebra of Opetopes

  [CiTO]
(10 Feb 1997)
by John C. Baez, James Dolan
posted to category-theory higher-dimensional-algebra operads opetopes by mstone on 2006-07-06 14:51:56 ** along with 2 people jrw Scis0000002
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Abstract

We give a definition of weak n-categories based on the theory of operads. We work with operads having an arbitrary set S of types, or `S-operads', and given such an operad O, we denote its set of operations by elt(O). Then for any S-operad O there is an elt(O)-operad O+ whose algebras are S-operads over O. Letting I be the initial operad with a one-element set of types, and defining I(0) = I, I(i+1) = I(i)+, we call the operations of I(n-1) the `n-dimensional opetopes'. Opetopes form ...
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