Homotopy theory, and change of base for groupoids and multiple groupoids

Brown, Ronald

doi:10.1007/bf00122251

Cited by 11 publications

(7 citation statements)

References 36 publications

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“…However, the recent development of different areas of mathematics demands a generalization of this notion to all higher dimensions [5,25,10,14].``It is clear that new ideas are needed to do so without a combinatorial explosion... . However, the potential payoffs of a good theory of weak n-categories should encourage us to persevere.''…”

Section: Introductionmentioning

confidence: 99%

Monoidal Globular Categories As a Natural Environment for the Theory of Weakn-Categories

Batanin¹

1998

Advances in Mathematics

191

450

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We present a definition of weak |-categories based on a higher-order generalization of apparatus of operads.1998 Academic Press Contents.1. From non-symmetric to higher order operads. Monoidal globular categories.3. Some examples. 4.Coherence for monoidal globular categories. Coproducts in monoidal globular categories.6. Collections in globular categories. 7.Higher-order operads and their algebras.8. Contractibility and weak categories. Higher fundamental groupoids.10. Appendix: 2-operads and bicategories.

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Section: Introductionmentioning

confidence: 99%

Monoidal Globular Categories As a Natural Environment for the Theory of Weakn-Categories

Batanin¹

1998

Advances in Mathematics

191

450

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“…See also an example (Douady and Douady 1979) that consists in an exposition of the relation of this approach with the Galois theory. Another paper, by Brown (1996), and Janelidze (1997, 2004), gives a general formulation of conditions for the theorem to hold in the case X 0 = X in terms of the map U t V ! X being an 'effective global descent morphism' (the theorem is given in the generality of lextensive categories).…”

Section: Wider Considerationsmentioning

confidence: 98%

“…For example, the opposite of the category of sets has objects but these have no structure from the categorical viewpoint. Other types of category are important as expressing useful relationships on structures, for example lextensive categories, which have been used to express a general van Kampen theorem by Brown (1996), and Janelidze (1997, 2004). This concrete categorical approach seems also to provide an elegant formalization that matches the ontological theory of levels briefly described above.…”

Section: From Object and Structure To Organismic Functions And Relatimentioning

confidence: 99%

A Conceptual Construction of Complexity Levels Theory in Spacetime Categorical Ontology: Non-Abelian Algebraic Topology, Many-Valued Logics and Dynamic Systems

2007

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A novel conceptual framework is introduced for the Complexity Levels Theory in a Categorical Ontology of Space and Time. This conceptual and formal construction is intended for ontological studies of Emergent Biosystems, Supercomplex Dynamics, Evolution and Human Consciousness. A claim is defended concerning the universal representation of an item's essence in categorical terms. As an essential example, relational structures of living organisms are well represented by applying the important categorical concept of natural transformations to biomolecular reactions and relational structures that emerge from the latter in living systems. Thus, several relational theories of living systems can be represented by natural transformations of organismic, relational structures. The ascent of man and other living organisms through adaptation, is viewed in novel categorical terms, such as variable biogroupoid representations of evolving species. Such precise but flexible evolutionary concepts will allow the further development of the unifying theme of local-to-global approaches to highly complex systems in order to represent novel patterns of relations that emerge in super-and ultra-complex systems in terms of compositions of local procedures. Solutions to such local-to-global problems in highly complex systems with 'broken symmetry' might be possible to be reached with the help of higher homotopy theorems in algebraic topology such as the generalized van Kampen theorems (HHvKT). Categories of many-valued, Łukas-iewicz-Moisil (LM) logic algebras provide useful concepts for representing the intrinsic dynamic 'asymmetry' of genetic networks in organismic development and evolution, as well as to derive novel results for (non-commutative) Quantum Logics. Furthermore, as recently pointed out by Baianu and Poli (Theory and applications of ontology, vol 1. Springer, Berlin, in press), LM-logic algebras may also provide the appropriate framework for future developments of the ontological theory of levels with its complex/entangled/intertwined ramifications in psychology, sociology and ecology. As shown in the preceding two papers in this issue, a paradigm shift towards non-commutative, or non-Abelian, theories of highly complex dynamics-which is presently unfolding in physics, mathematics, life and cognitive sciences-may be implemented through realizations of higher dimensional algebras in neurosciences and psychology, as well as in human genomics, bioinformatics and interactomics.Keywords Categorical ontology and the theory of levels Á Formal foundation and relational structure of categorical ontology and emergent complexity theories Á Ontological essence and Universal properties of items Á Mathematical categories, Groupoids, Locally Lie groupoids, Groupoid Atlas, Stacks, Fibred categories Á Relational biology principles Á Higher homotopy-General Van Kampen Theorems (HHvKT) and Non-Abelian algebraic topology (NAAT) Á Non-commutativity of diagrams and non-Abelian theories-Non-Abelian categorical ontology Á Non-commutative topological...

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“…This functor has a left adjoint ι * : (crossed modules over P ) → (crossed modules over Q) , which gives our induced crossed module. This construction can be described as a "change of base" [7]. To compute a colimit colim i (µ i : M i → P i ), one forms the group P = colim i P i , and uses the canonical morphisms φ i : P i → P to form the family of induced crossed Pmodules ((µ i ) * : (φ i ) * M i → P ).…”

Section: Computing Colimits Of Crossed Modulesmentioning

confidence: 99%