261 research outputs found
Weak n-categories: comparing opetopic foundations
We define the category of tidy symmetric multicategories. We construct for
each tidy symmetric multicategory Q a cartesian monad (E_Q,T_Q) and extend this
assignation to a functor. We exhibit a relationship between the slice
construction on symmetric multicategories, and the `free operad' monad
construction on suitable monads. We use this to give an explicit description of
the relationship between Baez-Dolan and Leinster opetopes.Comment: 31 page
The category of opetopes and the category of opetopic sets
We give an explicit construction of the category Opetope of opetopes. We
prove that the category of opetopic sets is equivalent to the category of
presheaves over Opetope.Comment: 23 page
Distributive laws for Lawvere theories
Distributive laws give a way of combining two algebraic structures expressed
as monads; in this paper we propose a theory of distributive laws for combining
algebraic structures expressed as Lawvere theories. We propose four approaches,
involving profunctors, monoidal profunctors, an extension of the free
finite-product category 2-monad from Cat to Prof, and factorisation systems
respectively. We exhibit comparison functors between CAT and each of these new
frameworks to show that the distributive laws between the Lawvere theories
correspond in a suitable way to distributive laws between their associated
finitary monads. The different but equivalent formulations then provide,
between them, a framework conducive to generalisation, but also an explicit
description of the composite theories arising from distributive laws.Comment: 30 pages, presented at CT2011, lightly edited 2019 for publication in
Compositionalit
Weak n-categories: opetopic and multitopic foundations
We generalise the concepts introduced by Baez and Dolan to define opetopes
constructed from symmetric operads with a category, rather than a set, of
objects. We describe the category of 1-level generalised multicategories, a
special case of the concept introduced by Hermida, Makkai and Power, and
exhibit a full embedding of this category in the category of symmetric operads
with a category of objects. As an analogy to the Baez-Dolan slice construction,
we exhibit a certain multicategory of function replacement as a slice
construction in the multitopic setting, and use it to construct multitopes. We
give an explicit description of the relationship between opetopes and
multitopes.Comment: 41 page
A note on the Penon definition of -category
We show that doubly degenerate Penon tricategories give symmetric rather than
braided monoidal categories. We prove that Penon tricategories cannot give all
tricategories, but we show that a slightly modified version of the definition
rectifies the situation. We give the modified definition, using non-reflexive
rather than reflexive globular sets, and show that the problem with doubly
degenerate tricategories does not arise.Comment: 14 pages, to appear in Cahiers de Topologie et Geometrie
Differentielle Categorique
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