3,180 research outputs found
The group of parenthesized braids
We investigate a group that includes Artin's braid group
and Thompson's group . The elements of are
represented by braids diagrams in which the distances between the strands are
not uniform and, besides the usual crossing generators, new rescaling operators
shrink or strech the distances between the strands. We prove that
is a group of fractions, that it is orderable, admits a non-trivial
self-distributive structure, i.e., one involving the law ,
embeds in the mapping class group of a sphere with a Cantor set of punctures,
and that Artin's representation of into the automorphisms of a free
group extends to
Higher-Dimensional Algebra III: n-Categories and the Algebra of Opetopes
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 a category, and presheaves on this category are called
`opetopic sets'. A weak n-category is defined as an opetopic set with certain
properties, in a manner reminiscent of Street's simplicial approach to weak
omega-categories. Similarly, starting from an arbitrary operad O instead of I,
we define `n-coherent O-algebras', which are n times categorified analogs of
algebras of O. Examples include `monoidal n-categories', `stable n-categories',
`virtual n-functors' and `representable n-prestacks'. We also describe how
n-coherent O-algebra objects may be defined in any (n+1)-coherent O-algebra.Comment: 59 pages LaTex, uses diagram.sty and auxdefs.sty macros, one
encapsulated Postscript figure, also available as a compressed Postscript
file at http://math.ucr.edu/home/baez/op.ps.Z or
ftp://math.ucr.edu/pub/baez/op.ps.
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