Results 1 
4 of
4
Higher dimensional algebra III: ncategories and the algebra of opetopes
, 1997
"... We give a definition of weak ncategories based on the theory of operads. We work with operads having an arbitrary set S of types, or ‘Soperads’, and given such an operad O, we denote its set of operations by elt(O). Then for any Soperad O there is an elt(O)operad O + whose algebras are Soperads ..."
Abstract

Cited by 109 (6 self)
 Add to MetaCart
We give a definition of weak ncategories based on the theory of operads. We work with operads having an arbitrary set S of types, or ‘Soperads’, and given such an operad O, we denote its set of operations by elt(O). Then for any Soperad O there is an elt(O)operad O + whose algebras are Soperads over O. Letting I be the initial operad with a oneelement set of types, and defining I 0+ = I, I (i+1)+ = (I i+) +, we call the operations of I (n−1)+ the ‘ndimensional opetopes’. Opetopes form a category, and presheaves on this category are called ‘opetopic sets’. A weak ncategory is defined as an opetopic set with certain properties, in a manner reminiscent of Street’s simplicial approach to weak ωcategories. In a similar manner, starting from an arbitrary operad O instead of I, we define ‘ncoherent Oalgebras’, which are n times categorified analogs of algebras of O. Examples include ‘monoidal ncategories’, ‘stable ncategories’, ‘virtual nfunctors ’ and ‘representable nprestacks’. We also describe how ncoherent Oalgebra objects may be defined in any (n + 1)coherent Oalgebra.
Simplicial Degrees Of Functors
"... this paper is to show that if G is a simplicial group of finite length, then H n G also has finite length. Here the length of a simplicial group means the length of the corresponding Moore normalization and H n G is a simplicial abelian group given by [k] 7! H n G k . A similar fact is true if we ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
(Show Context)
this paper is to show that if G is a simplicial group of finite length, then H n G also has finite length. Here the length of a simplicial group means the length of the corresponding Moore normalization and H n G is a simplicial abelian group given by [k] 7! H n G k . A similar fact is true if we replace G by a simplicial ring and we take the algebraic Kfunctors instead of group homology. The origin of such results goes back to the classical paper of Dold and Puppe (see Hilfsatz 4.23 of [DP]), where the following was proved: let
Projectives Are Free For Nilpotent Varieties
"... this paper was being written. Author partially supported by Grant No GM1115 of the U.S. CRDF. ..."
Abstract
 Add to MetaCart
(Show Context)
this paper was being written. Author partially supported by Grant No GM1115 of the U.S. CRDF.