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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 ..."
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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.
An introduction to ncategories
 Proc. 7th Conf. Category Theory and Computer Science
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On comparing definitions of "weak n–category"
, 2001
"... 1. My approach is "foundational". On the one hand, I am motivated by the problem of the foundations of mathematics (an unsolved problem as far as I am concerned). On the other hand and this is more relevant here, I start "from scratch", and thus what I say can be understood w ..."
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1. My approach is "foundational". On the one hand, I am motivated by the problem of the foundations of mathematics (an unsolved problem as far as I am concerned). On the other hand and this is more relevant here, I start "from scratch", and thus what I say can be understood with little technical knowledge. I only assume a modest amount of category theory as background. I will talk informally about technical matters that are written down formally elsewhere, where they can be studied further. [The text in square brackets [] is either some technical explanation, or a digression.] 2. Terminology First, some terminological conventions. I will use the word "category " in its most general sense: weak ωcategory. This is completely inclusive: all sorts of "categories " are categories ow. here are two extensions of the original meaning: "weak", and "omegadimensional". Weak " signifies an indeterminate notion; there are several different specific versions of weak ategory. It can also be used as a vague notion, when one is merely looking at what one would