Papers available

[Thorsten Palm <palm@pascal.math.yorku.ca>]

Dear categorists,

My Ph.D. dissertation, entitled `Dendrotopic Sets for Weak
Infinity-Categories', and a preliminary version of my paper
`Dendrotopic Sets' can now be downloaded from the web-page

   www.math.yorku.ca/Who/Grads/palm .

The content of the two works is summarized below, where they are
referred to as [A] and [B] respectively.

Best regards

Thorsten Palm


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In his unpublished paper [1], Makkai defined a notion of _weak
infinity-category_ (under the name `multitopic omega-category'). The
underlying geometric structures, called _multitopic sets_, are
described in the three-part paper [2]. Makkai takes a weak
infinity-category to be a multitopic set with the mere property that
_compositions_ exist, which are defined as equivalences of certain
_coslice_ objects.

[A] gives a definition of weak infinity-categories equivalent to
Makkai's. While this definition falls into the same two stages, it is
considerably shorter and more elementary in each of them. The
description of the underlying geometric structures, called
_dendrotopic sets_ here, is done purely combinatorially (whereas [2]
uses the algebraic machinery of multicategories). A coslice object
comes in two guises, in both of which it is a dendrotopic set with
mild extra structure in the form of a dendrotopic map (whereas in [1]
it is a model for a new signature for Makkai's first-order logic with
dependent sorts).

[A] also presents an alternative method of introducing composition to
a dendrotopic set. Here one has to impose the extra structure of a
_universality system_: a subset that behaves in such a way that each
member can be thought of as a universal arrow. The main theorem of [A]
states that a dendrotopic set containing a universality system is a
Makkai weak infinity-category.

The concept of a dendrotopic set is defined at a more leisurely
pace in [B], where the equivalence to the concept of a multitopic
set is, indirectly, established.

References:

[1] M. Makkai: `The multitopic omega-category of all multitopic
omega-categories'; mystic.biomed.mcgill.ca/Makkai

[2] C. Hermida, M. Makkai, A. J. Power: `On weak higher dimensional
categories'; Journal of Pure and Applied Algebra. Part 1: 154 (2000),
pp. 221-246; part 2: 157 (2001), pp. 247-277; part 3: 166 (2002), pp.
83-104