Coalgebraic cell complexes

• Main Author: Athorne, Thomas
• Other Authors: Gurski, Nick
• Published: University of Sheffield 2014
• Subjects: 510
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.605504

The main aim of this thesis is the definition of CellCx, the category of relative cell complexes generated from a given small category of generating maps. We establish sufficient conditions for this definition to work and give us a category that we can prove is equivalent to the left map category for the algebraic weak factorisation system (AWFS for short) generated by Garner's small object argument applied to the same generating maps. These sufficient conditions take the form of a special kind of nerve functor on the underlying category, and some properties the generating maps are required to satisfy with respect to that nerve functor. In particular, they isolate a special class of inclusion maps which we call typical inclusions; every cell complex will have an underlying map which is a typical inclusion. We also give a survey of the current understanding of the semantic structure (left and right maps) that an AWFS determines. This includes a theorem that left and right map structures are always detemined entirely by their lifting structures; this establishes that any AWFS is a fixed point for an adjunction which can be viewed as a higher order Galois connection. Additionally, we prove that computads for globular operads are a special case of cell complexes. This is a result that has been present in the folklore for a while, but could not be fully formalised without the definition of cell complexes we establish here. Finally we consider the question of when cell complexes can themselves be expressed as presheaves; we prove a number of results for different examples and find some positive and some negative cases of this property. At the end we will connect this to the work of Batanin (see [Bat02]) in which he establishes conditions for this property to hold for computads.