# Category Theory & Homotopy Theory

#### 05.01 GRATUS, J. & PORTER, T.

(combined with 05.02 to form 06.09)

### A geometry of information, I: Nerves, posets and differential forms

#### Abstract:

The main theme of this workshop is Spatial Representation: Continuous vs. Discrete. Spatial representation has two contrasting but interacting aspects (i) representation of spaces' and (ii) representation by spaces. In this paper we will examine two aspects that are common to both interpretations of the theme, namely nerve constructions and refinement. Representations change, data changes, spaces change. We will examine the possibility of a differential geometry of spatial representations of both types, and in the sequel give an algebra of differential forms that has the potential to handle the dynamical aspect of such a geometry. We will discuss briefly a conjectured class of spaces, generalising the Cantor set which would seem ideal as a test-bed for the set of tools we are developing.

http://drops.dagstuhl.de/portals/04351/

#### Presented at:

This paper is based on a talk at Schloss Dagstuhl (International Conference and Research Center for Computer Science) as part of the seminar Spatial Representation: Discrete vs. Continuous Computational Models (August 2004).
This paper, and the corresponding part II, evolved from the talks entitled Fractafolds, their geometry and topology: a test bed for spatial representation given at the Seminar, as a result of the insights gleaned by the authors during the excellent sessions of the week.
This research forms part of a project : \emph{Fractafolds, their geometry and topology}, partially supported by a grant from the Leverhulme Trust. This help is gratefully acknowledged.

#### 05.02 GRATUS, J. & PORTER, T.

(combined with 05.01 to form: 06.09)

### A geometry of information, II: Sorkin models, and biextensional collapses

#### Abstract:

In this second part of our contribution to the workshop, we look in more detail at the Sorkin model, its relationship to constructions in Chu space theory, and then compare it with the Nerve constructions given in the first part.

This research forms part of a project: Fractafolds, their geometry and topology, partially supported by a grant from the Leverhulme Trust.
This help is gratefully acknowledged.

http://drops.dagstuhl.de/portals/04351/

### Ordinal subdivision and special pasting in quasicategories

#### Abstract:

Quasicategories are simplicial sets with properties generalising those of the nerve of a category. They model weak infinity-categories. Using a combinatorially defined ordinal subdivision, we examine composition rules for certain special pasting diagrams in quasicategories. The subdivision is of combinatorial interest in its own right and is linked with various combinatorial constructions.

05_03.pdf

#### Published in:

Advances in Math. 217 (208) 489-518.

### Category Theory: an abstract setting for analogy and comparison

#### Abstract:

Comparison' and Analogy' are fundamental aspects of knowledge acquisition. We argue that one of the reasons for the usefulness and importance of Category Theory is that it gives an abstract mathematical setting for analogy and comparison, allowing an analysis of the process of abstracting and relating new concepts. This setting is one of the most important routes for the application of Mathematics to scientific problems. We explore the consequences of this through some examples and thought experiments.

05_10.pdf

#### Published in:

What is Category Theory? Advanced Studies in Mathematics and Logic,
Polimetrica Publisher, Italy, (2006) 257-274.

#### 05.11 PORTER, T. & TURAEV, V.

(revised as 07.08)

### Formal Homotopy Quantum Field Theories, II : Simplicial Formal Maps

#### Abstract:

Simplicial formal maps were introduced in the first paper of this series as a tool for studying Homotopy Quantum Field Theories with background a general homotopy 2-type. Here we continue their study, showing how a natural generalisation can handle much more general backgrounds. The question of the geometric interpretation of these formal maps is partially answered in terms of combinatorial bundles. This suggests new interpretations of HQFTs.

05_12.pdf

#### Published in:

Contemporary Math. 431: Categories in Algebra, Geometry and Mathematical Physics,
Conference and workshop in honor of Ross Street's 60th birthday,
eds. A.Davydon, M.Batanin, M.Johnson, S.Lack and A.Neeman,
AMS (2007) 375-403.

### Complex nonlinear biodynamics in categories

#### Abstract:

A categorical, higher dimensional algebra and generalized topos framework for \L ukasiewicz--Moisil Algebraic--Logic models of nonlinear dynamics in complex functional genomes and cell interactomes is proposed. \L ukasiewicz--Moisil Algebraic--Logic models of neural, genetic and neoplastic cell networks, as well as signaling pathways in cells are formulated in terms of nonlinear dynamic systems with n--state components that allow for the generalization of previous logical models of both genetic activities and neural networks. An algebraic formulation of variable `next-state functions' is extended to a \L ukasiewicz--Moisil Topos with an n--valued \L ukasiewicz--Moisil Algebraic Logic subobject classifier description that represents non-random and nonlinear network activities as well as their transformations in developmental processes and carcinogenesis. The unification of the theories of organismic sets, molecular sets and Robert Rosen's (M,R)--systems is also considered here in terms of natural transformations of organismal structures which generate higher dimensional algebras based on consistent axioms, thus avoiding well known logical paradoxes occuring with sets. Quantum bionetworks, such as quantum neural nets and quantum genetic networks, are also discussed and their underlying, noncommutative quantum logics are considered in the context of an emerging Quantum Relational Biology.