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Declaration The work of this thesis has been carried out by the candidate and contains the results of his own investigations. The work has not been already accepted in substance for any degree, and is not being concurrently submitted in candidature for any degree. All sources of information have been acknowledged in the text. Director of Studies Candidate… (More)

We give a small functorial algebraic model for the 2-stage Postnikov section of the K-theory spectrum of a Waldhausen category and use our presentation to describe the multiplicative structure with respect to biexact functors.

For any 1-reduced simplicial set K we define a canonical, coassociative coproduct on ΩC(K), the cobar construction applied to the normalized, integral chains on K, such that any canonical quasi-isomorphism of chain algebras from ΩC(K) to the normalized, integral chains on GK, the loop group of K, is a coalgebra map up to strong homotopy. Our proof relies on… (More)

- KATHRYN HESS, ANDREW TONKS
- 2009

We prove that for any 1-reduced simplicial set X, Adams’ cobar construction ΩCX on the normalised chain complex of X is naturally a strong deformation retract of the normalised chains CGX on the Kan loop group GX. In order to prove this result, we extend the definition of the cobar construction and actually obtain the existence of such a strong deformation… (More)

- Ronald Brown, Andrew Tonks
- J. Symb. Comput.
- 1994

Work on calculations with simplicial and cubical groups in AXIOM was carried out using loan equipment and software from IBM UK and guidance from L A Lambe. We report on the results of this work, and present the AXIOM code written by the second author during this period. This includes an implementation of the monoids which model cubes and simplices, together… (More)

We give a simple representation of all elements in K1 of a Waldhausen category and prove relations between these representatives which hold in K1.

- IMMA GÁLVEZ-CARRILLO, ANDREW TONKS
- 2011

We construct spectral sequences in the framework of Baues-Wirsching cohomology and homology for functors between small categories and analyze particular cases including Grothendieck fibrations. We also give applications to more classical cohomology and homology theories includingHochschild-Mitchell cohomology and those studied before by Watts, Roos, Quillen… (More)

- ANDREW TONKS
- 2006

We give a simple representation of all elements in K1 of a Waldhausen category and prove relations between these representatives which hold in K1.

- Kathryn Hess, Paul-Eugène Parent, Andrew Tonks, Krzysztof Worytkiewicz
- Electr. Notes Theor. Comput. Sci.
- 2004

We exhibit a model structure on 2-Cat, obtained by transfer from sSet across the adjunction C2 ◦ Sd a Ex ◦ N2. A certain class of homotopies in this model structure turns out to be in 1-to-1 correspondence with strong simulations among labeled transitions systems, formalising the geometric intuition of simulations as deformations. The correspondence still… (More)

- IMMA GÁLVEZ–CARRILLO, ANDREW TONKS
- 2016

We consider three a priori totally different setups for Hopf algebras from number theory, mathematical physics and algebraic topology. These are the Hopf algebras of Goncharov for multiple zeta values, that of Connes–Kreimer for renormalization, and a Hopf algebra constructed by Baues to study double loop spaces. We show that these examples can be… (More)

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