Tim Porter Tim Porter

Welcome to the personal area of Tim Porter within the nLab.

You can also find me on arXiv, ORCiD, Publons and Google scholar

Below you will find some indication of my research interests, past and present, with some indication of my contribution to the overall picture, together with lists of relevant papers, linked to copies where there are such online.

For convenience here is a link to a list of my papers together with numbers of times they have been cited (including some self citations, of course).

I also intend to put up here some sketches of some projects that I would love others to ‘help with’ or which are ‘in progress’, i.e., a snapshot of possible future work. Some of these projects are ‘work in progress’, incompleted papers, research grant applications that were unfunded because they were too much in advance of the field at the time of submission (my story and I’m sticking to it) and so on.

Some of these I mean to develop here in public, others may more appropriately be done via e-mail or on slightly restricted pages. We will see how it goes.

I may also put links to project pages and some preprints here, including a more up to date version of the Menagerie.

Finally I may put drafts for n-Lab entries that I hope, one day, to add to the general list.

Main Research Areas (past and present):

Simplicial foundations of homotopy coherence:

The problem of establishing simplicial foundations for homotopy coherence is discussed in more detail in a separate entry.

References

• (with J.-M. Cordier), Vogt’s Theorem on Categories of Homotopy Coherent Diagrams, Math. Proc. Camb. Phil. Soc. 100 (1986) pp. 65-90.

• (with J.-M. Cordier), Maps between homotopy coherent diagrams, Top. and its Applications, 28 (1988) 255-275.

• (with J.-M. Cordier), Fibrant diagrams, rectifications and a construction of Loday, J.Pure Appl. Algebra, 67 (1990) 111-124.

• (with J.-M. Cordier), Categorical Aspects of Equivariant Homotopy, Proc. ECCT, Applied Cat.Structures.4 (1996) 195-212.

• (with J.-M. Cordier), Homotopy Coherent Category Theory, Trans. Amer. Math. Soc. 349 (1997) 1-54.

• (with R. Brown, M. Golasinski and A. Tonks) Spaces of maps into classifying Spaces for equivariant crossed complexes, Indagationes Mathematica, 8 (1997) 157-172.

• (with R. Brown, M. Golasinski and A. Tonks), Spaces of maps into Classifying Spaces for equivariant crossed complexes, II: the general topological case, K-Theory, 23 (2001) 129-155.

• (with P.J. Ehlers) Joins for (augmented) simplicial sets, Jour. Pure Applied Algebra, 145 (2000) 37-44.

• (with P.J.Ehlers) Ordinal subdivision and special pasting in quasicategories, Advances in Math. 217 (2007), No 2. pp 489-518

• Abstract Homotopy Theory, the interaction of category theory and homotopy theory, (survey article, updated version of lecture notes from summer school course at Bressanone), Cubo, 5 (2003)115-165, 2003)

• (with K.H. Kamps), Abstract Homotopy and Simple Homotopy Theory, World Scientific, 462pp (ISBN 981-02-1602-5) June 1997

Algebraic models for homotopy $n$-types

and their applications in homological and homotopical algebra, and in the allied area of algebraic homotopy:

The finding of algebraic models for homotopy n-types is discussed in more detail in a separate entry.

References

• $n$-Types of Simplicial Groups and Crossed $n$-cubes, Topology, 32, (1993) 5-24.

• (with G. Donadze, and N. Inassaridze) N-fold Cech derived functors and generalised Hopf type formulas, K-Theory 35 (2005) 341 – 373.

• (with G.J. Ellis) Free and Projective Crossed Modules and the Second Homology Group of a Group, Jour. Pure Applied Alg., 40 (1986) pp. 27-32.

• (with P.J. Ehlers) Varieties of simplicial groupoids, I, Crossed complexes, Jour. Pure Applied Algebra, 120 (1997) 221 – 233. Erratum: Jour. Pure Applied Algebra, 134 (1999) 221-233.

• (with K.H.Kamps) 2-Groupoid Enrichments in Homotopy Theory and Algebra, K-Theory, 25 (2002) 373-409.

• (with R. Brown, and M. Bullejos), Crossed complexes, free crossed resolutions and graph products of groups, in Recent Advances in Group Theory and Low-Dimensional Topology , Research and Exposition in Mathematics vol 27, Helderman Verlag, 2003, pp. 11-26

• (with R. Brown, E. Moore and C.D. Wensley), Crossed complexes, and free crossed resolutions for amalgamated sums and HNN-extensions of groups, Georgian Math. J., Special issue for the 70th birthday of H. Inissaridze, 9 (2002) 623 - 644.89.

• (with A.Mutlu) Applications of Peiffer pairings in the Moore complex of a simplicial group,Theory and Applications of Categories, 4, No. 7, (1998) 148-173.

• (with A.Mutlu) Freeness conditions for 2-crossed modules and complexes,Theory and Applications of Categories, 4, No. 8, (1998) 174-194.

• (with A.Mutlu) Free crossed resolutions from simplicial resolutions with given $CW$-basis, Cahiers Top. Géom. Diff. Catégoriques, 50 (1999) 261-283.

• (with A.Mutlu) Freeness conditions for crossed squares and squared complexes, K-Theory, 20, (2000) 345 - 368.

• (with A.Mutlu) Iterated Peiffer pairings in the Moore complex of a simplicial group, Applied Categorical Structures, 9 (2001) 111-130.

• (with P. Carrasco) Coproduct of 2-crossed modules. Applications to a definition of a tensor product for 2-crossed complexes, Collectanea Mathematica, (DOI) 10.1007/s13348-015-0156-9.

Simplicial and categorical methods in TQFTs and HQFTs:

These are discussed in more detail in a separate entry.

References

• (with V. Turaev) Formal Homotopy Quantum Field Theories, I: Formal Maps and Crossed C-algebras, Journal of Homotopy and Related Structures 3(1), 2008, 113 - 159. (available also at ArXiv: math.QA/0512032

• Formal Homotopy Quantum Field Theories II: Simplicial Formal Maps, in Cont. Math. 431, p. 375 - 404 (Streetfest volume: Categories in Algebra, Geometry and Mathematical Physics - edited by A. Davydov, M. Batanin, and M. Johnson, S. Lack, and A. Neeman)(available also at ArXiv: math.QA/0512034).

• Interpretations of Yetter’s notion of G-coloring : simplicial fibre bundles and non-abelian cohomology, J. Knot Theory and its Ramifications 5 (1996) 687-720.

• Topological Quantum Field Theories from Homotopy n-types, J. London Math. Soc. (2) 58 (1998) 723-732.

• (with J. Faria Martins) On Yetter’s Invariant and an Extension of the Dijkgraaf-Witten Invariant to Categorical Groups, Theory and Applications of Categories,18, 2007, No. 4, pp 118-150; (available also at ArXiv: math.QA/0608484)

Strong shape, coherent prohomotopy, and the stability problem:

Strong shape, coherent prohomotopy, and the stability problem is discussed in a separate entry.

References

• Stability Results for Topological Spaces, Math. Zeit. 150, 1974, pp. 1-21.

• Abstract homotopy theory in procategories, Cahiers Top. Géom. Diff., 17, 1976, pp. 113-124.

• Coherent prohomotopical algebra, Cahiers Top.Géom.Diff., 18, (1978) pp. 139-179.

• Coherent prohomotopy theory, Cahiers Top. Géom. Diff., 19, (1978) pp. 3-46

• Cech and Steenrod homotopy and the Quigley exact couple in strong shape and proper homotopy theory, Jour. Pure Applied Alg., 24 (1982) pp. 303-312.

• Reduced Powers, Ultra Powers and Exactness of Limits, Jour. Pure Applied Alg., 26 (1982) pp. 325-330.

Resources

I have various survey articles and other things that could be useful. I will put them (or links to them) here in case they are:

Abstract Homotopy Theory: The Interaction of Category Theory and Homotopy theory

This article is an expanded version of notes for a series of lectures given at the Corso estivo Categorie e Topologia organised by the Gruppo Nazionale di Topologia del M.U.R.S.T. in Bressanone, 2 - 6 September 1991. Those notes have been partially brought up to date by the addition of new references and a summary of some of what has happen in the area in the last 20 years (inadequate at present!)

A copy can be found here.

$\mathcal{S}$-categories, $\mathcal{S}$-groupoids, Segal categories and quasicategories

The notes were prepared for a series of talks that I gave in Hagen in late June and early July 2003, and, with some changes, in the University of La Laguna, the Canary Islands, in September, 2003. They assume the audience knows some abstract homotopy theory and as Heiner Kamps was in the audience in Hagen, it is safe to assume that the notes assume a reasonable knowledge of our book, or any equivalent text if one can be found!

A copy can be found here.

Crossed Menagerie:

The crossed menagerie is intended as a set of notes outlining an approach to non-Abelian cohomology, stacks, etc., and Grothendieck’s conjectured extension of Galois-Poincaré theory. The title refers to the array of strange beasties that occur as generalisations of crossed modules. (The present version is 1006 pages long, but the above links to a much shorter 11 chapter 444 page version.)

Several ‘cut down’ versions have been prepared various workshops and as they are less long may be of use. For instance in February, 2011, there was a Workshop and School on Higher Gauge Theory, TQFT and Quantum Gravity in the IST, Lisbon, (7-13 February, 2011). A set of notes was prepared using the Menagerie as a ‘mine’ from which to extract the sections relevant to the theme of the workshop, which were then edited slightly. The result was entitled: Homotopy Quantum Field Theories meets the Crossed Menagerie: an introduction to HQFTs and their relationship with things simplicial and with lots of crossed gadgetry. A copy can be found here, whilst a table of contents and some more introductory material if HQFTs meet the Crossed Menagerie, which entry contains another copy of the link.

Another was for the LI2012 week on Algebra and computation (27 February – 2 March, 2012). A copy can be obtained by contacting me by e-mail or here.

Projects

Profinite Algebraic Homotopy

This is a project to extend the results of crossed homological algebra and algebraic homotopy to the profinite setting. There is a link to the first seven chapters of a draft ‘monograph’ that grew out of the thesis of a student (Fahmi Korkes) in the early 1980s. I have added a lot of extra material, and I hope this will form a book in the not too distant future. Anyone willing to have a look at the longer version (at present 1004 pages, and not yet finished!) as a proof reader and to give me comments, please e-mail me (at t dot porter dot maths AT gmail DOT com). (As it may be published, for the moment, I do not really want to make the full version freely available over the net.)

Generating and Extending Topological Quantum Field Theories:

The GTQFT project

This is a slightly edited version of an (unsuccessful) research proposal that I submitted in 2002. An earlier version had been refused funding a few years earlier. I would be interested in reviewing this project in the light of the recent results on the cobordism conjecture and the start of a classification of TQFTs.

In the report at the time of the refusal, I was told in no uncertain terms that the ideas in the proposal were rubbish, and that I was putting them forward just because I thought they could be done! Heigh-ho! That was the view of one referee. He was very wrong, but probably pushed another nail into the coffin of the mathematics department at Bangor. Like democracy, Peer review is the worst of all possible systems, except for all the others!

PS. I did completely know what I was proposing, and reactions of other referees were very positive. Shortly afterwards, Turaev put forward his Homotopy Quantum Field theories which were, to some extent, in a similar vein at about the same time as the earlier proposal. Note that many parts of the proposal have still not been done.

Some of what I planned to do then is now being put in the Menagerie. Some of that has been extracted as a cut-down version for use by participants at the Workshop and School on Higher Gauge Theory, TQFT and Quantum Gravity Lisbon, 10-13 February, 2011. I have a version of those notes here. Some things that I wanted to do, I still do not know how to do.

Quinn’s finite total homotopy TQFT

(Joint with Joao Faria Martins)

The main purpose of this project is to construct once-extended TQFTs derived from homotopy finite spaces $B$, and show how they can be calculated for the case when $B$ is the classifying space of a homotopy finite strict $\infty$-groupoid /$\omega$-groupoid, which we can take as being represented by a crossed complex.

Characterisation of homotopy types of $n$-crossed complexes.

Although the result giving a characterisation of those homotopy types representable by crossed complexes has appeared in several places in the literature, these have usually been slightly submerged in a mass of other results and so are difficult to find. The partial result of this is that the theorem itself is neglected in standard approaches to the subject. A stand-alone account of such a characterisation, however, would seem to be of interest, especially if it used as elementary an approach as was feasible, and so its seems a good idea to collect up some of the parts of the overall theory, which should lead to such a simple description and the initial aim of this project is to provide such an account which, hopefully, will be accessible to graduate students.

Beyond that is the challenge of extending that characterisation to those homotopy types representable by $n$-crossed complexes. Fairly little has been written / published about the various types of objects called $n$-crossed complexes, but the idea is simple. Crossed complexes can be thought of as crossed modules with a chain complex as a ‘tail’. Similarly a 2-crossed complex is a 2-crossed module with such a tail, and an $n$-crossed complex is any one of a set of equivalent forms of generalisation of this to higher dimensions.

Generalisations of the theory of Crowell’s derived modules.

This is partially for use in the previous project. Crowell’s derived modules were developed for use in knot theory, and relate to well known constructions in homological algebra. They have also occured in the area of arithmetic topology.

Higher syzygies, algebraic K-theory and the Kapranov-Saito conjectures

Work on homotopical syzygies in presentations of the Steinberg group $St(A)$ of a ring $A$ has indicated links with the higher $K$-groups of $A$. This work of Kapranov and Saito contains several interesting conjectures that link such syzygies with labelled polytopes, including the Stasheff polytopes (associahedra). The proposal was to approach some of these conjectures using a new combination of ideas and techniques from combinatorial group theory, the construction of resolutions, ideas of ‘higher generation by subgroups’, the methods of Volodin model for higher K-theory and the global actions of Tony Bak.

The project had the following objectives:

1. To prove the Kapranov-Saito conjecture that the space of $A$-labelled hieroglyphs is homotopically equivalent to the Volodin space of $A$, and to study non-stable analogues of this;

2. To analyse and generalise the methods of Stefan Holz, extending them to give detailed information on higher syzygies in families of groups such as classical linear groups, braid groups, etc.

3. To link homotopical and homological syzygies so as to extend the isomorphism $H_2(St(A), \mathbb{Z}) \cong K_3(A)$ to link higher $K$-groups with higher homology.

A separate line is to investigate the way in which this may be of interest for workers in rewriting theory.

Topological Data Analysis:

Topological Data Analysis (TDA) has recently emerged from the general area of Computational Topology. In this discussion, I try to examine not only TDA itself, but to put it in a historical context by identifying precursor theories both in algebraic topology and, surprisingly, within theoretical physics. The reason for doing this is not just to be scholarly, but is also to, hopefully, suggest some additions to the toolkit of TDA, which may widen its applicability. This, in turn, raises new questions both within TDA itself, and also within the related mathematical area of Shape Theory.

(The link gives a page extracted from a longer document. If you want a copy of that just ask.)

Multi-Agent Systems: Mathematical Models for Multi-Agent Systems.

There are several aspects that need to be looked at: algebraic, logical, coalgebraic and geometric/combinatorial. One project involving a categorical approach is: Categorical / coalgebraic perspectives on Multiagent Systems. This concentrates on the logics that result from MASs. The structures are combinatorial or more exactly coalgebraic and are closely related to those underlying global actions (see the mathematical research topics). The categorical structure of these objects is fascinating, especially as several known constructions from epistemic logic would seem to be analogues of well known homotopy theoretic constructions. Specific aims include (a) the construction of mapping space objects (cartesian closedness); (b) the study of interactions between communicating MASs with different classes of agents. This might possibly use the theory of Institutions (Burstall and Goguen) to account for the interaction between the models.

Grothendieck’s letters

Whilst Grothendieck was writing Pursuing Stacks, there was a correspondence, initially between himself and Ronnie Brown, and later for a short period, with myself. It is now nearly forty years since those letters were written. They include a discussion of the possible role of Kan complexes as a homotopy version of Grothendieck’s own intuition of $\infty$-groupoid. This became one aspect of what is called the Homotopy Hypothesis. This is discussed here in the next section.

The majority of these letters should shortly be published in a volume, edited by Georeges Maltsiniotis.

Spaces as infinity-groupoids.

• This is a draft chapter of the first part of a two volume collection, New Spaces for Mathematics and Physics. The accompanying talk is here in video form.