Homotopy and Higher Categories
Paris-Diderot PhilMath Seminar
Tuesday March 1st, 2011 : Homotopy and Higher Categories
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Université Paris Diderot - site Rive Gauche, bâtiment Condorcet (4, rue Elsa Morante, 75013, Paris), salle Mondrian (646A).
9h30-11h, Ralf Krömer (Universität Siegen) : "A history of the concept of fundamental groupoid"
11h15-12h45, David Corfield (University of Kent) : "Categorification and Logic"
14h30-16h, François Métayer (Paris-Diderot, PPS) : "Homotopy and rewriting"
See abstracts below.
Ralf Krömer, "A history of the concept of fundamental groupoid" : In the first part of the talk, we will investigate early conceptions of homotopy (and more generally, topology) and their role in the treatment of problems from analysis, starting with the work of Riemann and Jordan. This culminates in Poincaré ?s series of papers on "Analysis situs" where he introduced, along with many other topological tools, the fundamental group. These matters are well known and well described in the literature, but a recapitulation is prerequisite for the subsequent parts of the talk. In the second part, we take a look at the history of partial composition and the groupoid concept, starting with algebraic work by Heinrich Brandt on quadratic forms and ideals of algebras. Again focusing on the context of topology, notably on the concept of fundamental groupoid, we will examine the claim that the history of this latter concept should reasonably be divided in two periods in the first of which (from Poincaré via Reidemeister up to Crowell and Fox ?s book on knot theory) it didn ?t play a central role, and its uses were developed more or less independently of category theory, whereas things became different with Ronald Brown ?s stressing of the utility of the groupoid concept for algebraic topology. In the last part of the talk, we analyze the epistemological significance of the technical concept of homotopy for the work with the fundamental groupoid in the historical events described before, especially examining whether Wittgenstein ?s dictum recalled by the organizers of the seminar is significant in this case.
David Corfield, "Categorification and Logic" : Categorification designates the process of finding higher level constructions whose ?shadows ? are familiar constructions. The categorified construction generally contains information which has been lost in its shadow. As we move up the n-category ladder for increasing n, an increasing elaborate notion of sameness emerges : equality is replaced by isomorphism, isomorphism by equivalence, and so on. Now, categories of a certain kind will support a logic of an associated kind, so it is natural to expect that as we move up the n-category ladder, that we will find categorified versions of logic. In this talk I shall examine what sense we can make of a categorified logic.
François Métayer, "Homotopy and rewriting" : A result of 1987 by Craig Squier relates the topology of a monoid to the properties of its presentations by rewriting systems. Precisely, if a monoid can be presented by a finite, confluent and terminating system, then its third homology group is of finite type. I will show how the space of computations attached to such rewriting systems supports a structure of omega-category, and revisit Squier ?s theorem from this omega-categorical point of view. This approach is based on the construction of a Quillen model structure on strict higher-categories, a recent joint work with Yves Lafont and Krzysztof Worytkiewicz.
Presentation of the session : The algebraic formalization of homotopic notions, since the work of Daniel Quillen, has given rise to a huge amount of new category-theoretic constructions and concepts. In particular, model categories have been introduced as an abstract framework for homotopy theory and have recently been applied to type theory. Besides, higher categories have been introduced as a way to model higher-order homotopy in "iterated loop spaces" : Identity is replaced with homotopy, identity of homotopies is replaced with homotopy between homotopies, and so on. Since then, the algebraic expression of that idea has turned out to be very fruitful in many contexts outside the strict domain of topology, for example in rewriting theory, and has become a research topic in its own right. Category theory has clearly been instrumental in supporting homotopy theory and in articulating higher-order homotopy. The session will be devoted to that facet of category theory. Wittgenstein says in the Tractatus (5.303) that an identity statement (in particular, in an arithmetic context) is always meaningless ("Roughly speaking, to say of two things that they are identical is nonsense, and to say of one thing that it is identical with itself is to say nothing at all"). As a way to take that argument into account, one can view a calculation as a path, and an identity as the exhibition of an homotopy betweeen two different paths. Accordingly, identity between two calculations is never a fact, but always itself the result of a (higher-order) calculation. Quite generally, conceiving of identity of mathematical objects as an identity up to some homotopy could contribute towards an understanding of that issue in the framework of category theory. The goal of this session is to provide an interdisciplinary forum in which to further explore those issues. In terms of format, it will have three speakers, embracing respectively an historical, a philosophical and a mathematical point of view.