Periodic Reporting for period 1 - HotCoalgebras (Homotopy theory of cosimplicial unstable (co-)algebras over the Steenrod algebra)
Reporting period: 2015-05-01 to 2017-04-30
One important invariant is singular homology. It yields a functor (ie. a natural construction) from topological spaces (ie. geometric objects) to the category of unstable coalgebras over the Steenrod algebra. The general theory of such homotopy-invariant functors is now called Goodwillie calculus. The original proposal laid out the importance of deepening our understanding of the homotopy theory of cosimplicial unstable (co-)algebras over the Steenrod algebra and its relation to the homotopy theory of cosimplicial spaces. It also mentioned the desirability of bringing methods from Goodwillie calculus to bear on this question.
Higher Topos Theory was introduced by Rezk, Joyal, and Lurie in order to produce a highly structured environment where homotopy invariant constructions can be carried out. It is the part of modern higher category theory that is most relevant for classical Homotopy Theory. Type theory is the theory of programming languages. Recently, deep and unexpected relations between type theory and homotopy theory have been discovered by the work of Voevodsky, opening up a new subject in mathematics and theoretical computer science now called Homotopy Type Theory. It is believed that there is great potential for future research with considerable impact on pure and applied mathematics.
Using these techniques we produce a generalized Blakers-Massey theorem with the following features: it is valid in any higher topos; it axiomatizes and generalizes earlier results in the literature; its proof is elementary in the following two senses: classical Homotopy Theory is reduced to a minimum, the proof (in work in progress by Finster) will be machine-verified. In a second joint paper we manage to apply the generalized Blakers-Massey theorem to Goodwillie calculus. Our result proves a conjecture by Goodwillie and simplifies many of his arguments in setting up the theory. Another strong point of our approach is that it is especially adapted to the unpointed setting which is mostly avoided by other authors. Taken together the output of the two papers provide a cutting-edge application of Higher Topos Theory to classical problems in Homotopy Theory. We hope that both papers demonstrate how useful and easy Higher Category Theory is and will help leading the subject to become main stream mathematics.
Homotopy nilpotent groups were introduced by earlier joint work by Dwyer and the researcher. They arise from the study of the relation of the Goodwillie tower of the identity and the lower central series of the loop group on connected spaces. It provides a new invariant for loop spaces; their Biedermann-Dwyer nilpotence degree. A recent article by Costoya-Scherer-Viruel relates this invariant to other classical invariants, the Berstein-Ganea nilpotence degree and Ganea's cocategory. Their arguments rely on a statement proved in the preprint “Homotopy nilpotent groups and their associated functors”. Further properties of those functors will be deduced in an updated version that is currently being written by the researcher. The finished article will an advertisement of the importance of homotopy nilpotent groups in Homotopy Theory.
The collaboration Anel-Biedermann-Finster-Joyal was highly successful. Taken together the output of the two papers provide a cutting-edge application of Higher Topos Theory with ideas stemming from Homotopy Type Theory to classical problems in Homotopy Theory. We expect that both papers demonstrate how useful and easy Higher Category Theory is and will help leading the subject to become main stream mathematics. Eventually, our work aims at bringing together the different communities of classical Homotopy Theory and Homotopy Type Theory. We also hope to provide in the near future further evidence of the strength of our methods through applications to other questions in Homotopy Theory.
The theory of homotopy nilpotent groups is still in its infancy but has already attracted the interest of mathematicians other than its two inventors. Its exact role is still open.