## 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

## Summary of the context and overall objectives of the project

"In algebraic topology one studies algebraic invariants of geometric objects up to continuous deformation. The latter is what is called a ""homotopy"" giving the subject its name: Homotopy Theory. Homotopy theorists engage in defining, interpreting, and computing such invariants. Since its ""invention"" around 1900 the importance of the field within mathematics has steadily grown. Very recent developments extend its reach to physics and computer science.

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."

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."

## Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

The researcher together with his coauthors Raptis and Stelzer has finished the monograph “The realization space for an unstable coalgebra”. Let us recall briefly the content of the monograph. Singular homology yields a functor from topological spaces to the category of unstable coalgebras over the Steenrod algebra. One asks whether a given unstable coalgebra possesses a realization, i.e. whether it is in fact the homology of some topological space: an inverse problem. More generally, one wants to describe a moduli space of realizations of a particular unstable coalgebra. In the monograph we construct a general framework to deal with the realization and moduli problems associated to singular homology with prime field coefficients. The researcher and his coauthors have significantly improved and strengthened an earlier preprint with the same name. The monograph has been accepted for publication in Astérisque, a major monograph series in mathematics published by the Société Mathématique de France. The new emphasis on Goodwillie calculus is seen by the new collaboration the researcher has started with Anel, Finster, and Joyal about Generalized Blakers-Massey theorems. The results by the team are obtained by joining classical Homotopy Theory with two very recent developments in Mathematics and Computer Science: Higher Topos Theory and Homotopy Type Theory. It reflects clearly the individual strengths and interest of the authors and constitutes the main new skills acquired by the researcher.

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.

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.

## Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

The collaboration Biedermann-Raptis-Stelzer constitutes the first conceptual treatment of the moduli problem associated to singular homology theory with prime field coefficients in positive characteristic. The obstruction theory for realizing unstable coalgebras derived from our general approach (Biedermann, Raptis, Stelzer) supersedes earlier work in the literature. Our work facilitates and organizes the use of cosimplicial techniques in the subject. The monograph will become a standard reference for people working on (non-)realization problems associated to singular homology.

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.

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.