Workshop on Functor Calculus

[The talks were recorded by Carmen Rovi and are available as a playlist on Youtube.]

University of Münster

Monday 22 June ­– Friday 26 June 2015

The theme of this workshop was Calculus of Functors, meant in a very broad sense, covering some of the families of Goodwillie–derived  functor calculi as well as that coming more directly from the work of Eilenberg and MacLane.

The roots of the notion of a calculus of functors goes back to work extending the construction of a derived functor to those functors which are non-additive (see fundamental work of Eilenberg–MacLane (1954) as well as Dold–Puppe (1961)).  This required defining the notion of the cross effects of a functor, which measure its failure to be additive and which was used to define the degree of a functor.

As interest among topologists and algebraists developed in understanding K–theory (of a ring, or Waldhausen's algebraic K–theory of a space), related methods were developed to study the relationship of (stable) K–theory to THH and TC. See, for instance, Fiedorowicz–Pirashvili–Schwaenzl–Vogt–Waldhausen.

This continued to grow and develop, in multiple directions. The more algebraic track kept cross-effects and the related notion of degree in the forefront, and employs methods of homological algebra. See, for instance, work of Hartl–Pirashvili–Vespa, Djament–Vespa, or the strict polynomial functors of Friedlander–Suslin, Touzé, etc. This is also related closely to functor homology (see, e.g. Pirashvili–Richter) and (twisted) coefficient systems, e.g. Remark 5.5 of Wahl.

Goodwillie (1990, 1991, 2003) took the notion of stabilization and derivation of a functor to the setting of functors of spaces, where homological stabilization was replaced by the stable homotopy category – linear functors are then determined by spectra, developing a calculus of homotopy functors to aid further study of Waldhausen's algebraic K–theory and its relationship to TC (see, e.g. Bökstedt–Carlsson–Cohen–Goodwillie–Hsiang–Madsen and the recent book of Dundas–Goodwillie–McCarthy for more on the TC–K–theory story).

A family of related theories grew out of this work; the manifold (or embedding) calculus of GoodwillieWeiss, the Orthogonal Calculus of Weiss, the algebraic (or discrete) calculus of Johnson–McCarthy (which is strongly linked to the algebraic/homological calculus, being based again on cross-effects) and, more recently, two sorts of G–equivariant calculus (Blumberg and later Dotto–Moi).

Invited Speakers

Short courses were given by:

Gregory Arone, University of Virginia
Pascal Lambrechts, Université catholique de Louvain
Teimuraz Pirashvili, University of Leicester

Additional talks were given by:

Michael Ching, Amherst College
Oliver Röndigs, University of Osnabrück
Victor Turchin, Kansas State University
Nathalie Wahl, University of Copenhagen
Michael Weiss
, University of Münster

Organisers:  Chris BraunFederico Cantero, Rosona Eldred, Geoffroy Horel, Martin Palmer