Generalizations of Loday's assembly maps for Lawvere's algebraic theories

Loday’s assembly maps approximate the K-theory of group rings by the K-theory of the coefficient ring and the corresponding homology of the group. We present a generalization that places both ingredients on the same footing. Building on Elmendorf–Mandell’s multiplicativity results and our earlier work, we show that the K-theory of Lawvere theories is lax monoidal. This result makes it possible to present our theory in a user-friendly way without using higher categorical language. It also allows us to extend the idea to new contexts and set up a non-abelian interpolation scheme, raising novel questions. Numerous examples illustrate the scope of our extension.

[1]  F. W. Lawvere,et al.  FUNCTORIAL SEMANTICS OF ALGEBRAIC THEORIES. , 1963, Proceedings of the National Academy of Sciences of the United States of America.

[2]  Boolean algebras, Morita invariance, and the algebraic K-theory of Lawvere theories , 2020, 2011.11755.

[3]  Ib Madsen,et al.  The cyclotomic trace and algebraic K-theory of spaces , 1993 .

[4]  A. Connes,et al.  Segal’s Gamma rings and universal arithmetic , 2020, 2004.08879.

[5]  Francesco Matucci,et al.  Presentations of generalisations of Thompson’s group V , 2016, Pacific Journal of Mathematics.

[6]  Y. Neretin,et al.  Diffeomorphism groups of tame Cantor sets and Thompson-like groups , 2014, Compositio Mathematica.

[7]  E. Pedersen,et al.  Identifying assembly maps in K- and L-theory , 2004 .

[8]  P. Freyd,et al.  Algebra valued functors in general and tensor products in particular , 1966 .

[9]  D. Puppe,et al.  Kategorien und Funktoren , 1966 .

[10]  Michael A. Mandell,et al.  Permutative categories, multicategories and algebraicK–theory , 2007, Algebraic & Geometric Topology.

[11]  P. Ara,et al.  $K$-theory of Leavitt path algebras , 2009, 0903.0056.

[12]  B. Nucinkis,et al.  Bredon cohomological finiteness conditions for generalisations of Thompson's groups , 2011, 1105.0189.

[13]  G. Segal,et al.  Categories and cohomology theories , 1974 .

[14]  Søren Galatius.  Stable homology of automorphism groups of free groups , 2006, math/0610216.

[15]  B. Dundas,et al.  Ring completion of rig categories , 2007, 0706.0531.

[16]  V. E. Cazanescu.  Algebraic theories , 2004 .

[17]  N. Durov.  New Approach to Arakelov Geometry , 2007, 0704.2030.

[18]  Rings, modules, and algebras in infinite loop space theory , 2004, math/0403403.

[19]  James F. Davis,et al.  Spaces over a category and assembly maps in isomorphism conjectures in K- and L-theory. , 1998 .

[20]  The homology of the Higman–Thompson groups , 2014, 1411.5035.

[21]  Sally Popkorn,et al.  A Handbook of Categorical Algebra , 2009 .

[22]  S. Hu.  Assembly , 2019, CIRP Encyclopedia of Production Engineering.

[23]  J.-L. Loday.  $K$-théorie algébrique et représentations de groupes , 1976 .

[24]  Donald Yau,et al.  Bimonoidal Categories, $E_n$-Monoidal Categories, and Algebraic $K$-Theory , 2021, 2107.10526.

[25]  F. Quinn.  Ends of Maps, I , 1979 .

[26]  F. Farrell,et al.  Isomorphism conjectures in algebraic $K$-theory , 1993 .

[27]  Jirí Adámek,et al.  Algebraic Theories: A Categorical Introduction to General Algebra , 2010 .

[28]  Warren Dicks,et al.  Isomorphisms of Brin-Higman-Thompson groups , 2011, 1112.1606.

[29]  I. Shafarevich,et al.  Abelian and nonabelian mathematics , 1991 .

[30]  Markus Szymik.  Twisted homological stability for extensions and automorphism groups of free nilpotent groups , 2014, 1401.5215.

[31]  W. Lueck.  Assembly maps , 2018, Handbook of Homotopy Theory.

[32]  A. Kock.  Strong functors and monoidal monads , 1972 .

[33]  Tomáš Zeman.  On the quotients of mapping class groups of surfaces by the Johnson subgroups , 2019, Mathematical Proceedings of the Cambridge Philosophical Society.

[34]  A. Kock.  Monads on symmetric monoidal closed categories , 1970 .

[35]  A. Connes,et al.  Absolute algebra and Segal's Γ-rings: Au dessous de Spec(Z)‾ , 2016 .

[36]  Anders Kock,et al.  Closed categories generated by commutative monads , 1971, Journal of the Australian Mathematical Society.

[37]  F. W. Lawvere,et al.  Some algebraic problems in the context of functorial semantics of algebraic theories , 1968 .

[38]  Dominic R. Verity,et al.  ∞-Categories for the Working Mathematician , 2018 .

[39]  Markus Szymik.  The rational stable homology of mapping class groups of universal nil-manifolds , 2016, Annales de l'Institut Fourier.

[40]  Comparing Assembly Maps in Algebraic K -Theory , 2011 .

[41]  John Power,et al.  The Category Theoretic Understanding of Universal Algebra: Lawvere Theories and Monads , 2007, Computation, Meaning, and Logic.

[42]  Matthew G. Brin.  Higher Dimensional Thompson Groups , 2004, math/0406046.

[43]  John Power,et al.  Pseudo-commutative monads and pseudo-closed 2-categories , 2002 .