Homology, Homotopy and Applications

Volume 23 (2021)

Number 2

The Tamarkin–Tsygan calculus of an associative algebra à la Stasheff

Pages: 257 – 282

DOI: https://dx.doi.org/10.4310/HHA.2021.v23.n2.a14


Pedro Tamaroff (School of Mathematics, Trinity College Dublin, Ireland)


We show how to compute the Tamarkin–Tsygan calculus of an associative algebra by providing, for a given cofibrant replacement of it, a ‘small’ $\mathsf{Calc}_\infty$-model of its calculus, which we make somewhat explicit at the level of $\mathsf{Calc}$-algebras. To do this, we prove that the operad $\mathsf{Calc}$ is inhomogeneous Koszul; to our best knowledge, this result is new. We illustrate our technique by carrying out some computations for two monomial associative algebras using the cofibrant replacement obtained by the author in [39].


monomial algebras, minimal models, $A_\infty$-algebras, rewriting theory, higher structures

2010 Mathematics Subject Classification

16E05, 16E40, 16E45, 18G15, 18G55

Copyright © 2021, Pedro Tamaroff. Permission to copy for private use granted.

Received 3 March 2020

Received revised 7 April 2021

Accepted 17 September 2020

Published 7 July 2021