Regular patterns, substitudes, feynman categories and operads

Michael Batanin, Joachim Kock, Mark Weber

Research output: Contribution to journalArticleResearchpeer-review

4 Citations (Scopus)


© Michael Batanin, Joachim Kock, and Mark Weber, 2018. We show that the regular patterns of Getzler (2009) form a 2-category biequivalent to the 2-category of substitudes of Day and Street (2003), and that the Feynman categories of Kaufmann and Ward (2013) form a 2-category biequivalent to the 2-category of coloured operads (with invertible 2-cells). These biequivalences induce equivalences between the corresponding categories of algebras. There are three main ingredients in establishing these biequivalences. The first is a strictification theorem (exploiting Power’s General Coherence Result) which allows to reduce to the case where the structure maps are identity-on-objects functors and strict monoidal. Second, we subsume the Getzler and Kaufmann–Ward hereditary axioms into the notion of Guitart exactness, a general condition ensuring compatibility between certain left Kan extensions and a given monad, in this case the free-symmetric-monoidal-category monad. Finally we set up a biadjunction between substitudes and what we call pinned symmetric monoidal categories, from which the results follow as a consequence of the fact that the hereditary map is precisely the counit of this biadjunction.
Original languageEnglish
Pages (from-to)148-192
JournalTheory and Applications of Categories
Publication statusPublished - 19 Feb 2018


  • Operads
  • Symmetric monoidal categories


Dive into the research topics of 'Regular patterns, substitudes, feynman categories and operads'. Together they form a unique fingerprint.

Cite this