1,886 research outputs found
Regular patterns, substitudes, Feynman categories and operads
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
Polynomial functors from Algebras over a set-operad and non-linear Mackey functors
In this paper, we give a description of polynomial functors from (finitely
generated free) groups to abelian groups in terms of non-linear Mackey functors
generalizing those given in a paper of Baues-Dreckmann-Franjou-Pirashvili
published in 2001. This description is a consequence of our two main results: a
description of functors from (fi nitely generated free) P-algebras (for P a
set-operad) to abelian groups in terms of non-linear Mackey functors and the
isomorphism between polynomial functors on (finitely generated free) monoids
and those on (finitely generated free) groups. Polynomial functors from
(finitely generated free) P-algebras to abelian groups and from (finitely
generated free) groups to abelian groups are described explicitely by their
cross-e ffects and maps relating them which satisfy a list of relations.Comment: 58 page
Laplaza Sets, or How to Select Coherence Diagrams for Pseudo Algebras
We define a general concept of pseudo algebras over theories and 2-theories.
A more restrictive such notion was introduced by Hu and Kriz, but as noticed by
M. Gould, did not capture the desired examples. The approach taken in this
paper corrects the mistake by introducing a more general concept, allowing more
flexibility in selecting coherence diagrams for pseudo algebras.Comment: 21 pages. To appear in the Advances of Mathematics. Exposition
improved, notion of operad with degeneracies removed as a simplification,
definition of pseudo algebra improved
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