### Abstract

Motivated by the second author’s construction of a classifying space for the group of pure symmetric automorphisms of a free product, we introduce and study a family of topological operads, the operads of based cacti, defined for every pointed simplicial set (Y,p). These operads also admit linear versions, which are defined for every augmented graded cocommutative coalgebra C. We show that the homology of the topological operad of based Y–cacti is the linear operad of based H_{*}(Y)–cacti. In addition, we show that for every coalgebra C the operad of based C–cacti is Koszul. To prove the latter result, we use the criterion of Koszulness for operads due to the first author, utilising the notion of a filtered distributive law between two quadratic operads. We also present a new proof of that criterion, which works over a ground field of arbitrary characteristic.

Original language | English |
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Pages (from-to) | 3185-3225 |

Number of pages | 41 |

Journal | Algebraic and Geometric Topology |

Volume | 14 |

Issue number | 6 |

DOIs | |

Publication status | Published - 15 Jan 2015 |

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### Keywords

- Based cactus products
- Distributive law
- Gröbner basis
- Koszul operad

### ASJC Scopus subject areas

- Geometry and Topology

### Cite this

*Algebraic and Geometric Topology*,

*14*(6), 3185-3225. https://doi.org/10.2140/agt.2014.14.3185

**Cacti and filtered distributive laws.** / Dotsenko, Vladimir; Griffin, James.

Research output: Contribution to journal › Article

*Algebraic and Geometric Topology*, vol. 14, no. 6, pp. 3185-3225. https://doi.org/10.2140/agt.2014.14.3185

}

TY - JOUR

T1 - Cacti and filtered distributive laws

AU - Dotsenko, Vladimir

AU - Griffin, James

PY - 2015/1/15

Y1 - 2015/1/15

N2 - Motivated by the second author’s construction of a classifying space for the group of pure symmetric automorphisms of a free product, we introduce and study a family of topological operads, the operads of based cacti, defined for every pointed simplicial set (Y,p). These operads also admit linear versions, which are defined for every augmented graded cocommutative coalgebra C. We show that the homology of the topological operad of based Y–cacti is the linear operad of based H*(Y)–cacti. In addition, we show that for every coalgebra C the operad of based C–cacti is Koszul. To prove the latter result, we use the criterion of Koszulness for operads due to the first author, utilising the notion of a filtered distributive law between two quadratic operads. We also present a new proof of that criterion, which works over a ground field of arbitrary characteristic.

AB - Motivated by the second author’s construction of a classifying space for the group of pure symmetric automorphisms of a free product, we introduce and study a family of topological operads, the operads of based cacti, defined for every pointed simplicial set (Y,p). These operads also admit linear versions, which are defined for every augmented graded cocommutative coalgebra C. We show that the homology of the topological operad of based Y–cacti is the linear operad of based H*(Y)–cacti. In addition, we show that for every coalgebra C the operad of based C–cacti is Koszul. To prove the latter result, we use the criterion of Koszulness for operads due to the first author, utilising the notion of a filtered distributive law between two quadratic operads. We also present a new proof of that criterion, which works over a ground field of arbitrary characteristic.

KW - Based cactus products

KW - Distributive law

KW - Gröbner basis

KW - Koszul operad

UR - http://www.scopus.com/inward/record.url?scp=84922324428&partnerID=8YFLogxK

U2 - 10.2140/agt.2014.14.3185

DO - 10.2140/agt.2014.14.3185

M3 - Article

VL - 14

SP - 3185

EP - 3225

JO - Algebraic and Geometric Topology

JF - Algebraic and Geometric Topology

SN - 1472-2747

IS - 6

ER -