Polynomial Functors and Trees

Research output: Contribution to journal › Journal article › Research › peer-review

Standard

Polynomial Functors and Trees. / Kock, Joachim.

In: International Mathematics Research Notices, Vol. 2011, No. 3, 2011, p. 609-673.

Research output: Contribution to journal › Journal article › Research › peer-review

Harvard

Kock, J 2011, 'Polynomial Functors and Trees', International Mathematics Research Notices, vol. 2011, no. 3, pp. 609-673. https://doi.org/10.1093/imrn/rnq068

APA

Kock, J. (2011). Polynomial Functors and Trees. International Mathematics Research Notices, 2011(3), 609-673. https://doi.org/10.1093/imrn/rnq068

Vancouver

Kock J. Polynomial Functors and Trees. International Mathematics Research Notices. 2011;2011(3):609-673. https://doi.org/10.1093/imrn/rnq068

Author

Kock, Joachim. / Polynomial Functors and Trees. In: International Mathematics Research Notices. 2011 ; Vol. 2011, No. 3. pp. 609-673.

Bibtex

@article{0d656d3a1a15410f8536869fa787c14e,
title = "Polynomial Functors and Trees",
abstract = "We explore the relationship between polynomial functors and (rooted) trees. In the first part, we use polynomial functors to derive a new convenient formalism for trees, and obtain a natural and conceptual construction of the category Omega of Moerdijk and Weiss; its main properties are described in terms of some factorization systems. Although the constructions are motivated and explained in terms of polynomial functors, they all amount to elementary manipulations with finite sets. In the second part, we describe polynomial endofunctors and monads as structures built from trees, characterizing the images of several nerve functors from polynomial endofunctors and monads into presheaves on categories of trees. Polynomial endofunctors and monads over a base are characterized by a sheaf condition on categories of decorated trees. In the absolute case, one further condition is needed, a certain projectivity condition, which serves also to characterize polynomial endofunctors and monads among (colored) collections and operads.",
keywords = "WELLFOUNDED TREES",
author = "Joachim Kock",
year = "2011",
doi = "10.1093/imrn/rnq068",
language = "English",
volume = "2011",
pages = "609--673",
journal = "International Mathematics Research Notices",
issn = "1073-7928",
publisher = "Oxford University Press",
number = "3",

}

RIS

TY - JOUR

T1 - Polynomial Functors and Trees

AU - Kock, Joachim

PY - 2011

Y1 - 2011

N2 - We explore the relationship between polynomial functors and (rooted) trees. In the first part, we use polynomial functors to derive a new convenient formalism for trees, and obtain a natural and conceptual construction of the category Omega of Moerdijk and Weiss; its main properties are described in terms of some factorization systems. Although the constructions are motivated and explained in terms of polynomial functors, they all amount to elementary manipulations with finite sets. In the second part, we describe polynomial endofunctors and monads as structures built from trees, characterizing the images of several nerve functors from polynomial endofunctors and monads into presheaves on categories of trees. Polynomial endofunctors and monads over a base are characterized by a sheaf condition on categories of decorated trees. In the absolute case, one further condition is needed, a certain projectivity condition, which serves also to characterize polynomial endofunctors and monads among (colored) collections and operads.

AB - We explore the relationship between polynomial functors and (rooted) trees. In the first part, we use polynomial functors to derive a new convenient formalism for trees, and obtain a natural and conceptual construction of the category Omega of Moerdijk and Weiss; its main properties are described in terms of some factorization systems. Although the constructions are motivated and explained in terms of polynomial functors, they all amount to elementary manipulations with finite sets. In the second part, we describe polynomial endofunctors and monads as structures built from trees, characterizing the images of several nerve functors from polynomial endofunctors and monads into presheaves on categories of trees. Polynomial endofunctors and monads over a base are characterized by a sheaf condition on categories of decorated trees. In the absolute case, one further condition is needed, a certain projectivity condition, which serves also to characterize polynomial endofunctors and monads among (colored) collections and operads.

KW - WELLFOUNDED TREES

U2 - 10.1093/imrn/rnq068

DO - 10.1093/imrn/rnq068

M3 - Journal article

VL - 2011

SP - 609

EP - 673

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

SN - 1073-7928

IS - 3

ER -

ID: 331502104