ISOMORPHISM OF THE CUBICAL AND CATEGORICAL COHOMOLOGY GROUPS OF A HIGHER-RANK GRAPH

Elizabeth Gillaspy; Jianchao Wu

doi:10.1090/btran/38

ISOMORPHISM OF THE CUBICAL AND CATEGORICAL COHOMOLOGY GROUPS OF A HIGHER-RANK GRAPH

Elizabeth Gillaspy
, Jianchao Wu

Mathematical Sciences

Texas A&M University

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

Abstract

We use category-theoretic techniques to provide two proofs show-ing that for a higher-rank graph Λ, its cubical (co-)homology and categorical (co-)homology groups are isomorphic in all degrees, thus answering a question of Kumjian, Pask and Sims in the positive. Our first proof uses the topological realization of a higher-rank graph, which was introduced by Kaliszewski, Kumjian, Quigg, and Sims. In our more combinatorial second proof, we con-struct, explicitly and in both directions, maps on the level of (co-)chain com-plexes that implement said isomorphism. Along the way, we extend the defi-nition of cubical (co-)homology to allow arbitrary coefficient modules.

Original language	English
Pages (from-to)	442-480
Number of pages	39
Journal	Transactions of the American Mathematical Society Series B
Volume	8
Issue number	16
DOIs	https://doi.org/10.1090/btran/38
State	Published - Jun 10 2021

Funding

Received by the editors July 19, 2018, and, in revised form, January 26, 2019. 2020 Mathematics Subject Classification. Primary 18G90; Secondary 55N10. The first author was partially supported by the Deutsches Forschungsgemeinschaft via the SFB 878 “Groups, Geometry, and Actions.” The second author was partially supported by NSF grant #DMS–1564401.

Funder number
–1564401
SFB 878

Access to Document

10.1090/btran/38

Cite this

@article{4b4793fef46044e6b43077d3649a8ae4,

title = "ISOMORPHISM OF THE CUBICAL AND CATEGORICAL COHOMOLOGY GROUPS OF A HIGHER-RANK GRAPH",

abstract = "We use category-theoretic techniques to provide two proofs show-ing that for a higher-rank graph Λ, its cubical (co-)homology and categorical (co-)homology groups are isomorphic in all degrees, thus answering a question of Kumjian, Pask and Sims in the positive. Our first proof uses the topological realization of a higher-rank graph, which was introduced by Kaliszewski, Kumjian, Quigg, and Sims. In our more combinatorial second proof, we con-struct, explicitly and in both directions, maps on the level of (co-)chain com-plexes that implement said isomorphism. Along the way, we extend the defi-nition of cubical (co-)homology to allow arbitrary coefficient modules.",

author = "Elizabeth Gillaspy and Jianchao Wu",

note = "Publisher Copyright: {\textcopyright} 2021 by the authors.",

year = "2021",

month = jun,

day = "10",

doi = "10.1090/btran/38",

language = "English",

volume = "8",

pages = "442--480",

journal = "Transactions of the American Mathematical Society Series B",

issn = "2330-0000",

publisher = "American Mathematical Society",

number = "16",

}

TY - JOUR

T1 - ISOMORPHISM OF THE CUBICAL AND CATEGORICAL COHOMOLOGY GROUPS OF A HIGHER-RANK GRAPH

AU - Gillaspy, Elizabeth

AU - Wu, Jianchao

PY - 2021/6/10

Y1 - 2021/6/10

N2 - We use category-theoretic techniques to provide two proofs show-ing that for a higher-rank graph Λ, its cubical (co-)homology and categorical (co-)homology groups are isomorphic in all degrees, thus answering a question of Kumjian, Pask and Sims in the positive. Our first proof uses the topological realization of a higher-rank graph, which was introduced by Kaliszewski, Kumjian, Quigg, and Sims. In our more combinatorial second proof, we con-struct, explicitly and in both directions, maps on the level of (co-)chain com-plexes that implement said isomorphism. Along the way, we extend the defi-nition of cubical (co-)homology to allow arbitrary coefficient modules.

AB - We use category-theoretic techniques to provide two proofs show-ing that for a higher-rank graph Λ, its cubical (co-)homology and categorical (co-)homology groups are isomorphic in all degrees, thus answering a question of Kumjian, Pask and Sims in the positive. Our first proof uses the topological realization of a higher-rank graph, which was introduced by Kaliszewski, Kumjian, Quigg, and Sims. In our more combinatorial second proof, we con-struct, explicitly and in both directions, maps on the level of (co-)chain com-plexes that implement said isomorphism. Along the way, we extend the defi-nition of cubical (co-)homology to allow arbitrary coefficient modules.

UR - https://www.scopus.com/pages/publications/85148034221

U2 - 10.1090/btran/38

DO - 10.1090/btran/38

M3 - Article

AN - SCOPUS:85148034221

SN - 2330-0000

VL - 8

SP - 442

EP - 480

JO - Transactions of the American Mathematical Society Series B

JF - Transactions of the American Mathematical Society Series B

IS - 16

ER -

ISOMORPHISM OF THE CUBICAL AND CATEGORICAL COHOMOLOGY GROUPS OF A HIGHER-RANK GRAPH

Abstract

Funding

Access to Document

Other files and links

Fingerprint

Cite this