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The Catalan simplicial set

Published online by Cambridge University Press:  03 December 2014

MITCHELL BUCKLEY ,
RICHARD GARNER ,
STEPHEN LACK  and
ROSS STREET
MITCHELL BUCKLEY
Affiliation:
Department of Mathematics, Macquarie University NSW 2109, Australia. e-mails: mitchell.buckley@mq.edu.au; richard.garner@mq.edu.au; steve.lack@mq.edu.au; ross.street@mq.edu.au
RICHARD GARNER
Affiliation:
Department of Mathematics, Macquarie University NSW 2109, Australia. e-mails: mitchell.buckley@mq.edu.au; richard.garner@mq.edu.au; steve.lack@mq.edu.au; ross.street@mq.edu.au
STEPHEN LACK
Affiliation:
Department of Mathematics, Macquarie University NSW 2109, Australia. e-mails: mitchell.buckley@mq.edu.au; richard.garner@mq.edu.au; steve.lack@mq.edu.au; ross.street@mq.edu.au
ROSS STREET
Affiliation:
Department of Mathematics, Macquarie University NSW 2109, Australia. e-mails: mitchell.buckley@mq.edu.au; richard.garner@mq.edu.au; steve.lack@mq.edu.au; ross.street@mq.edu.au
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Abstract

The Catalan numbers are well known to be the answer to many different counting problems, and so there are many different families of sets whose cardinalities are the Catalan numbers. We show how such a family can be given the structure of a simplicial set. We show how the low-dimensional parts of this simplicial set classify, in a precise sense, the structures of monoid and of monoidal category. This involves aspects of combinatorics, algebraic topology, quantum groups, logic, and category theory.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 158 , Issue 2 , March 2015 , pp. 211 - 222
Copyright
Copyright © Cambridge Philosophical Society 2014 

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References

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