*Extended* cubical sets (with connections and interchanges) are
presheaves on a ground category, the *extended cubical site
***K**, corresponding to the (augmented) simplicial site, the
category of finite ordinals. We prove here that **K** has
characterisations similar to the classical ones for the simplicial
analogue, by generators and relations, or by the existence of a universal
*symmetric cubical monoid*; in fact, **K** is the classifying
category of a *monoidal * algebraic theory of such monoids.
Analogous results are given for the * restricted cubical * site}
**I**, of * ordinary * cubical sets (just faces and
degeneracies) and for the * intermediate * site **J**
(including connections). We also
consider briefly the *reversible* analogue,
!**K**.

Keywords: Simplicial sets, cubical sets, monoidal categories, algebraic theories, generators and relations, word problem, classifying categories

2000 MSC: 18G30, 55U10, 18D10, 18C10, 20F05, 20F10

*Theory and Applications of Categories*
, Vol. 11, 2003,
No. 8, pp 185-211.

http://www.tac.mta.ca/tac/volumes/11/8/11-08.dvi

http://www.tac.mta.ca/tac/volumes/11/8/11-08.ps

http://www.tac.mta.ca/tac/volumes/11/8/11-08.pdf

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/11/8/11-08.dvi

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/11/8/11-08.ps