A study of Penon weak n-categories Part 2: A multisimplicial nerve construction

In this paper we take the first step towards a comparison between
an algebraic and a non-algebraic definition of weak n-category. This comparison takes the form of a nerve functor, the established method of moving from the algebraic setting to the non-algebraic setting. The algebraic definition we use is that due to Penon, and the non-algebraic definition we use is Simpson’s variant of Tamsamani’s definition. As a prototype for our nerve
construction, we recall a nerve construction for bicategories proposed by Leinster, and prove that the nerve of a bicategory given by this construction is
a Tamsamani–Simpson weak 2-category. We then define our nerve functor
for Penon weak n-categories. We prove that the nerve of a Penon weak 2-
category is a Tamsamani–Simpson weak 2-category, and conjecture that this
result holds for higher n.