## An n-Categorical Pasting Theorem

**A J Power**
*Abstract:*In order to facilitate the study of 2-categories
with structure, we state and prove an n-categorical pasting
theorem. This is based upon a new definition of n-pasting scheme
that generalises Johnson's definition of a well-formed loop-free
pasting scheme by weakening his no direct loops condition. We
define n-pasting, prove the theorem, and show that for n=3, it
incorporates all possible composites of n-cells. We show that that
is not true for higher n. We define the horizontal n-category of an
(n+1)-category to generalise that of a 2-category, we define
horizontal and vertical composition for an (n+1)-category and we
state and prove an interchange law. We also study further
conditions on a pasting diagram and their impact upon how one may
evaluate a composite, and we express Street's free n-categories in
terms of left adjoints.

*LFCS report ECS-LFCS-91-190*

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