179 \todo{This is just a copy and paste of the statements of the axioms. We need to rewrite this into something that's both compact and comprehensible! The first few at least aren't that terrifying, but we definitely don't want to derail the reader with the actual product axiom, for example.}

   179 \nn{rough draft of n-cat stuff...}

   180 

   181 \nn{maybe say something about goals: well-suited to TQFTs; avoid proliferation of coherency axioms;

   182 non-recursive (n-cats not defined n terms of (n-1)-cats; easy to show that the motivating

   183 examples satisfy the axioms; strong duality; both plain and infty case;

   184 (?) easy to see that axioms are correct, in the sense of nothing missing (need

   185 to say this better if we keep it)}

   186 

   187 \nn{maybe: the typical n-cat definition tries to do two things at once: (1) give a list of basic properties

   188 which are weak enough to include the basic examples and strong enough to support the proofs

   189 of the main theorems; and (2) specify a minimal set of generators and/or axioms.

   190 We separate these two tasks, and address only the first, which becomes much easier when not burdened by the second.

   191 More specifically, life is easier when working with maximal, rather than minimal, collections of axioms.}

   192 

   193 \nn{say something about defining plain and infty cases simultaneously}

   194 

   195 There are five basic ingredients of an $n$-category definition:

   196 $k$-morphisms (for $0\le k \le n$), domain and range, composition,

   197 identity morphisms, and special behavior in dimension $n$ (e.g. enrichment

   198 in some auxiliary category, or strict associativity instead of weak associativity).

   199 We will treat each of these it turn.

   200 

   201 To motivate our morphism axiom, consider the venerable notion of the Moore loop space

   202 \nn{need citation}.

   203 In the standard definition of a loop space, loops are always parameterized by the unit interval $I = [0,1]$,

   204 so composition of loops requires a reparameterization $I\cup I \cong I$, and this leads to a proliferation

   205 of higher associativity relations.

   206 While this proliferation is manageable for 1-categories (and indeed leads to an elegant theory

   207 of Stasheff polyhedra and $A_\infty$ categories), it becomes undesirably complex for higher categories.

   208 In a Moore loop space, we have a separate space $\Omega_r$ for each interval $[0,r]$, and a 

   209 {\it strictly associative} composition $\Omega_r\times \Omega_s\to \Omega_{r+s}$.

   210 Thus we can have the simplicity of strict associativity in exchange for more morphisms.

   211 We wish to imitate this strategy in higher categories.

   212 Because we are mainly interested in the case of strong duality, we replace the intervals $[0,r]$ not with

   213 a product of $n$ intervals \nn{cf xxxx} but rather with any $n$-ball, that is, any $n$-manifold which is homeomorphic

   214 to the standard $n$-ball $B^n$.

   215 

   216 \nn{...}