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

   249 In this section we give a definition of $n$-categories designed to work well with TQFTs.

   250 

   250 The main idea is to base the definition on actual balls, rather combinatorial models of them.

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

   251 This has the advantages of avoiding a proliferation of coherency axioms and building in a strong

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

   252 version of duality from the start.

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

   253 

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

   254 

   255 to say this better if we keep it)}

   255 %\nn{maybe say something about goals: well-suited to TQFTs; avoid proliferation of coherency axioms;

   256 

   256 %non-recursive (n-cats not defined n terms of (n-1)-cats; easy to show that the motivating

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

   257 %examples satisfy the axioms; strong duality; both plain and infty case;

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

   258 %(?) easy to see that axioms are correct, in the sense of nothing missing (need

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

   259 %to say this better if we keep it)}

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

   260 %

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

   261 %\nn{maybe: the typical n-cat definition tries to do two things at once: (1) give a list of basic properties

   262 

   262 %which are weak enough to include the basic examples and strong enough to support the proofs

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

   263 %of the main theorems; and (2) specify a minimal set of generators and/or axioms.

   264 %We separate these two tasks, and address only the first, which becomes much easier when not burdened by the second.

   265 %More specifically, life is easier when working with maximal, rather than minimal, collections of axioms.}

   266 

   267 We will define plain and $A_\infty$ $n$-categories simultaneously, as all but one of the axioms are identical

   268 in the two cases.

   269