495 \subsection{\texorpdfstring{$n$}{n}-category terminology}

   496 \label{n-cat-names}

   497 

   498 Section \S \ref{sec:ncats} adds to the zoo of $n$-category definitions, and the new creatures need names.

   499 Unfortunately, we have found it difficult to come up with terminology which satisfies all

   500 of the colleagues whom we have consulted, or even satisfies just ourselves.

   501 

   502 One distinction we need to make is between $n$-categories which are associative in dimension $n$ and those

   503 that are associative only up to higher homotopies.

   504 The latter are closely related to $(\infty, n)$-categories (i.e.\ $\infty$-categories where all morphisms

   505 of dimension greater than $n$ are invertible), but we don't want to use that name

   506 since we think of the higher homotopies not as morphisms of the $n$-category but

   507 rather as belonging to some auxiliary category (like chain complexes)

   508 that we are enriching in.

   509 We have decided to call them ``$A_\infty$ $n$-categories", since they are a natural generalization 

   510 of the familiar $A_\infty$ 1-categories.

   511 Other possible names include ``homotopy $n$-categories" and ``infinity $n$-categories".

   512 When we need to emphasize that we are talking about an $n$-category which is not $A_\infty$

   513 we will say ``ordinary $n$-category".

   514 % small problem: our n-cats are of course strictly associative, since we have more morphisms.

   515 % when we say ``associative only up to homotopy" above we are thinking about

   516 % what would happen we we tried to convert to a more traditional n-cat with fewer morphisms

   517 

   518 Another distinction we need to make is between our style of definition of $n$-categories and

   519 more traditional and combinatorial definitions.

   520 We will call instances of our definition ``disk-like $n$-categories", since $n$-dimensional disks

   521 play a prominent role in the definition.

   522 (In general we prefer to ``$k$-ball" to ``$k$-disk", but ``ball-like" doesn't roll off 

   523 the tongue as well as "disk-like".)

   524 

   525 Another thing we need a name for is the ability to rotate morphisms around in various ways.

   526 For 2-categories, ``pivotal" is a standard term for what we mean.

   527 A more general term is ``duality", but duality comes in various flavors and degrees.

   528 We are mainly interested in a very strong version of duality, where the available ways of

   529 rotating $k$-morphisms correspond to all the ways of rotating $k$-balls.

   530 We sometimes refer to this as ``strong duality", and sometimes we consider it to be implied

   531 by ``disk-like".

   532 (But beware: disks can come in various flavors, and some of them (such as framed disks)

   533 don't actually imply much duality.)

   534 Another possibility here is ``pivotal $n$-category".

   535 

   536 Finally, we need a general name for isomorphisms between balls, where the balls could be

   537 piecewise linear or smooth or topological or Spin or framed or etc., or some combination thereof.

   538 We have chosen to use ``homeomorphism" for the appropriate sort of isomorphism, so the reader should

   539 keep in mind that ``homeomorphism" could mean PL homeomorphism or diffeomorphism (and so on)

   540 depending on context.

   541