# HG changeset patch # User Kevin Walker # Date 1294438790 28800 # Node ID 4e3a152f49364169cc710f787076ed47d8c99e2e # Parent 240e4abfb405ac76aa5edeb30f13848fc31a91aa added subsection to intro about n-cat terminology; have not yet actually changed terminology in the rest of the paper diff -r 240e4abfb405 -r 4e3a152f4936 text/intro.tex --- a/text/intro.tex Fri Jan 07 12:41:45 2011 -0800 +++ b/text/intro.tex Fri Jan 07 14:19:50 2011 -0800 @@ -492,6 +492,53 @@ Most importantly, however, \nn{applications!} \nn{cyclic homology, $n=2$ cases, contact, Kh} \nn{stabilization} \nn{stable categories, generalized cohomology theories} } %%% end \noop %%%%%%%%%%%%%%%%%%%%% +\subsection{\texorpdfstring{$n$}{n}-category terminology} +\label{n-cat-names} + +Section \S \ref{sec:ncats} adds to the zoo of $n$-category definitions, and the new creatures need names. +Unfortunately, we have found it difficult to come up with terminology which satisfies all +of the colleagues whom we have consulted, or even satisfies just ourselves. + +One distinction we need to make is between $n$-categories which are associative in dimension $n$ and those +that are associative only up to higher homotopies. +The latter are closely related to $(\infty, n)$-categories (i.e.\ $\infty$-categories where all morphisms +of dimension greater than $n$ are invertible), but we don't want to use that name +since we think of the higher homotopies not as morphisms of the $n$-category but +rather as belonging to some auxiliary category (like chain complexes) +that we are enriching in. +We have decided to call them ``$A_\infty$ $n$-categories", since they are a natural generalization +of the familiar $A_\infty$ 1-categories. +Other possible names include ``homotopy $n$-categories" and ``infinity $n$-categories". +When we need to emphasize that we are talking about an $n$-category which is not $A_\infty$ +we will say ``ordinary $n$-category". +% small problem: our n-cats are of course strictly associative, since we have more morphisms. +% when we say ``associative only up to homotopy" above we are thinking about +% what would happen we we tried to convert to a more traditional n-cat with fewer morphisms + +Another distinction we need to make is between our style of definition of $n$-categories and +more traditional and combinatorial definitions. +We will call instances of our definition ``disk-like $n$-categories", since $n$-dimensional disks +play a prominent role in the definition. +(In general we prefer to ``$k$-ball" to ``$k$-disk", but ``ball-like" doesn't roll off +the tongue as well as "disk-like".) + +Another thing we need a name for is the ability to rotate morphisms around in various ways. +For 2-categories, ``pivotal" is a standard term for what we mean. +A more general term is ``duality", but duality comes in various flavors and degrees. +We are mainly interested in a very strong version of duality, where the available ways of +rotating $k$-morphisms correspond to all the ways of rotating $k$-balls. +We sometimes refer to this as ``strong duality", and sometimes we consider it to be implied +by ``disk-like". +(But beware: disks can come in various flavors, and some of them (such as framed disks) +don't actually imply much duality.) +Another possibility here is ``pivotal $n$-category". + +Finally, we need a general name for isomorphisms between balls, where the balls could be +piecewise linear or smooth or topological or Spin or framed or etc., or some combination thereof. +We have chosen to use ``homeomorphism" for the appropriate sort of isomorphism, so the reader should +keep in mind that ``homeomorphism" could mean PL homeomorphism or diffeomorphism (and so on) +depending on context. + \subsection{Thanks and acknowledgements} % attempting to make this chronological rather than alphabetical We'd like to thank @@ -508,6 +555,5 @@ Alexander Kirillov for many interesting and useful conversations. During this work, Kevin Walker has been at Microsoft Station Q, and Scott Morrison has been at Microsoft Station Q and the Miller Institute for Basic Research at UC Berkeley. We'd like to thank the Aspen Center for Physics for the pleasant and productive -% "conducive" needs an object; "conducive to blah" environment provided there during the final preparation of this manuscript.