507 since we think of the higher homotopies not as morphisms of the $n$-category but |
507 since we think of the higher homotopies not as morphisms of the $n$-category but |
508 rather as belonging to some auxiliary category (like chain complexes) |
508 rather as belonging to some auxiliary category (like chain complexes) |
509 that we are enriching in. |
509 that we are enriching in. |
510 We have decided to call them ``$A_\infty$ $n$-categories", since they are a natural generalization |
510 We have decided to call them ``$A_\infty$ $n$-categories", since they are a natural generalization |
511 of the familiar $A_\infty$ 1-categories. |
511 of the familiar $A_\infty$ 1-categories. |
512 Other possible names include ``homotopy $n$-categories" and ``infinity $n$-categories". |
512 We also considered the names ``homotopy $n$-categories" and ``infinity $n$-categories". |
513 When we need to emphasize that we are talking about an $n$-category which is not $A_\infty$ |
513 When we need to emphasize that we are talking about an $n$-category which is not $A_\infty$ in this sense |
514 we will say ``ordinary $n$-category". |
514 we will say ``ordinary $n$-category". |
515 % small problem: our n-cats are of course strictly associative, since we have more morphisms. |
515 % small problem: our n-cats are of course strictly associative, since we have more morphisms. |
516 % when we say ``associative only up to homotopy" above we are thinking about |
516 % when we say ``associative only up to homotopy" above we are thinking about |
517 % what would happen we we tried to convert to a more traditional n-cat with fewer morphisms |
517 % what would happen we we tried to convert to a more traditional n-cat with fewer morphisms |
518 |
518 |
519 Another distinction we need to make is between our style of definition of $n$-categories and |
519 Another distinction we need to make is between our style of definition of $n$-categories and |
520 more traditional and combinatorial definitions. |
520 more traditional and combinatorial definitions. |
521 We will call instances of our definition ``disk-like $n$-categories", since $n$-dimensional disks |
521 We will call instances of our definition ``disk-like $n$-categories", since $n$-dimensional disks |
522 play a prominent role in the definition. |
522 play a prominent role in the definition. |
523 (In general we prefer to ``$k$-ball" to ``$k$-disk", but ``ball-like" doesn't roll off |
523 (In general we prefer ``$k$-ball" to ``$k$-disk", but ``ball-like" doesn't roll off |
524 the tongue as well as "disk-like".) |
524 the tongue as well as ``disk-like''.) |
525 |
525 |
526 Another thing we need a name for is the ability to rotate morphisms around in various ways. |
526 Another thing we need a name for is the ability to rotate morphisms around in various ways. |
527 For 2-categories, ``pivotal" is a standard term for what we mean. |
527 For 2-categories, ``pivotal" is a standard term for what we mean. |
528 A more general term is ``duality", but duality comes in various flavors and degrees. |
528 A more general term is ``duality", but duality comes in various flavors and degrees. |
529 We are mainly interested in a very strong version of duality, where the available ways of |
529 We are mainly interested in a very strong version of duality, where the available ways of |
530 rotating $k$-morphisms correspond to all the ways of rotating $k$-balls. |
530 rotating $k$-morphisms correspond to all the ways of rotating $k$-balls. |
531 We sometimes refer to this as ``strong duality", and sometimes we consider it to be implied |
531 We sometimes refer to this as ``strong duality", and sometimes we consider it to be implied |
532 by ``disk-like". |
532 by ``disk-like". |
533 (But beware: disks can come in various flavors, and some of them (such as framed disks) |
533 (But beware: disks can come in various flavors, and some of them, such as framed disks, |
534 don't actually imply much duality.) |
534 don't actually imply much duality.) |
535 Another possibility here is ``pivotal $n$-category". |
535 Another possibility considered here was ``pivotal $n$-category", but we prefer to preserve pivotal for its usual sense. It will thus be a theorem that our disk-like 2-categories are equivalent to pivotal 2-categories, c.f. \S \ref{ssec:2-cats}. |
536 |
536 |
537 Finally, we need a general name for isomorphisms between balls, where the balls could be |
537 Finally, we need a general name for isomorphisms between balls, where the balls could be |
538 piecewise linear or smooth or topological or Spin or framed or etc., or some combination thereof. |
538 piecewise linear or smooth or topological or Spin or framed or etc., or some combination thereof. |
539 We have chosen to use ``homeomorphism" for the appropriate sort of isomorphism, so the reader should |
539 We have chosen to use ``homeomorphism" for the appropriate sort of isomorphism, so the reader should |
540 keep in mind that ``homeomorphism" could mean PL homeomorphism or diffeomorphism (and so on) |
540 keep in mind that ``homeomorphism" could mean PL homeomorphism or diffeomorphism (and so on) |