55 and so on. |
55 and so on. |
56 (This allows for strict associativity; see \cite{ulrike-tillmann-2008,0909.2212}.) |
56 (This allows for strict associativity; see \cite{ulrike-tillmann-2008,0909.2212}.) |
57 Still other definitions (see, for example, \cite{MR2094071}) |
57 Still other definitions (see, for example, \cite{MR2094071}) |
58 model the $k$-morphisms on more complicated combinatorial polyhedra. |
58 model the $k$-morphisms on more complicated combinatorial polyhedra. |
59 |
59 |
60 For our definition, we will allow our $k$-morphisms to have any shape, so long as it is |
60 For our definition, we will allow our $k$-morphisms to have {\it any} shape, so long as it is |
61 homeomorphic to the standard $k$-ball. |
61 homeomorphic to the standard $k$-ball. |
62 Thus we associate a set of $k$-morphisms $\cC_k(X)$ to any $k$-manifold $X$ homeomorphic |
62 Thus we associate a set of $k$-morphisms $\cC_k(X)$ to any $k$-manifold $X$ homeomorphic |
63 to the standard $k$-ball. |
63 to the standard $k$-ball. |
64 By ``a $k$-ball" we mean any $k$-manifold which is homeomorphic to the |
64 |
|
65 Below, we will use ``a $k$-ball" to mean any $k$-manifold which is homeomorphic to the |
65 standard $k$-ball. |
66 standard $k$-ball. |
66 We {\it do not} assume that it is equipped with a |
67 We {\it do not} assume that such $k$-balls are equipped with a |
67 preferred homeomorphism to the standard $k$-ball, and the same applies to ``a $k$-sphere" below. |
68 preferred homeomorphism to the standard $k$-ball. |
|
69 The same applies to ``a $k$-sphere" below. |
68 |
70 |
69 Given a homeomorphism $f:X\to Y$ between $k$-balls (not necessarily fixed on |
71 Given a homeomorphism $f:X\to Y$ between $k$-balls (not necessarily fixed on |
70 the boundary), we want a corresponding |
72 the boundary), we want a corresponding |
71 bijection of sets $f:\cC_k(X)\to \cC_k(Y)$. |
73 bijection of sets $f:\cC_k(X)\to \cC_k(Y)$. |
72 (This will imply ``strong duality", among other things.) Putting these together, we have |
74 (This will imply ``strong duality", among other things.) Putting these together, we have |
238 $$ |
240 $$ |
239 \cC(X) \trans E \xrightarrow{\bdy} \cl{\cC}(\bdy X)\trans E \xrightarrow{\res} \cC(B_i) |
241 \cC(X) \trans E \xrightarrow{\bdy} \cl{\cC}(\bdy X)\trans E \xrightarrow{\res} \cC(B_i) |
240 .$$ |
242 .$$ |
241 These restriction maps can be thought of as |
243 These restriction maps can be thought of as |
242 domain and range maps, relative to the choice of splitting $\bd X = B_1 \cup_E B_2$. |
244 domain and range maps, relative to the choice of splitting $\bd X = B_1 \cup_E B_2$. |
243 These restriction maps in fact have their image in the subset $\cC(B_i)\trans E$, |
245 %%%% the next sentence makes no sense to me, even though I'm probably the one who wrote it -- KW |
244 and so to emphasize this we will sometimes write the restriction map as $\cC(X)\trans E \to \cC(B_i)\trans E$. |
246 \noop{These restriction maps in fact have their image in the subset $\cC(B_i)\trans E$, |
|
247 and so to emphasize this we will sometimes write the restriction map as $\cC(X)\trans E \to \cC(B_i)\trans E$.} |
245 |
248 |
246 |
249 |
247 Next we consider composition of morphisms. |
250 Next we consider composition of morphisms. |
248 For $n$-categories which lack strong duality, one usually considers |
251 For $n$-categories which lack strong duality, one usually considers |
249 $k$ different types of composition of $k$-morphisms, each associated to a different ``direction". |
252 $k$ different types of composition of $k$-morphisms, each associated to a different ``direction". |
407 \caption{Examples of pinched products}\label{pinched_prods} |
410 \caption{Examples of pinched products}\label{pinched_prods} |
408 \end{figure} |
411 \end{figure} |
409 The need for a strengthened version will become apparent in Appendix \ref{sec:comparing-defs} |
412 The need for a strengthened version will become apparent in Appendix \ref{sec:comparing-defs} |
410 where we construct a traditional 2-category from a disk-like 2-category. |
413 where we construct a traditional 2-category from a disk-like 2-category. |
411 For example, ``half-pinched" products of 1-balls are used to construct weak identities for 1-morphisms |
414 For example, ``half-pinched" products of 1-balls are used to construct weak identities for 1-morphisms |
412 in 2-categories. |
415 in 2-categories (see \S\ref{ssec:2-cats}). |
413 We also need fully-pinched products to define collar maps below (see Figure \ref{glue-collar}). |
416 We also need fully-pinched products to define collar maps below (see Figure \ref{glue-collar}). |
414 |
417 |
415 Define a {\it pinched product} to be a map |
418 Define a {\it pinched product} to be a map |
416 \[ |
419 \[ |
417 \pi: E\to X |
420 \pi: E\to X |