diff -r 91d32d0cb2ef -r c0cdde54913a text/ncat.tex --- a/text/ncat.tex Wed May 25 11:08:16 2011 -0600 +++ b/text/ncat.tex Fri May 27 13:43:20 2011 -0600 @@ -33,11 +33,16 @@ \medskip The axioms for an $n$-category are spread throughout this section. -Collecting these together, an $n$-category is a gadget satisfying Axioms \ref{axiom:morphisms}, \ref{nca-boundary}, \ref{axiom:composition}, \ref{nca-assoc}, \ref{axiom:product} and \ref{axiom:extended-isotopies}; for an $A_\infty$ $n$-category, we replace Axiom \ref{axiom:extended-isotopies} with Axiom \ref{axiom:families}. +Collecting these together, an $n$-category is a gadget satisfying Axioms \ref{axiom:morphisms}, +\ref{nca-boundary}, \ref{axiom:composition}, \ref{nca-assoc}, \ref{axiom:product} and +\ref{axiom:extended-isotopies}; for an $A_\infty$ $n$-category, we replace +Axiom \ref{axiom:extended-isotopies} with Axiom \ref{axiom:families}. +\nn{need to revise this after we're done rearranging the a-inf and enriched stuff} Strictly speaking, before we can state the axioms for $k$-morphisms we need all the axioms for $k{-}1$-morphisms. -Readers who prefer things to be presented in a strictly logical order should read this subsection $n+1$ times, first setting $k=0$, then $k=1$, and so on until they reach $k=n$. +Readers who prefer things to be presented in a strictly logical order should read this +subsection $n+1$ times, first setting $k=0$, then $k=1$, and so on until they reach $k=n$. \medskip @@ -52,7 +57,8 @@ Still other definitions (see, for example, \cite{MR2094071}) model the $k$-morphisms on more complicated combinatorial polyhedra. -For our definition, we will allow our $k$-morphisms to have any shape, so long as it is homeomorphic to the standard $k$-ball. +For our definition, we will allow our $k$-morphisms to have any shape, so long as it is +homeomorphic to the standard $k$-ball. Thus we associate a set of $k$-morphisms $\cC_k(X)$ to any $k$-manifold $X$ homeomorphic to the standard $k$-ball. By ``a $k$-ball" we mean any $k$-manifold which is homeomorphic to the @@ -141,17 +147,6 @@ while the second is the ordinary boundary of manifolds. Given $c\in\cl{\cC}(\bd(X))$, we will write $\cC(X; c)$ for $\bd^{-1}(c)$, those morphisms with specified boundary $c$. -Most of the examples of $n$-categories we are interested in are enriched in the following sense. -The various sets of $n$-morphisms $\cC(X; c)$, for all $n$-balls $X$ and -all $c\in \cl{\cC}(\bd X)$, have the structure of an object in some auxiliary symmetric monoidal category -with sufficient limits and colimits -(e.g.\ vector spaces, or modules over some ring, or chain complexes), -%\nn{actually, need both disj-union/sum and product/tensor-product; what's the name for this sort of cat?} -and all the structure maps of the $n$-category should be compatible with the auxiliary -category structure. -Note that this auxiliary structure is only in dimension $n$; if $\dim(Y) < n$ then -$\cC(Y; c)$ is just a plain set. - \medskip In order to simplify the exposition we have concentrated on the case of @@ -239,10 +234,10 @@ Next we consider composition of morphisms. For $n$-categories which lack strong duality, one usually considers -$k$ different types of composition of $k$-morphisms, each associated to a different direction. +$k$ different types of composition of $k$-morphisms, each associated to a different ``direction". (For example, vertical and horizontal composition of 2-morphisms.) In the presence of strong duality, these $k$ distinct compositions are subsumed into -one general type of composition which can be in any ``direction". +one general type of composition which can be in any direction. \begin{axiom}[Composition] \label{axiom:composition} @@ -258,10 +253,9 @@ \] which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions to the intersection of the boundaries of $B$ and $B_i$. -If $k < n$, -or if $k=n$ and we are in the $A_\infty$ case, +If $k < n$ we require that $\gl_Y$ is injective. -(For $k=n$ in the ordinary (non-$A_\infty$) case, see below.) +%(For $k=n$ see below.) \end{axiom} \begin{figure}[t] \centering @@ -401,7 +395,7 @@ \caption{Examples of pinched products}\label{pinched_prods} \end{figure} The need for a strengthened version will become apparent in Appendix \ref{sec:comparing-defs} -where we construct a traditional category from a disk-like category. +where we construct a traditional 2-category from a disk-like 2-category. For example, ``half-pinched" products of 1-balls are used to construct weak identities for 1-morphisms in 2-categories. We also need fully-pinched products to define collar maps below (see Figure \ref{glue-collar}). @@ -660,6 +654,24 @@ Let $\Homeo_\bd(X)$ denote homeomorphisms of $X$ which fix $\bd X$ and $C_*(\Homeo_\bd(X))$ denote the singular chains on this space. +\nn{begin temp relocation} + +Most of the examples of $n$-categories we are interested in are enriched in the following sense. +The various sets of $n$-morphisms $\cC(X; c)$, for all $n$-balls $X$ and +all $c\in \cl{\cC}(\bd X)$, have the structure of an object in some auxiliary symmetric monoidal category +with sufficient limits and colimits +(e.g.\ vector spaces, or modules over some ring, or chain complexes), +%\nn{actually, need both disj-union/sum and product/tensor-product; what's the name for this sort of cat?} +and all the structure maps of the $n$-category should be compatible with the auxiliary +category structure. +Note that this auxiliary structure is only in dimension $n$; if $\dim(Y) < n$ then +$\cC(Y; c)$ is just a plain set. + +\nn{$k=n$ injectivity for a-inf (necessary?)} +or if $k=n$ and we are in the $A_\infty$ case, + + +\nn{end temp relocation} %\addtocounter{axiom}{-1} \begin{axiom}[\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act in dimension $n$.]