35 The axioms for an $n$-category are spread throughout this section. |
35 The axioms for an $n$-category are spread throughout this section. |
36 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}. |
36 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}. |
37 |
37 |
38 Strictly speaking, before we can state the axioms for $k$-morphisms we need all the axioms |
38 Strictly speaking, before we can state the axioms for $k$-morphisms we need all the axioms |
39 for $k{-}1$-morphisms. |
39 for $k{-}1$-morphisms. |
40 So readers who prefer things to be presented in a strictly logical order should read this subsection $n$ times, first imagining that $k=0$, then that $k=1$, and so on until they reach $k=n$. |
40 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 $k=n$. |
41 |
41 |
42 \medskip |
42 \medskip |
43 |
43 |
44 There are many existing definitions of $n$-categories, with various intended uses. |
44 There are many existing definitions of $n$-categories, with various intended uses. |
45 In any such definition, there are sets of $k$-morphisms for each $0 \leq k \leq n$. |
45 In any such definition, there are sets of $k$-morphisms for each $0 \leq k \leq n$. |
832 $W \to W'$ which restricts to the identity on the boundary. |
832 $W \to W'$ which restricts to the identity on the boundary. |
833 For $n=1$ we have the familiar bordism 1-category of $d$-manifolds. |
833 For $n=1$ we have the familiar bordism 1-category of $d$-manifolds. |
834 The case $n=d$ captures the $n$-categorical nature of bordisms. |
834 The case $n=d$ captures the $n$-categorical nature of bordisms. |
835 The case $n > 2d$ captures the full symmetric monoidal $n$-category structure. |
835 The case $n > 2d$ captures the full symmetric monoidal $n$-category structure. |
836 \end{example} |
836 \end{example} |
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837 \begin{remark} |
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838 Working with the smooth bordism category would require careful attention to either collars, corners or halos. |
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839 \end{remark} |
837 |
840 |
838 %\nn{the next example might be an unnecessary distraction. consider deleting it.} |
841 %\nn{the next example might be an unnecessary distraction. consider deleting it.} |
839 |
842 |
840 %\begin{example}[Variation on the above examples] |
843 %\begin{example}[Variation on the above examples] |
841 %We could allow $F$ to have boundary and specify boundary conditions on $X\times \bd F$, |
844 %We could allow $F$ to have boundary and specify boundary conditions on $X\times \bd F$, |