blob: comparison text/ncat.tex

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-:4e3a152f4936
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 It is easy to show that examples of topological origin
 (e.g.\ categories whose morphisms are maps into spaces or decorated balls),
 satisfy our axioms.
 For examples of a more purely algebraic origin, one would typically need the combinatorial
 results that we have avoided here.
+See \S\ref{n-cat-names} for a discussion of $n$-category terminology.
 %\nn{Say something explicit about Lurie's work here?
 %It seems like this was something that Dan Freed wanted explaining when we talked to him in Aspen}
 \medskip
 \end{tikzpicture}
 $$
 \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 topological category.)
+where we construct a traditional category from a disk-like category.)
 Define a {\it pinched product} to be a map
 \[
 	\pi: E\to X
 \]
 such that $E$ is a $k{+}m$-ball, $X$ is a $k$-ball ($m\ge 1$), and $\pi$ is locally modeled
 balls and, at level $n$, quotienting out by the local relations:
 \begin{align*}
 \cC_{\cF,U}(B^k) & = \begin{cases}\cF(B) & \text{when $k<n$,} \\ \cF(B) / U(B) & \text{when $k=n$.}\end{cases}
 \end{align*}
 This $n$-category can be thought of as the local part of the fields.
-Conversely, given a topological $n$-category we can construct a system of fields via
+Conversely, given a disk-like $n$-category we can construct a system of fields via
 a colimit construction; see \S \ref{ss:ncat_fields} below.
 In the $n$-category axioms above we have intermingled data and properties for expository reasons.
 Here's a summary of the definition which segregates the data from the properties.
 define $\cC(X; c) = \bc^\cE_*(X\times F; c)$
 where $\bc^\cE_*$ denotes the blob complex based on $\cE$.
 \end{example}
 This example will be used in Theorem \ref{thm:product} below, which allows us to compute the blob complex of a product.
-Notice that with $F$ a point, the above example is a construction turning a topological
+Notice that with $F$ a point, the above example is a construction turning an ordinary
 $n$-category $\cC$ into an $A_\infty$ $n$-category.
 We think of this as providing a ``free resolution"
-of the topological $n$-category.
+of the ordinary $n$-category.
 %\nn{say something about cofibrant replacements?}
 In fact, there is also a trivial, but mostly uninteresting, way to do this:
 we can think of each vector space associated to an $n$-ball as a chain complex concentrated in degree $0$,
 and take $\CD{B}$ to act trivially.
-Beware that the ``free resolution" of the topological $n$-category $\pi_{\leq n}(T)$
+Beware that the ``free resolution" of the ordinary $n$-category $\pi_{\leq n}(T)$
 is not the $A_\infty$ $n$-category $\pi^\infty_{\leq n}(T)$.
 It's easy to see that with $n=0$, the corresponding system of fields is just
 linear combinations of connected components of $T$, and the local relations are trivial.
 There's no way for the blob complex to magically recover all the data of $\pi^\infty_{\leq 0}(T) \iso C_* T$.
 The remaining data for the $A_\infty$ $n$-category
 --- composition and $\Diff(X\to X')$ action ---
 also comes from the $\cE\cB_n$ action on $A$.
 %\nn{should we spell this out?}
-Conversely, one can show that a topological $A_\infty$ $n$-category $\cC$, where the $k$-morphisms
+Conversely, one can show that a disk-like $A_\infty$ $n$-category $\cC$, where the $k$-morphisms
 $\cC(X)$ are trivial (single point) for $k<n$, gives rise to
 an $\cE\cB_n$-algebra.
 %\nn{The paper is already long; is it worth giving details here?}
 If we apply the homotopy colimit construction of the next subsection to this example,
 in the context of an $m{+}1$-dimensional TQFT.
 Such a $W$ gives rise to a module for the $n$-category associated to $\bd W$.
 This will be explained in more detail as we present the axioms.
 Throughout, we fix an $n$-category $\cC$.
-For all but one axiom, it doesn't matter whether $\cC$ is a topological $n$-category or an $A_\infty$ $n$-category.
+For all but one axiom, it doesn't matter whether $\cC$ is an ordinary $n$-category or an $A_\infty$ $n$-category.
 We state the final axiom, regarding actions of homeomorphisms, differently in the two cases.
 Define a {\it marked $k$-ball} to be a pair $(B, N)$ homeomorphic to the pair
 $$(\text{standard $k$-ball}, \text{northern hemisphere in boundary of standard $k$-ball}).$$
 We call $B$ the ball and $N$ the marking.
 In all other cases ($k>1$ or unoriented or $\text{Pin}_\pm$),
 there is no left/right module distinction.
 \medskip
-We now give some examples of modules over topological and $A_\infty$ $n$-categories.
+We now give some examples of modules over ordinary and $A_\infty$ $n$-categories.
 \begin{example}[Examples from TQFTs]
 \rm
 Continuing Example \ref{ex:ncats-from-tqfts}, with $\cF$ a TQFT, $W$ an $n{-}j$-manifold,
 and $\cF(W)$ the $j$-category associated to $W$.
 \subsection{Modules as boundary labels (colimits for decorated manifolds)}
 \label{moddecss}
-Fix a topological $n$-category or $A_\infty$ $n$-category  $\cC$.
+Fix an ordinary $n$-category or $A_\infty$ $n$-category  $\cC$.
 Let $W$ be a $k$-manifold ($k\le n$),
 let $\{Y_i\}$ be a collection of disjoint codimension 0 submanifolds of $\bd W$,
 and let $\cN = (\cN_i)$ be an assignment of a $\cC$ module $\cN_i$ to $Y_i$.
 We will define a set $\cC(W, \cN)$ using a colimit construction very similar to
 this is much less true for higher dimensional spheres,
 so we prefer the term ``sphere module" for the general case.
 %The results of this subsection are not needed for the rest of the paper,
 %so we will skimp on details in a couple of places. We have included this mostly
-%for the sake of comparing our notion of a topological $n$-category to other definitions.
+%for the sake of comparing our notion of a disk-like $n$-category to other definitions.
 For simplicity, we will assume that $n$-categories are enriched over $\c$-vector spaces.
 The $0$- through $n$-dimensional parts of $\cS$ are various sorts of modules, and we describe
 these first.

changeset 685	8efbd2730ef9
parent 683	240e4abfb405
child 686	fa7042811d84