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77 of morphisms). |
77 of morphisms). |
78 The 0-sphere is unusual among spheres in that it is disconnected. |
78 The 0-sphere is unusual among spheres in that it is disconnected. |
79 Correspondingly, for 1-morphisms it makes sense to distinguish between domain and range. |
79 Correspondingly, for 1-morphisms it makes sense to distinguish between domain and range. |
80 (Actually, this is only true in the oriented case, with 1-morphsims parameterized |
80 (Actually, this is only true in the oriented case, with 1-morphsims parameterized |
81 by oriented 1-balls.) |
81 by oriented 1-balls.) |
82 For $k>1$ and in the presence of strong duality the division into domain and range makes less sense. For example, in a pivotal tensor category, there are natural isomorphisms $\Hom{}{A}{B \tensor C} \isoto \Hom{}{A \tensor B^*}{C}$, etc. (sometimes called ``Frobenius reciprocity''), which canonically identify all the morphism spaces which have the same boundary. We prefer to not make the distinction in the first place. |
82 For $k>1$ and in the presence of strong duality the division into domain and range makes less sense. For example, in a pivotal tensor category, there are natural isomorphisms $\Hom{}{A}{B \tensor C} \isoto \Hom{}{B^* \tensor A}{C}$, etc. (sometimes called ``Frobenius reciprocity''), which canonically identify all the morphism spaces which have the same boundary. We prefer to not make the distinction in the first place. |
83 |
83 |
84 Instead, we combine the domain and range into a single entity which we call the |
84 Instead, we combine the domain and range into a single entity which we call the |
85 boundary of a morphism. |
85 boundary of a morphism. |
86 Morphisms are modeled on balls, so their boundaries are modeled on spheres: |
86 Morphisms are modeled on balls, so their boundaries are modeled on spheres: |
87 |
87 |
478 %The universal (colimit) construction becomes our generalized definition of blob homology. |
478 %The universal (colimit) construction becomes our generalized definition of blob homology. |
479 %Need to explain how it relates to the old definition.} |
479 %Need to explain how it relates to the old definition.} |
480 |
480 |
481 \medskip |
481 \medskip |
482 |
482 |
483 \subsection{Examples of $n$-categories}\ \ |
483 \subsection{Examples of $n$-categories} |
484 |
484 \label{ss:ncat-examples} |
485 \nn{these examples need to be fleshed out a bit more} |
485 |
486 |
486 |
487 We now describe several classes of examples of $n$-categories satisfying our axioms. |
487 We now describe several classes of examples of $n$-categories satisfying our axioms. |
488 |
488 |
489 \begin{example}[Maps to a space] |
489 \begin{example}[Maps to a space] |
490 \rm |
490 \rm |
543 Finally, we describe a version of the bordism $n$-category suitable to our definitions. |
543 Finally, we describe a version of the bordism $n$-category suitable to our definitions. |
544 |
544 |
545 \nn{should also include example of ncats coming from TQFTs, or refer ahead to where we discuss that example} |
545 \nn{should also include example of ncats coming from TQFTs, or refer ahead to where we discuss that example} |
546 |
546 |
547 \newcommand{\Bord}{\operatorname{Bord}} |
547 \newcommand{\Bord}{\operatorname{Bord}} |
548 \begin{example}[The bordism $n$-category] |
548 \begin{example}[The bordism $n$-category, plain version] |
549 \rm |
549 \rm |
550 \label{ex:bordism-category} |
550 \label{ex:bordism-category} |
551 For a $k$-ball or $k$-sphere $X$, $k<n$, define $\Bord^n(X)$ to be the set of all $k$-dimensional |
551 For a $k$-ball or $k$-sphere $X$, $k<n$, define $\Bord^n(X)$ to be the set of all $k$-dimensional |
552 submanifolds $W$ of $X\times \Real^\infty$ such that the projection $W \to X$ is transverse |
552 submanifolds $W$ of $X\times \Real^\infty$ such that the projection $W \to X$ is transverse |
553 to $\bd X$. |
553 to $\bd X$. |
565 %%to think of these guys as affording a representation |
565 %%to think of these guys as affording a representation |
566 %%of the $n{+}1$-category associated to $\bd F$.} |
566 %%of the $n{+}1$-category associated to $\bd F$.} |
567 %\end{example} |
567 %\end{example} |
568 |
568 |
569 |
569 |
570 We have two main examples of $A_\infty$ $n$-categories, coming from maps to a target space and from the blob complex. |
570 %We have two main examples of $A_\infty$ $n$-categories, coming from maps to a target space and from the blob complex. |
571 |
571 |
572 \begin{example}[Chains of maps to a space] |
572 \begin{example}[Chains of maps to a space] |
573 \rm |
573 \rm |
574 \label{ex:chains-of-maps-to-a-space} |
574 \label{ex:chains-of-maps-to-a-space} |
575 We can modify Example \ref{ex:maps-to-a-space} above to define the fundamental $A_\infty$ $n$-category $\pi^\infty_{\le n}(T)$ of a topological space $T$. |
575 We can modify Example \ref{ex:maps-to-a-space} above to define the fundamental $A_\infty$ $n$-category $\pi^\infty_{\le n}(T)$ of a topological space $T$. |
595 |
595 |
596 This example will be essential for Theorem \ref{product_thm} 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 $n$-category into an $A_\infty$ $n$-category. We think of this as providing a `free resolution' of the topological $n$-category. \todo{Say more here!} In fact, there is also a trivial 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. |
596 This example will be essential for Theorem \ref{product_thm} 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 $n$-category into an $A_\infty$ $n$-category. We think of this as providing a `free resolution' of the topological $n$-category. \todo{Say more here!} In fact, there is also a trivial 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. |
597 |
597 |
598 Be careful that the `free resolution' of the topological $n$-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 $\pi^\infty_{\leq 0}(T) \iso C_* T$. |
598 Be careful that the `free resolution' of the topological $n$-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 $\pi^\infty_{\leq 0}(T) \iso C_* T$. |
599 |
599 |
600 \begin{example} |
600 \begin{example}[The bordism $n$-category, $A_\infty$ version] |
601 \nn{should add $\infty$ version of bordism $n$-cat} |
601 \rm |
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602 \label{ex:bordism-category-ainf} |
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603 blah blah \nn{to do...} |
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604 \end{example} |
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605 |
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606 |
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607 \begin{example}[$E_n$ algebras] |
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608 \rm |
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609 \label{ex:e-n-alg} |
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610 Let $\cE\cB_n$ be the operad of smooth embeddings of $k$ (little) |
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611 copies of the standard $n$-ball $B^n$ into another (big) copy of $B^n$. |
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612 $\cE\cB_n$ is homotopy equivalent to the standard framed little $n$-ball operad. |
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613 (By shrining the little balls, we see that both are homotopic to the space of $k$ framed points |
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614 in $B^n$.) |
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615 |
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616 Let $A$ be an $\cE\cB_n$-algebra. |
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617 We will define an $A_\infty$ $n$-category $\cC^A$. |
602 \end{example} |
618 \end{example} |
603 |
619 |
604 |
620 |
605 |
621 |
606 |
622 |