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96 %% For titles, only capitalize the first letter |
96 %% For titles, only capitalize the first letter |
97 %% \title{Almost sharp fronts for the surface quasi-geostrophic equation} |
97 %% \title{Almost sharp fronts for the surface quasi-geostrophic equation} |
98 |
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99 \title{Higher categories, colimits and the blob complex} |
99 \title{Higher categories, colimits, and the blob complex} |
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101 |
102 %% Enter authors via the \author command. |
102 %% Enter authors via the \author command. |
103 %% Use \affil to define affiliations. |
103 %% Use \affil to define affiliations. |
104 %% (Leave no spaces between author name and \affil command) |
104 %% (Leave no spaces between author name and \affil command) |
169 %% \subsubsection{} |
169 %% \subsubsection{} |
170 |
170 |
171 \dropcap{T}he aim of this paper is to describe a derived category analogue of topological quantum field theories. |
171 \dropcap{T}he aim of this paper is to describe a derived category analogue of topological quantum field theories. |
172 |
172 |
173 For our purposes, an $n{+}1$-dimensional TQFT is a locally defined system of |
173 For our purposes, an $n{+}1$-dimensional TQFT is a locally defined system of |
174 invariants of manifolds of dimensions 0 through $n+1$. In particular, |
174 invariants of manifolds of dimensions 0 through $n{+}1$. In particular, |
175 the TQFT invariant $A(Y)$ of a closed $k$-manifold $Y$ is a linear $(n{-}k)$-category. |
175 the TQFT invariant $A(Y)$ of a closed $k$-manifold $Y$ is a linear $(n{-}k)$-category. |
176 If $Y$ has boundary then $A(Y)$ is a collection of $(n{-}k)$-categories which afford |
176 If $Y$ has boundary then $A(Y)$ is a collection of $(n{-}k)$-categories which afford |
177 a representation of the $(n{-}k{+}1)$-category $A(\bd Y)$. |
177 a representation of the $(n{-}k{+}1)$-category $A(\bd Y)$. |
178 (See \cite{1009.5025} and \cite{kw:tqft}; |
178 (See \cite{1009.5025} and \cite{kw:tqft}; |
179 for a more homotopy-theoretic point of view see \cite{0905.0465}.) |
179 for a more homotopy-theoretic point of view see \cite{0905.0465}.) |
237 conjecture on Hochschild cochains and the little 2-disks operad. |
237 conjecture on Hochschild cochains and the little 2-disks operad. |
238 |
238 |
239 Of course, there are currently many interesting alternative notions of $n$-category and of TQFT. |
239 Of course, there are currently many interesting alternative notions of $n$-category and of TQFT. |
240 We note that our $n$-categories are both more and less general |
240 We note that our $n$-categories are both more and less general |
241 than the ``fully dualizable" ones which play a prominent role in \cite{0905.0465}. |
241 than the ``fully dualizable" ones which play a prominent role in \cite{0905.0465}. |
242 They are more general in that we make no duality assumptions in the top dimension $n+1$. |
242 They are more general in that we make no duality assumptions in the top dimension $n{+}1$. |
243 They are less general in that we impose stronger duality requirements in dimensions 0 through $n$. |
243 They are less general in that we impose stronger duality requirements in dimensions 0 through $n$. |
244 Thus our $n$-categories correspond to $(n{+}\epsilon)$-dimensional unoriented or oriented TQFTs, while |
244 Thus our $n$-categories correspond to $(n{+}\epsilon)$-dimensional unoriented or oriented TQFTs, while |
245 Lurie's (fully dualizable) $n$-categories correspond to $(n{+}1)$-dimensional framed TQFTs. |
245 Lurie's (fully dualizable) $n$-categories correspond to $(n{+}1)$-dimensional framed TQFTs. |
246 |
246 |
247 At several points we only sketch an argument briefly; full details can be found in \cite{1009.5025}. |
247 At several points we only sketch an argument briefly; full details can be found in \cite{1009.5025}. |
544 $W_1$ and $W_2$ to obtain a permissible decomposition of $W_1 \sqcup W_2$. |
544 $W_1$ and $W_2$ to obtain a permissible decomposition of $W_1 \sqcup W_2$. |
545 |
545 |
546 An $n$-category $\cC$ determines |
546 An $n$-category $\cC$ determines |
547 a functor $\psi_{\cC;W}$ from $\cell(W)$ to the category of sets |
547 a functor $\psi_{\cC;W}$ from $\cell(W)$ to the category of sets |
548 (possibly with additional structure if $k=n$). |
548 (possibly with additional structure if $k=n$). |
549 Each $k$-ball $X$ of a decomposition $y$ of $W$ has its boundary decomposed into $k{-}1$-balls, |
549 Each $k$-ball $X$ of a decomposition $y$ of $W$ has its boundary decomposed into $k{-}1$-manifolds, |
550 and there is a subset $\cC(X)\spl \subset \cC(X)$ of morphisms whose boundaries |
550 and there is a subset $\cC(X)\spl \subset \cC(X)$ of morphisms whose boundaries |
551 are splittable along this decomposition. |
551 are splittable along this decomposition. |
552 |
552 |
553 \begin{defn} |
553 \begin{defn} |
554 Define the functor $\psi_{\cC;W} : \cell(W) \to \Set$ as follows. |
554 Define the functor $\psi_{\cC;W} : \cell(W) \to \Set$ as follows. |
555 For a decomposition $x = \bigsqcup_a X_a$ in $\cell(W)$, $\psi_{\cC;W}(x)$ is the subset |
555 For a decomposition $x = \bigsqcup_a X_a$ in $\cell(W)$, $\psi_{\cC;W}(x)$ is the subset |
556 \begin{equation*} |
556 \begin{equation*} |
557 %\label{eq:psi-C} |
557 %\label{eq:psi-C} |
558 \psi_{\cC;W}(x) \subset \prod_a \cC(X_a)\spl |
558 \psi_{\cC;W}(x) \subset \prod_a \cC(X_a)\spl |
559 \end{equation*} |
559 \end{equation*} |
560 where the restrictions to the various pieces of shared boundaries amongst the cells |
560 where the restrictions to the various pieces of shared boundaries amongst the balls |
561 $X_a$ all agree (this is a fibered product of all the labels of $k$-cells over the labels of $k-1$-cells). |
561 $X_a$ all agree (this is a fibered product of all the labels of $k$-balls over the labels of $k-1$-manifolds). |
562 When $k=n$, the `subset' and `product' in the above formula should be |
562 When $k=n$, the `subset' and `product' in the above formula should be |
563 interpreted in the appropriate enriching category. |
563 interpreted in the appropriate enriching category. |
564 If $x$ is a refinement of $y$, the map $\psi_{\cC;W}(x) \to \psi_{\cC;W}(y)$ is given by the composition maps of $\cC$. |
564 If $x$ is a refinement of $y$, the map $\psi_{\cC;W}(x) \to \psi_{\cC;W}(y)$ is given by the composition maps of $\cC$. |
565 \end{defn} |
565 \end{defn} |
566 |
566 |