372 \gl_Y : \cC(B_1)_E \times_{\cC(Y)} \cC(B_2)_E \to \cC(B)_E |
372 \gl_Y : \cC(B_1)_E \times_{\cC(Y)} \cC(B_2)_E \to \cC(B)_E |
373 \] |
373 \] |
374 which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
374 which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
375 to the intersection of the boundaries of $B$ and $B_i$. |
375 to the intersection of the boundaries of $B$ and $B_i$. |
376 If $k < n$, |
376 If $k < n$, |
377 or if $k=n$ and we are in the $A_\infty$ case, |
377 or if $k=n$ and we are in the $A_\infty$ case \nn{Kevin: remind me why we ask this?}, |
378 we require that $\gl_Y$ is injective. |
378 we require that $\gl_Y$ is injective. |
379 (For $k=n$ in the isotopy $n$-category case, see below.) |
379 (For $k=n$ in the isotopy $n$-category case, see below. \nn{where?}) |
380 \end{axiom} |
380 \end{axiom} |
381 |
381 |
382 \begin{axiom}[Strict associativity] \label{nca-assoc}\label{axiom:associativity} |
382 \begin{axiom}[Strict associativity] \label{nca-assoc}\label{axiom:associativity} |
383 The gluing maps above are strictly associative. |
383 The gluing maps above are strictly associative. |
384 Given any decomposition of a ball $B$ into smaller balls |
384 Given any decomposition of a ball $B$ into smaller balls |
579 larger categories of all $k$-manifolds (again, with homeomorphisms). |
579 larger categories of all $k$-manifolds (again, with homeomorphisms). |
580 In fact, the axioms stated above already require such an extension to $k$-spheres for $k<n$. |
580 In fact, the axioms stated above already require such an extension to $k$-spheres for $k<n$. |
581 |
581 |
582 The natural construction achieving this is a colimit along the poset of permissible decompositions. |
582 The natural construction achieving this is a colimit along the poset of permissible decompositions. |
583 For an isotopy $n$-category $\cC$, |
583 For an isotopy $n$-category $\cC$, |
584 we denote the extension to all manifolds by $\cl{\cC}$. On a $k$-manifold $W$, with $k \leq n$, |
584 we will denote the extension to all manifolds by $\cl{\cC}$. On a $k$-manifold $W$, with $k \leq n$, |
585 this is defined to be the colimit of the functor $\psi_{\cC;W}$. |
585 this is defined to be the colimit along $\cell(W)$ of the functor $\psi_{\cC;W}$. |
586 Note that Axioms \ref{axiom:composition} and \ref{axiom:associativity} |
586 Note that Axioms \ref{axiom:composition} and \ref{axiom:associativity} |
587 imply that $\cl{\cC}(X) \iso \cC(X)$ when $X$ is a $k$-ball with $k<n$. |
587 imply that $\cl{\cC}(X) \iso \cC(X)$ when $X$ is a $k$-ball with $k<n$. |
588 Recall that given boundary conditions $c \in \cl{\cC}(\bdy X)$, for $X$ an $n$-ball, |
588 Recall that given boundary conditions $c \in \cl{\cC}(\bdy X)$, for $X$ an $n$-ball, |
589 the set $\cC(X;c)$ is a vector space. Using this, we note that for $c \in \cl{\cC}(\bdy X)$, |
589 the set $\cC(X;c)$ is a vector space. Using this, we note that for $c \in \cl{\cC}(\bdy W)$, |
590 for $X$ an arbitrary $n$-manifold, the set $\cl{\cC}(X;c) = \bdy^{-1} (c)$ inherits the structure of a vector space. |
590 for $W$ an arbitrary $n$-manifold, the set $\cl{\cC}(W;c) = \bdy^{-1} (c)$ inherits the structure of a vector space. |
591 These are the usual TQFT skein module invariants on $n$-manifolds. |
591 These are the usual TQFT skein module invariants on $n$-manifolds. |
592 |
592 |
593 We can now give a straightforward but rather abstract definition of the blob complex of an $n$-manifold $W$ |
593 We can now give a straightforward but rather abstract definition of the blob complex of an $n$-manifold $W$ |
594 with coefficients in the $n$-category $\cC$ as the homotopy colimit along $\cell(W)$ |
594 with coefficients in the $n$-category $\cC$ as the homotopy colimit along $\cell(W)$ |
595 of the functor $\psi_{\cC; W}$ described above. We write this as $\clh{\cC}(W)$. |
595 of the functor $\psi_{\cC; W}$ described above. We write this as $\clh{\cC}(W)$. |
607 product with another cone-product polyhedron. Just as simplices correspond to linear directed graphs, |
607 product with another cone-product polyhedron. Just as simplices correspond to linear directed graphs, |
608 cone-product polyheda correspond to directed trees: taking cone adds a new root before the existing root, |
608 cone-product polyheda correspond to directed trees: taking cone adds a new root before the existing root, |
609 and taking product identifies the roots of several trees. |
609 and taking product identifies the roots of several trees. |
610 The `local homotopy colimit' is then defined according to the same formula as above, but with $\bar{x}$ a cone-product polyhedron in $\cell(W)$. |
610 The `local homotopy colimit' is then defined according to the same formula as above, but with $\bar{x}$ a cone-product polyhedron in $\cell(W)$. |
611 The differential acts on $(\bar{x},a)$ both on $a$ and on $\bar{x}$, applying the appropriate gluing map to $a$ when required. |
611 The differential acts on $(\bar{x},a)$ both on $a$ and on $\bar{x}$, applying the appropriate gluing map to $a$ when required. |
612 A Eilenberg-Zilber subdivision argument shows this is the same as the usual realization. |
612 A Eilenberg-Zilber subdivision argument shows this is the same as the usual realization. In the local homotopy colimit we can further require that any composition of morphisms in a directed tree is not expressible as a product. |
613 |
613 |
614 %When $\cC$ is a topological $n$-category, |
614 %When $\cC$ is a topological $n$-category, |
615 %the flexibility available in the construction of a homotopy colimit allows |
615 %the flexibility available in the construction of a homotopy colimit allows |
616 %us to give a much more explicit description of the blob complex which we'll write as $\bc_*(W; \cC)$. |
616 %us to give a much more explicit description of the blob complex which we'll write as $\bc_*(W; \cC)$. |
617 %\todo{either need to explain why this is the same, or significantly rewrite this section} |
617 %\todo{either need to explain why this is the same, or significantly rewrite this section} |
618 When $\cC$ is the topological $n$-category based on string diagrams for a traditional |
618 When $\cC$ is the isotopy $n$-category based on string diagrams for a traditional |
619 $n$-category $C$, |
619 $n$-category $C$, |
620 one can show \cite{1009.5025} that the above two constructions of the homotopy colimit |
620 one can show \cite{1009.5025} that the above two constructions of the homotopy colimit |
621 are equivalent to the more concrete construction which we describe next, and which we denote $\bc_*(W; \cC)$. |
621 are equivalent to the more concrete construction which we describe next, and which we denote $\bc_*(W; \cC)$. |
622 Roughly speaking, the generators of $\bc_k(W; \cC)$ are string diagrams on $W$ together with |
622 Roughly speaking, the generators of $\bc_k(W; \cC)$ are string diagrams on $W$ together with |
623 a configuration of $k$ balls (or ``blobs") in $W$ whose interiors are pairwise disjoint or nested. |
623 a configuration of $k$ balls (or ``blobs") in $W$ whose interiors are pairwise disjoint or nested. |