387 |
387 |
388 \todo{ |
388 \todo{ |
389 Decide if we need a friendlier, skein-module version. |
389 Decide if we need a friendlier, skein-module version. |
390 } |
390 } |
391 |
391 |
392 \subsubsection{Examples} |
392 \subsection{Example (the fundamental $n$-groupoid)} |
393 |
393 We will define $\pi_{\le n}(T)$, the fundamental $n$-groupoid of a topological space $T$. |
394 \nn{can't figure out environment stuff; want no italics} |
394 When $X$ is a $k$-ball with $k<n$, define $\pi_{\le n}(T)(X)$ |
395 |
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396 \noindent |
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397 Example. [Fundamental $n$-groupoid of a space] |
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398 Let $T$ be a topological space. |
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399 Define $\pi_{\le n}(T)(X)$, for $X$ a $k$-ball and $k<n$, |
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400 to be the set of continuous maps from $X$ to $T$. |
395 to be the set of continuous maps from $X$ to $T$. |
401 If $X$ is an $n$-ball, define $\pi_{\le n}(T)(X)$ to be homotopy classes (rel boundary) of such maps. |
396 When $X$ is an $n$-ball, define $\pi_{\le n}(T)(X)$ to be homotopy classes (rel boundary) of such maps. |
402 Define boundary restrictions and gluing in the obvious way. |
397 Define boundary restrictions and gluing in the obvious way. |
403 If $\rho:E\to X$ is a pinched product and $f:X\to T$ is a $k$-morphism, |
398 If $\rho:E\to X$ is a pinched product and $f:X\to T$ is a $k$-morphism, |
404 define the product morphism $\rho^*(f)$ to be $f\circ\rho$. |
399 define the product morphism $\rho^*(f)$ to be $f\circ\rho$. |
405 |
400 |
406 We can also define an $A_\infty$ version $\pi_{\le n}^\infty(T)$ of the fundamental $n$-groupoid. |
401 We can also define an $A_\infty$ version $\pi_{\le n}^\infty(T)$ of the fundamental $n$-groupoid. |
407 Most of the definition is the same as above. |
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408 For $X$ an $n$-ball define $\pi_{\le n}^\infty(T)(X)$ to be the space of all maps from $X$ to $T$ |
402 For $X$ an $n$-ball define $\pi_{\le n}^\infty(T)(X)$ to be the space of all maps from $X$ to $T$ |
409 (if we are enriching over spaces) or the singular chains on that space (if we are enriching over chain complexes). |
403 (if we are enriching over spaces) or the singular chains on that space (if we are enriching over chain complexes). |
410 |
404 |
411 |
405 |
412 \noindent |
406 \subsection{Example (string diagrams)} |
413 Example. [String diagrams] |
407 Fix a `traditional' $n$-category $C$ with strong duality (e.g.\ a pivotal 2-category). |
414 Fix a traditional $n$-category $C$ with strong duality (e.g.\ a pivotal 2-category). |
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415 Let $X$ be a $k$-ball and define $\cS_C(X)$ to be the set of $C$ string diagrams drawn on $X$; |
408 Let $X$ be a $k$-ball and define $\cS_C(X)$ to be the set of $C$ string diagrams drawn on $X$; |
416 that is, certain cell complexes embedded in $X$, with the codimension-$j$ cells labeled by $j$-morphisms of $C$. |
409 that is, certain cell complexes embedded in $X$, with the codimension-$j$ cells labeled by $j$-morphisms of $C$. |
417 If $X$ is an $n$-ball, identify two such string diagrams if they evaluate to the same $n$-morphism of $C$. |
410 If $X$ is an $n$-ball, identify two such string diagrams if they evaluate to the same $n$-morphism of $C$. |
418 Boundary restrictions and gluing are again straightforward to define. |
411 Boundary restrictions and gluing are again straightforward to define. |
419 Define product morphisms via product cell decompositions. |
412 Define product morphisms via product cell decompositions. |
488 |
481 |
489 An explicit realization of the homotopy colimit is provided by the simplices of the functor $\psi_{\cC; W}$. That is, $$\clh{\cC}(W) = \DirectSum_{\bar{x}} \psi_{\cC; W}(x_0)[m],$$ where $\bar{x} = x_0 \leq \cdots \leq x_m$ is a simplex in $\cell(W)$. The differential acts on $(\bar{x},a)$ (here $a \in \psi_{\cC; W}(x_0)$) as |
482 An explicit realization of the homotopy colimit is provided by the simplices of the functor $\psi_{\cC; W}$. That is, $$\clh{\cC}(W) = \DirectSum_{\bar{x}} \psi_{\cC; W}(x_0)[m],$$ where $\bar{x} = x_0 \leq \cdots \leq x_m$ is a simplex in $\cell(W)$. The differential acts on $(\bar{x},a)$ (here $a \in \psi_{\cC; W}(x_0)$) as |
490 $$\bdy (\bar{x},a) = (\bar{x}, \bdy a) + (-1)^{\deg a} \left( (d_0 \bar{x}, g(a)) + \sum_{i=1}^m (-1)^i (d_i \bar{x}, a) \right)$$ |
483 $$\bdy (\bar{x},a) = (\bar{x}, \bdy a) + (-1)^{\deg a} \left( (d_0 \bar{x}, g(a)) + \sum_{i=1}^m (-1)^i (d_i \bar{x}, a) \right)$$ |
491 where $g$ is the gluing map from $x_0$ to $x_1$, and $d_i \bar{x}$ denotes the $i$-th face of the simplex $\bar{x}$. |
484 where $g$ is the gluing map from $x_0$ to $x_1$, and $d_i \bar{x}$ denotes the $i$-th face of the simplex $\bar{x}$. |
492 |
485 |
493 Alternatively, we can take advantage of the product structure on $\cell(W)$ to realize the homotopy colimit as the cone-product polyhedra of the functor $\psi_{\cC;W}$. (A cone-product polyhedra is obtained from a point by successively taking the cone or taking the product with another cone-product polyhedron; just as simplices correspond to linear graphs, cone-product polyheda correspond to directed trees.) A Eilenberg-Zilber subdivision argument shows this is the same as the usual realization. |
486 Alternatively, we can take advantage of the product structure on $\cell(W)$ to realize the homotopy colimit via the cone-product polyhedra in $\cell(W)$. A cone-product polyhedra is obtained from a point by successively taking the cone or taking the product with another cone-product polyhedron. Just as simplices correspond to linear graphs, cone-product polyheda correspond to directed trees: taking cone adds a new root before the existing root, and taking product identifies the roots of several trees. The `local homotopy colimit' is then defined according to the same formula as above, but with $x$ a cone-product polyhedron in $\cell(W)$. |
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487 A Eilenberg-Zilber subdivision argument shows this is the same as the usual realization. |
494 |
488 |
495 When $\cC$ is a topological $n$-category, |
489 When $\cC$ is a topological $n$-category, |
496 the flexibility available in the construction of a homotopy colimit allows |
490 the flexibility available in the construction of a homotopy colimit allows |
497 us to give a much more explicit description of the blob complex which we'll write as $\bc_*(W; \cC)$. |
491 us to give a much more explicit description of the blob complex which we'll write as $\bc_*(W; \cC)$. |
498 |
492 |