6 In this section, we extend the definition of blob homology to allow \emph{homological systems of fields}. |
6 In this section, we extend the definition of blob homology to allow \emph{homological systems of fields}. |
7 |
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8 We begin with a definition of a \emph{topological $A_\infty$ category}, and then introduce the notion of a homological system of fields. A topological $A_\infty$ category gives a $1$-dimensional homological system of fields. We'll suggest that any good definition of a topological $A_\infty$ $n$-category with duals should allow construction of an $n$-dimensional homological system of fields, but we won't propose any such definition here. Later, we extend the definition of blob homology to allow homological fields as input. These definitions allow us to state and prove a theorem about the blob homology of a product manifold, and an intermediate theorem about gluing, in preparation for the proof of Property \ref{property:gluing}. |
8 We begin with a definition of a \emph{topological $A_\infty$ category}, and then introduce the notion of a homological system of fields. A topological $A_\infty$ category gives a $1$-dimensional homological system of fields. We'll suggest that any good definition of a topological $A_\infty$ $n$-category with duals should allow construction of an $n$-dimensional homological system of fields, but we won't propose any such definition here. Later, we extend the definition of blob homology to allow homological fields as input. These definitions allow us to state and prove a theorem about the blob homology of a product manifold, and an intermediate theorem about gluing, in preparation for the proof of Property \ref{property:gluing}. |
9 |
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10 \subsection{Topological $A_\infty$ categories} |
10 \subsection{Topological $A_\infty$ categories} |
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11 In this section we define a notion of `topological $A_\infty$ category' and sketch an equivalence with the usual definition of $A_\infty$ category. We then define `topological $A_\infty$ modules', and their morphisms and tensor products. |
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12 |
12 First recall the \emph{coloured little intervals operad}. Given a set of labels $\cL$, the operations are indexed by \emph{decompositions of the interval}, each of which is a collection of disjoint subintervals $\{(a_i,b_i)\}_{i=1}^k$ of $[0,1]$, along with a labeling of the complementary regions by $\cL$, $\{l_0, \ldots, l_k\}$. Given two decompositions $\cJ^{(1)}$ and $\cJ^{(2)}$, and an index $m$ such that $l^{(1)}_{m-1} = l^{(2)}_0$ and $l^{(1)}_{m} = l^{(2)}_{k^{(2)}}$, we can form a new decomposition by inserting the intervals of $\cJ^{(2)}$ linearly inside the $m$-th interval of $\cJ^{(1)}$. We call the resulting decomposition $\cJ^{(1)} \circ_m \cJ^{(2)}$. |
13 \paragraph{Categories} |
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14 \begin{defn} |
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15 \label{defn:topological-Ainfty-category}% |
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16 A \emph{topological $A_\infty$ category} $\cC$ has a set of objects $\Obj(\cC)$, and for each interval $J$ and objects $a,b \in \Obj(\cC)$, a chain complex $\cC(J;a,b)$, along with |
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17 \begin{itemize} |
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18 \item for each pair of intervals $J_1$, $J_2$ so that $J_1 \cup_{\text{pt}} J_2$ is also an interval, `gluing' chain maps |
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19 $$gl: \cC(J_1;a,b) \tensor \cC(J_2;b,c) \to \cC(J_1 \cup J_2;a,c),$$ |
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20 \item and `evaluation' chain maps $\CD{J \to J'} \tensor \cC(J;a,b) \to \cC(J';a,b)$ |
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21 \end{itemize} |
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22 such that |
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23 \begin{itemize} |
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24 \item the gluing maps compose strictly associatively, |
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25 \item the evaluation maps compose, up to a weakly unique homotopy, |
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26 \item and the evaluation maps are compatible with the gluing maps, up to a weakly unique homotopy. |
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27 \end{itemize} |
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28 \end{defn} |
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29 |
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30 \paragraph{Modules} |
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31 We now define left-modules, right-modules and bimodules over a topological $A_\infty category$. We'll say that a right-marked interval is a pair $(J,p)$, diffeomorphic to the pair $([0,1],1)$, and similarly for a left-marked interval. Recall in what follows that when we write a union of interval $J \cup J'$, we're implicitly assuming that both intervals are oriented, and that the union glues together the `highest' point of $J$ with the `lowest' point of $J'$. |
13 |
32 |
14 \begin{defn} |
33 \begin{defn} |
15 A \emph{topological $A_\infty$ category} $\cC$ has a set of objects $\Obj(\cC)$ and for each $a,b \in \Obj(\cC)$ a chain complex $\cC_{a,b}$, along with a compatible `composition map' and an `action of families of diffeomorphisms'. |
34 \label{defn:topological-Ainfty-module}% |
16 |
35 A \emph{topological $A_\infty$ left-module} $\cM$ over a topological $A_\infty$ category $\cC$ has for each right-marked interval $(J,p)$ and object $a \in \Obj(\cM)$ a chain complex $\cM(J,p; a)$, along with |
17 A \emph{composition map} $f$ is a family of chain maps, one for each decomposition of the interval, $f_\cJ : A^{\tensor k} \to A$, making $\cC$ into a category over the coloured little intervals operad, with labels $\cL = \Obj(\cC)$. Thus the chain maps satisfy the identity |
36 \begin{itemize} |
18 \begin{equation*} |
37 \item for each right-marked interval $(J,p)$, and interval $J'$ so that $J' \cup J$ is also right-marked interval, `gluing' chain maps |
19 f_{\cJ^{(1)} \circ_m \cJ^{(2)}} = f_{\cJ^{(1)}} \circ (\id^{\tensor m-1} \tensor f_{\cJ^{(2)}} \tensor \id^{\tensor k^{(1)} - m}). |
38 $$gl: \cC(J';a,b) \tensor \cM(J,p;b) \to \cM(J' \cup J,p;a),$$ |
20 \end{equation*} |
39 \item and `evaluation' chain maps $\CD{(J,p) \to (J',p')} \tensor \cM(J,p;a) \to \cM(J',p';a)$ |
21 |
40 \end{itemize} |
22 An \emph{action of families of diffeomorphisms} is a chain map $ev: \CD{[0,1]} \tensor A \to A$, such that \todo{What goes here, if anything?} |
41 satisfying the same axioms given for a topological $A_\infty$ category in Definition \ref{defn:topological-Ainfty-category}. |
23 \begin{enumerate} |
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24 \item The diagram |
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25 \begin{equation*} |
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26 \xymatrix{ |
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27 \CD{[0,1]} \tensor \CD{[0,1]} \tensor A \ar[r]^{\id \tensor ev} \ar[d]^{\circ \tensor \id} & \CD{[0,1]} \tensor A \ar[d]^{ev} \\ |
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28 \CD{[0,1]} \tensor A \ar[r]^{ev} & A |
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29 } |
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30 \end{equation*} |
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31 commutes up to weakly unique \todo{???} homotopy. |
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32 \item If $\phi \in \Diff([0,1])$ and $\cJ$ is a decomposition of the interval, we obtain a new decomposition $\phi(\cJ)$ and a collection $\phi_m \in \Diff([0,1])$ of diffeomorphisms obtained by taking the restrictions $\restrict{\phi}{[a_m,b_m]} : [a_m,b_m] \to [\phi(a_m),\phi(b_m)]$ and pre- and post-composing these with the linear diffeomorphisms $[0,1] \to [a_m,b_m]$ and $[\phi(a_m),\phi(b_m)] \to [0,1]$. We require that |
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33 \begin{equation*} |
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34 \phi(f_\cJ(a_1, \cdots, a_k)) = f_{\phi(\cJ)}(\phi_1(a_1), \cdots, \phi_k(a_k)). |
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35 \end{equation*} |
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36 \end{enumerate} |
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37 \end{defn} |
42 \end{defn} |
38 |
43 |
39 From a topological $A_\infty$ category $\cC$ we can produce a `conventional' $A_\infty$ category $(A, \{m_k\})$ as defined in, for example, \cite{MR1854636}. We'll just describe the algebra case (that is, a category with only one object), as the modifications required to deal with multiple objects are trivial. Define $A = \cC$ as a chain complex (so $m_1 = d$). Define $m_2 : A\tensor A \to A$ by $f_{\{(0,\frac{1}{2}),(\frac{1}{2},1)\}}$. To define $m_3$, we begin by taking the one parameter family $\phi_3$ of diffeomorphisms of $[0,1]$ that interpolates linearly between the identity and the piecewise linear diffeomorphism taking $\frac{1}{4}$ to $\frac{1}{2}$ and $\frac{1}{2}$ to $\frac{3}{4}$, and then define |
44 A right module is the same, replacing right-marked intervals with left-marked intervals, and changing the order of the factors in the gluing maps. |
40 \begin{equation*} |
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41 m_3(a,b,c) = ev(\phi_3, m_2(m_2(a,b), c)). |
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42 \end{equation*} |
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43 |
45 |
44 It's then easy to calculate that |
46 \begin{defn} |
45 \begin{align*} |
47 \label{defn:topological-Ainfty-bimodule}% |
46 d(m_3(a,b,c)) & = ev(d \phi_3, m_2(m_2(a,b),c)) - ev(\phi_3 d m_2(m_2(a,b), c)) \\ |
48 A \emph{topological $A_\infty$ bimodule} $\cM$ over a topological $A_\infty$ category $\cC$ has for each pair of a right-marked interval $(J,p)$ and a left-marked interval $(K,q)$ and object $a,b \in \Obj(\cM)$ a chain complex $\cM(J,p,K,q; a,b)$, along with |
47 & = ev( \phi_3(1), m_2(m_2(a,b),c)) - ev(\phi_3(0), m_2 (m_2(a,b),c)) - \\ & \qquad - ev(\phi_3, m_2(m_2(da, b), c) + (-1)^{\deg a} m_2(m_2(a, db), c) + \\ & \qquad \quad + (-1)^{\deg a+\deg b} m_2(m_2(a, b), dc) \\ |
49 \begin{itemize} |
48 & = m_2(a , m_2(b,c)) - m_2(m_2(a,b),c) - \\ & \qquad - m_3(da,b,c) + (-1)^{\deg a + 1} m_3(a,db,c) + \\ & \qquad \quad + (-1)^{\deg a + \deg b + 1} m_3(a,b,dc), \\ |
50 \item for each pair of marked intervals $(J,p)$ and $(K,q)$, for each interval $J'$ so that $J' \cup J$ is also right-marked interval, a `gluing' chain maps |
49 \intertext{and thus that} |
51 $$gl: \cC(J';a',a) \tensor \cM(J,p,K,q;a,b) \to \cM(J' \cup J,p,K,q;a',b),$$ |
50 m_1 \circ m_3 & = m_2 \circ (\id \tensor m_2) - m_2 \circ (m_2 \tensor \id) - \\ & \qquad - m_3 \circ (m_1 \tensor \id \tensor \id) - m_3 \circ (\id \tensor m_1 \tensor \id) - m_3 \circ (\id \tensor \id \tensor m_1) |
52 and for each interval $K'$ so that $K \cup K'$ is also a left-marked interval, maps |
51 \end{align*} |
53 $$gl: \cM(J,p,K,q;a,b) \tensor \cC(K';b,b') \to \cM(J,p,K \cup K',q;a,b'),$$ |
52 as required (c.f. \cite[p. 6]{MR1854636}). |
54 \item and `evaluation' chain maps $\CD{(J,p) \to (J',p')} \tensor \cM(J,p,K,q;a,b) \to \cM(J',p',K,q;a,b)$ and |
53 \todo{then the general case.} |
55 \end{itemize} |
54 We won't describe a reverse construction (producing a topological $A_\infty$ category from a `conventional' $A_\infty$ category), but we presume that this will be easy for the experts. |
56 satisfying the same axioms given for a topological $A_\infty$ category in Definition \ref{defn:topological-Ainfty-category}. |
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57 \end{defn} |
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58 |
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59 |
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60 \paragraph{Morphisms} |
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61 |
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62 \paragraph{Tensor products} |
55 |
63 |
56 \subsection{Homological systems of fields} |
64 \subsection{Homological systems of fields} |
57 A homological system of fields $\cF$ is nothing more than a system of fields in the category $\Kom$ of complexes of vector spaces; that is, the set of top level fields with given boundary conditions is always a complex. |
65 A homological system of fields $\cF$ is nothing more than a system of fields in the category $\Kom$ of complexes of vector spaces; that is, the set of top level fields with given boundary conditions is always a complex. |
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66 |
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67 |
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68 |
59 A topological $A_\infty$ category $\cC$ gives rise to a one dimensional homological system of fields. The functor $\cF_0$ simply assigns the set of objects of $\cC$ to a point. |
69 A topological $A_\infty$ category $\cC$ gives rise to a one dimensional homological system of fields. The functor $\cF_0$ simply assigns the set of objects of $\cC$ to a point. |
60 For a $1$-manifold $X$, define a \emph{decomposition of $X$} with labels in $\cL$ as a (possibly empty) set of disjoint closed intervals $\{J\}$ in $X$, and a labeling of the complementary regions by elements of $\cL$. |
70 For a $1$-manifold $X$, define a \emph{decomposition of $X$} with labels in $\cL$ as a (possibly empty) set of disjoint closed intervals $\{J\}$ in $X$, and a labeling of the complementary regions by elements of $\cL$. |
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71 |
62 The functor $\cF_1$ assigns to a $1$-manifold $X$ the vector space |
72 The functor $\cF_1$ assigns to a $1$-manifold $X$ the vector space |