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 |
9 |
10 \subsection{Topological $A_\infty$ categories} |
10 \subsection{Topological $A_\infty$ categories} |
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. |
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. |
12 |
12 |
13 \paragraph{Categories} |
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14 \begin{defn} |
13 \begin{defn} |
15 \label{defn:topological-Ainfty-category}% |
14 \label{defn:topological-Ainfty-category}% |
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 |
15 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 |
17 \begin{itemize} |
16 \begin{itemize} |
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 |
17 \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 |
25 \item the evaluation maps compose, up to a weakly unique homotopy, |
24 \item the evaluation maps compose, up to a weakly unique homotopy, |
26 \item and the evaluation maps are compatible with the gluing maps, up to a weakly unique homotopy. |
25 \item and the evaluation maps are compatible with the gluing maps, up to a weakly unique homotopy. |
27 \end{itemize} |
26 \end{itemize} |
28 \end{defn} |
27 \end{defn} |
29 |
28 |
30 \paragraph{Modules} |
29 Appendix \ref{sec:comparing-A-infty} explains the translation between this definition and the usual one expressed in terms of `associativity up to higher homotopy', as in \cite{MR1854636}. (In this version of the paper, that appendix is incomplete, however.) |
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 $J'$. |
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31 The motivating example is `chains of maps to $M$' for some fixed target space $M$. This is a topological $A_\infty$ category $\Xi_M$ with $\Xi_M(J) = C_*(\Maps(J \to M))$. The gluing maps $\Xi_M(J) \tensor \Xi_M(J') \to \Xi_M(J \cup J')$ takes the product of singular chains, then glues maps to $M$ together; the associativity condition is automatically satisfied. The evaluation map $\ev_{J,J'} : \CD{J \to J'} \tensor \Xi_M(J) \to \Xi_M(J')$ is the composition |
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32 \begin{align*} |
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33 \CD{J \to J'} \tensor C_*(\Maps(J \to M)) & \to C_*(\Diff(J \to J') \times \Maps(J \to M)) \\ & \to C_*(\Maps(J' \to M)), |
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34 \end{align*} |
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35 where the first map is the product of singular chains, and the second is precomposition by the inverse of a diffeomorphism. |
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36 |
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37 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 $J'$. |
32 |
38 |
33 \begin{defn} |
39 \begin{defn} |
34 \label{defn:topological-Ainfty-module}% |
40 \label{defn:topological-Ainfty-module}% |
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 |
41 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 |
36 \begin{itemize} |
42 \begin{itemize} |
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 |
60 \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 |
55 \end{itemize} |
61 \end{itemize} |
56 satisfying the same axioms given for a topological $A_\infty$ category in Definition \ref{defn:topological-Ainfty-category}. |
62 satisfying the same axioms given for a topological $A_\infty$ category in Definition \ref{defn:topological-Ainfty-category}. |
57 \end{defn} |
63 \end{defn} |
58 |
64 |
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65 We now define the tensor product of a left module with a right module. The notion of the self-tensor product of a bimodule is a minor variation which we'll leave to the reader. |
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66 Our definition requires choosing a `fixed' interval, and for simplicity we'll use $[0,1]$, but you should note that the definition is equivariant with respect to diffeomorphisms of this interval. |
59 |
67 |
60 \paragraph{Morphisms} |
68 \begin{defn} |
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69 The tensor product of a left module $\cM$ and a right module $\cN$ over a topological $A_\infty$ category $\cC$, denoted $\cM \tensor_{\cC} \cN$, is a vector space, which we'll specify as the limit of a certain commutative diagram. This (infinite) diagram has vertices indexed by partitions $$[0,1] = [0,x_1] \cup \cdots \cup [x_k,1]$$ and boundary conditions $$a_1, \ldots, a_k \in \Obj(\cC),$$ and arrows labeled by refinements. At each vertex put the vector space $$\cM([0,x_1],0; a_1) \tensor \cC([x_1,x_2];a_1,a_2]) \tensor \cdots \tensor \cC([x_{k-1},x_k];a_{k-1},a_k) \tensor \cN([x_k,1],1;a_k),$$ and on each arrow the corresponding gluing map. Faces of this diagram commute because the gluing maps compose associatively. |
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70 \end{defn} |
61 |
71 |
62 \paragraph{Tensor products} |
72 \newcommand{\lmod}[1]{{}_{#1}{\operatorname{mod}}} |
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73 For completeness, we still need to define morphisms between modules and duals of modules, and explain how the functors $\hom_{\lmod{\cC}}\left(\cM \to -\right)$ and $\cM^* \tensor_{\cC} -$ from $\lmod{\cC}$ to $\Vect$ are naturally isomorphic. We don't actually need this for the present version of the paper, so the half-written discussion has been banished to Appendix \ref{sec:A-infty-hom-and-duals}. |
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74 |
64 \subsection{Homological systems of fields} |
75 \subsection{Homological systems of fields} |
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. |
76 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. |
66 |
77 |
67 |
78 |
170 \end{equation*} |
181 \end{equation*} |
171 |
182 |
172 We remark that if $\cF$ takes values in vector spaces, not chain complexes, then the $d_v$ terms vanish, and this coincides with our earlier definition of blob homomology for (non-homological) systems of fields. |
183 We remark that if $\cF$ takes values in vector spaces, not chain complexes, then the $d_v$ terms vanish, and this coincides with our earlier definition of blob homomology for (non-homological) systems of fields. |
173 |
184 |
174 \todo{We'll quickly check $d^2=0$.} |
185 \todo{We'll quickly check $d^2=0$.} |
175 |
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176 |
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177 \subsection{An intermediate gluing theorem} |
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178 |
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179 \begin{thm}[Gluing, intermediate form] |
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180 Suppose $M = M_1 \cup_Y M_2$ is the union of two submanifolds $M_1$ and $M_2$ along a codimension $1$ manifold $Y$. The blob homology of $M$ can be computed as |
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181 \begin{equation*} |
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182 \cB_*(M, \cF) = \cB_*(([0,1], \{0\}, \{1\}), (\cB_*(Y, \cF), \cB_*(M_1, \cF), \cB_*(M_2, \cF))). |
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183 \end{equation*} |
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184 The right hand side is the blob homology of the interval, using ... |
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185 \end{thm} |
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