equal
deleted
inserted
replaced
2077 \label{jun23c} |
2077 \label{jun23c} |
2078 \end{figure} |
2078 \end{figure} |
2079 Let $D' = B\cap C$. |
2079 Let $D' = B\cap C$. |
2080 It is not hard too show that the above two maps are mutually inverse. |
2080 It is not hard too show that the above two maps are mutually inverse. |
2081 |
2081 |
2082 \begin{lem} |
2082 \begin{lem} \label{equator-lemma} |
2083 Any two choices of $E$ and $E'$ are related by a series of modifications as above. |
2083 Any two choices of $E$ and $E'$ are related by a series of modifications as above. |
2084 \end{lem} |
2084 \end{lem} |
2085 |
2085 |
2086 \begin{proof} |
2086 \begin{proof} |
2087 (Sketch) |
2087 (Sketch) |
2235 |
2235 |
2236 We define product $n{+}1$-morphisms to be identity maps of modules. |
2236 We define product $n{+}1$-morphisms to be identity maps of modules. |
2237 |
2237 |
2238 To define (binary) composition of $n{+}1$-morphisms, choose the obvious common equator |
2238 To define (binary) composition of $n{+}1$-morphisms, choose the obvious common equator |
2239 then compose the module maps. |
2239 then compose the module maps. |
2240 Associativity of this composition rules follows from repeated application of the adjoint identity between |
2240 The proof that this composition rule is associative is similar to the proof of Lemma \ref{equator-lemma}. |
2241 the maps of Figures \ref{jun23b} and \ref{jun23c}. |
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2242 |
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2243 |
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2244 %\nn{still to do: associativity} |
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2245 |
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2246 \medskip |
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2247 |
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2248 %\nn{Stuff that remains to be done (either below or in an appendix or in a separate section or in |
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2249 %a separate paper): discuss Morita equivalence; functors} |
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2250 |
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2251 |
|