2016 \cite{MR1718089}. |
2016 \cite{MR1718089}. |
2017 |
2017 |
2018 \medskip |
2018 \medskip |
2019 |
2019 |
2020 We can define marked pinched products $\pi:E\to M$ of marked balls similarly to the |
2020 We can define marked pinched products $\pi:E\to M$ of marked balls similarly to the |
2021 plain ball case. A marked pinched product $\pi: E \to M$ is a pinched product (that is, locally modeled on degeneracy maps) which restricts to a map between the markings which is also a pinched product, and in a neighborhood of the markings is the product of the map between the markings with an interval. |
2021 plain ball case. A marked pinched product $\pi: E \to M$ is a pinched product (that is, locally modeled on degeneracy maps) which restricts to a map between the markings which is also a pinched product, and in a neighborhood of the markings is the product of the map between the markings with an interval. (See Figure \ref{fig:marked-pinched-products}.) |
2022 \nn{figure, 2 examples} |
2022 |
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2023 \begin{figure}[ht] |
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2024 \begin{equation*} |
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2025 \begin{tikzpicture} |
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2026 \draw (0,2) -- (2,2.5); |
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2027 \draw (0,2) -- (2,1.5); |
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2028 \draw[line width=2pt] (2,1.5) -- (2,2.5); |
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2029 \draw (0,0) -- (2,0) node[circle,fill,inner sep=2pt] {}; |
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2030 \draw[->] (1,1.5) -- (1,0.25); |
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2031 \end{tikzpicture} |
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2032 \qquad \qquad \qquad |
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2033 \begin{tikzpicture} |
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2034 \draw (2,2.5) -- (0,2.5) -- (0,1.5) -- (2,1.5); |
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2035 \draw[line width=2pt] (2,1.5) -- (2,2.5); |
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2036 \draw (0,0) -- (2,0) node[circle,fill,inner sep=2pt] {}; |
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2037 \draw[->] (1,1.2) -- (1,0.25); |
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2038 \end{tikzpicture} |
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2039 \end{equation*} |
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2040 \caption{Two examples of marked pinched products.} |
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2041 \label{fig:marked-pinched-products} |
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2042 \end{figure} |
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2043 |
2023 Note that a marked pinched product can be decomposed into either |
2044 Note that a marked pinched product can be decomposed into either |
2024 two marked pinched products or a plain pinched product and a marked pinched product. |
2045 two marked pinched products or a plain pinched product and a marked pinched product. |
2025 \nn{should give figure} |
2046 (See Figure \ref{fig:decomposing-marked-pinched-products}.) |
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2047 \begin{figure}[ht] |
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2048 \begin{equation*} |
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2049 \begin{tikzpicture} |
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2050 \draw (0,2) -- (2,2.5); |
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2051 \draw (0,2) -- (2,1.5); |
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2052 \draw[dashed] (1.333,2.333) -- (1.333,1.666); |
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2053 \draw[line width=2pt] (2,1.5) -- (2,2.5); |
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2054 \draw (0,0) -- (2,0) node[circle,fill,inner sep=2pt] {}; |
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2055 \draw[->] (1,1.5) -- (1,0.25); |
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2056 \end{tikzpicture} |
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2057 \qquad \qquad \qquad |
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2058 \begin{tikzpicture} |
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2059 \draw (0,2) -- (2,2.5); |
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2060 \draw (0,2) -- (2,1.5); |
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2061 \draw[dashed] (0.666,2.166) -- (2,1.833); |
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2062 \draw[line width=2pt] (2,1.5) -- (2,2.5); |
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2063 \draw (0,0) -- (2,0) node[circle,fill,inner sep=2pt] {}; |
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2064 \draw[->] (1,1.5) -- (1,0.25); |
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2065 \end{tikzpicture} |
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2066 \end{equation*} |
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2067 \caption{Two examples of decompositions of marked pinched products.} |
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2068 \label{fig:decomposing-marked-pinched-products} |
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2069 \end{figure} |
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2070 |
2026 |
2071 |
2027 \begin{module-axiom}[Product (identity) morphisms] |
2072 \begin{module-axiom}[Product (identity) morphisms] |
2028 For each pinched product $\pi:E\to M$, with $M$ a marked $k$-ball and $E$ a marked |
2073 For each pinched product $\pi:E\to M$, with $M$ a marked $k$-ball and $E$ a marked |
2029 $k{+}m$-ball ($m\ge 1$), |
2074 $k{+}m$-ball ($m\ge 1$), |
2030 there is a map $\pi^*:\cM(M)\to \cM(E)$. |
2075 there is a map $\pi^*:\cM(M)\to \cM(E)$. |