1518 the above antirefinements of the fixed interval $J$, but with the rightmost subinterval $I_m$ always |
1518 the above antirefinements of the fixed interval $J$, but with the rightmost subinterval $I_m$ always |
1519 omitted. |
1519 omitted. |
1520 More specifically, $D\to D'$ is an antirefinement if $D'$ is obtained from $D$ by |
1520 More specifically, $D\to D'$ is an antirefinement if $D'$ is obtained from $D$ by |
1521 gluing subintervals together and/or omitting some of the rightmost subintervals. |
1521 gluing subintervals together and/or omitting some of the rightmost subintervals. |
1522 (See Figure \ref{fig:lmar}.) |
1522 (See Figure \ref{fig:lmar}.) |
1523 \begin{figure}[t]\begin{equation*} |
1523 \begin{figure}[t]$$ |
1524 \mathfig{.6}{tempkw/left-marked-antirefinements} |
1524 \begin{tikzpicture} |
1525 \end{equation*}\caption{Antirefinements of left-marked intervals}\label{fig:lmar}\end{figure} |
1525 \fill (0,0) circle (.1); |
|
1526 \draw (0,0) -- (2,0); |
|
1527 \draw (1,0.1) -- (1,-0.1); |
|
1528 |
|
1529 \draw [->,red] (1,0.25) -- (1,0.75); |
|
1530 |
|
1531 \fill (0,1) circle (.1); |
|
1532 \draw (0,1) -- (2,1); |
|
1533 \end{tikzpicture} |
|
1534 \qquad |
|
1535 \begin{tikzpicture} |
|
1536 \fill (0,0) circle (.1); |
|
1537 \draw (0,0) -- (2,0); |
|
1538 \draw (1,0.1) -- (1,-0.1); |
|
1539 |
|
1540 \draw [->,red] (1,0.25) -- (1,0.75); |
|
1541 |
|
1542 \fill (0,1) circle (.1); |
|
1543 \draw (0,1) -- (1,1); |
|
1544 \end{tikzpicture} |
|
1545 \qquad |
|
1546 \begin{tikzpicture} |
|
1547 \fill (0,0) circle (.1); |
|
1548 \draw (0,0) -- (3,0); |
|
1549 \foreach \x in {0.5, 1.0, 1.25, 1.5, 2.0, 2.5} { |
|
1550 \draw (\x,0.1) -- (\x,-0.1); |
|
1551 } |
|
1552 |
|
1553 \draw [->,red] (1,0.25) -- (1,0.75); |
|
1554 |
|
1555 \fill (0,1) circle (.1); |
|
1556 \draw (0,1) -- (2,1); |
|
1557 \foreach \x in {1.0, 1.5} { |
|
1558 \draw (\x,1.1) -- (\x,0.9); |
|
1559 } |
|
1560 |
|
1561 \end{tikzpicture} |
|
1562 $$ |
|
1563 \caption{Antirefinements of left-marked intervals}\label{fig:lmar}\end{figure} |
1526 |
1564 |
1527 Now we define the chain complex $\hom_\cC(\cX_\cC \to \cY_\cC)$. |
1565 Now we define the chain complex $\hom_\cC(\cX_\cC \to \cY_\cC)$. |
1528 The underlying vector space is |
1566 The underlying vector space is |
1529 \[ |
1567 \[ |
1530 \prod_l \prod_{\olD} \hom[l]\left( |
1568 \prod_l \prod_{\olD} \hom[l]\left( |