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nodes/rx-gph-constructions.tex
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nodes/rx-gph-constructions.tex
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\begin{definition}[Opposite reflexive graph]\label[definition]{def:op-rx-gph}
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We define the \DefEmph{opposite} of a reflexive graph $\gA$ as follows:
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\begin{align*}
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\vrt{\gA\Op}
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&:\equiv
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\vrt{\gA\Op}
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&:\equiv
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\vrt{\gA}
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\\
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x \Edge{\gA\Op} y
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&:\equiv
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\\
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x \Edge{\gA\Op} y
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&:\equiv
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y\Edge{\gA} x
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\\
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\\
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\Rx{\gA\Op}{x}
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&:\equiv
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&:\equiv
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\Rx{\gA}{x}
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\end{align*}
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\end{definition}
···
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\begin{definition}[Total opposite of a displayed reflexive graph]\label[definition]{def:tot-op-disp-rx-gph}
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Let $\gA$ be a reflexive graph, and let $\gB$ be a displayed reflexive graph over $\gA$. We define the \DefEmph{total opposite} $\gB\TotOp$ of $\gB$ to be the following displayed reflexive graph over $\gA\Op$:
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\begin{align*}
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\vrt{\gB\TotOp}\prn{x}
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&:\equiv
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\vrt{\gB\TotOp}\prn{x}
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&:\equiv
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\vrt{\gB}\prn{x}
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\\
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u \Edge{\gB\TotOp}[p] v
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&:\equiv
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u \Edge{\gB\TotOp}[p] v
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&:\equiv
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v\Edge{\gB}[p]u
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\\
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\DRx{\gB\TotOp}{x}{u}
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&:\equiv
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\\
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\DRx{\gB\TotOp}{x}{u}
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&:\equiv
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\DRx{\gB}{x}{u}
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\end{align*}
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\end{definition}
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Note that \cref{def:tot-op-disp-rx-gph} does not define the actual ``opposite'' of a displayed reflexive graph $\gB$, which would naturally have the same base as $\gB$; opposites of arbitrary displayed reflexive graphs do not make sense (for the same reason that B\'enabou's definition of opposites applies only to displayed categories that are additionally fibrations).
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Note that \cref{def:tot-op-disp-rx-gph} does not define the actual ``opposite'' of a displayed reflexive graph $\gB$, which would naturally have the same base as $\gB$; opposites of arbitrary displayed reflexive graphs do not make sense (for the same reason that B\'enabou's definition of opposites applies only to displayed categories that are additionally fibrations).
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\cref{obs:op-rx-gph-involution} extends to the following duality involution on displayed reflexive graphs vs \cref{def:tot-op-disp-rx-gph}.
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\begin{observation}[Duality involution for displayed reflexive graphs]\label[observation]{def:disp-rx-gph-duality}
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The operation sending a displayed reflexive graph to its total opposite is definitionally involutive, \ie we have $\prn{\gB\TotOp}\TotOp \equiv \gB$.
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\end{observation}
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\end{xsect}