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lower adjoints preserve sups
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For those familiar with (covariant) Galois connections, you may have noticed that they can be viewed as categorical adjunctions. A Galois connection is a pair of maps between posets $X$ and $Y$ $$ f_{\bullet}: X \rightleftharpoons Y: f^{\bullet}$$ such that $f_{\bullet} x \preceq y$ if and only if $x \preceq f^{\bullet} y$. If we view $X$ and $Y$ as small categories, then order-preserving maps are simply functors between such categories as $$Hom(f_\bullet x, y) \cong Hom(x, f^\bullet y)$$ is equivalent to the condition above (naturality is trivial).

In the case of posets, there is an explicit formula to calculate the right adjoint from the left adjoint if one exists (sim. left adjoint from right adjoint).

You can check that $$f^\bullet (y) = \bigvee f_{\bullet}^{-1}(\downarrow y)$$ is a right adjoint (if $f_\bullet$ preserves meetsjoins I believe??).

Now here is my question: Are there other areas of mathematics or other examples where an explicit formula for a right adjoint in terms of the left adjoint appears?

References most welcome. Thanks!

For those familiar with (covariant) Galois connections, you may have noticed that they can be viewed as categorical adjunctions. A Galois connection is a pair of maps between posets $X$ and $Y$ $$ f_{\bullet}: X \rightleftharpoons Y: f^{\bullet}$$ such that $f_{\bullet} x \preceq y$ if and only if $x \preceq f^{\bullet} y$. If we view $X$ and $Y$ as small categories, then order-preserving maps are simply functors between such categories as $$Hom(f_\bullet x, y) \cong Hom(x, f^\bullet y)$$ is equivalent to the condition above (naturality is trivial).

In the case of posets, there is an explicit formula to calculate the right adjoint from the left adjoint if one exists (sim. left adjoint from right adjoint).

You can check that $$f^\bullet (y) = \bigvee f_{\bullet}^{-1}(\downarrow y)$$ is a right adjoint (if $f_\bullet$ preserves meets I believe??).

Now here is my question: Are there other areas of mathematics or other examples where an explicit formula for a right adjoint in terms of the left adjoint appears?

References most welcome. Thanks!

For those familiar with (covariant) Galois connections, you may have noticed that they can be viewed as categorical adjunctions. A Galois connection is a pair of maps between posets $X$ and $Y$ $$ f_{\bullet}: X \rightleftharpoons Y: f^{\bullet}$$ such that $f_{\bullet} x \preceq y$ if and only if $x \preceq f^{\bullet} y$. If we view $X$ and $Y$ as small categories, then order-preserving maps are simply functors between such categories as $$Hom(f_\bullet x, y) \cong Hom(x, f^\bullet y)$$ is equivalent to the condition above (naturality is trivial).

In the case of posets, there is an explicit formula to calculate the right adjoint from the left adjoint if one exists (sim. left adjoint from right adjoint).

You can check that $$f^\bullet (y) = \bigvee f_{\bullet}^{-1}(\downarrow y)$$ is a right adjoint (if $f_\bullet$ preserves joins I believe??).

Now here is my question: Are there other areas of mathematics or other examples where an explicit formula for a right adjoint in terms of the left adjoint appears?

References most welcome. Thanks!

Source Link
Hans
Hans
  • 147
  • 8

A formula for a right adjoint in terms of a left

For those familiar with (covariant) Galois connections, you may have noticed that they can be viewed as categorical adjunctions. A Galois connection is a pair of maps between posets $X$ and $Y$ $$ f_{\bullet}: X \rightleftharpoons Y: f^{\bullet}$$ such that $f_{\bullet} x \preceq y$ if and only if $x \preceq f^{\bullet} y$. If we view $X$ and $Y$ as small categories, then order-preserving maps are simply functors between such categories as $$Hom(f_\bullet x, y) \cong Hom(x, f^\bullet y)$$ is equivalent to the condition above (naturality is trivial).

In the case of posets, there is an explicit formula to calculate the right adjoint from the left adjoint if one exists (sim. left adjoint from right adjoint).

You can check that $$f^\bullet (y) = \bigvee f_{\bullet}^{-1}(\downarrow y)$$ is a right adjoint (if $f_\bullet$ preserves meets I believe??).

Now here is my question: Are there other areas of mathematics or other examples where an explicit formula for a right adjoint in terms of the left adjoint appears?

References most welcome. Thanks!