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</script></head><body><div id="package-header"><ul class="links" id="page-menu"><li><a href="src/Control-Comonad-Density.html">Source</a></li><li><a href="index.html">Contents</a></li><li><a href="doc-index.html">Index</a></li></ul><p class="caption">kan-extensions-4.2.3: Kan extensions, Kan lifts, various forms of the Yoneda lemma, and (co)density (co)monads</p></div><div id="content"><div id="module-header"><table class="info"><tr><th>Copyright</th><td>(C) 2008-2011 Edward Kmett</td></tr><tr><th>License</th><td>BSD-style (see the file LICENSE)</td></tr><tr><th>Maintainer</th><td>Edward Kmett <ekmett@gmail.com></td></tr><tr><th>Stability</th><td>experimental</td></tr><tr><th>Portability</th><td>non-portable (GADTs, MPTCs)</td></tr><tr><th>Safe Haskell</th><td>Safe</td></tr><tr><th>Language</th><td>Haskell98</td></tr></table><p class="caption">Control.Comonad.Density</p></div><div id="description"><p class="caption">Description</p><div class="doc"><p>The <code><a href="Control-Comonad-Density.html#t:Density">Density</a></code> <code><a href="file:///usr/share/doc/libghc-comonad-doc/html/Control-Comonad.html#t:Comonad">Comonad</a></code> for a <code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.8.2.0/Data-Functor.html#t:Functor">Functor</a></code> (aka the 'Comonad generated by a <code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.8.2.0/Data-Functor.html#t:Functor">Functor</a></code>)
The <code><a href="Control-Comonad-Density.html#t:Density">Density</a></code> term dates back to Dubuc''s 1974 thesis. The term
<code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.8.2.0/Control-Monad.html#t:Monad">Monad</a></code> genererated by a <code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.8.2.0/Data-Functor.html#t:Functor">Functor</a></code> dates back to 1972 in Street''s
''Formal Theory of Monads''.</p><p>The left Kan extension of a <code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.8.2.0/Data-Functor.html#t:Functor">Functor</a></code> along itself (<code><code><a href="Data-Functor-Kan-Lan.html#t:Lan">Lan</a></code> f f</code>) forms a <code><a href="file:///usr/share/doc/libghc-comonad-doc/html/Control-Comonad.html#t:Comonad">Comonad</a></code>. This is
that <code><a href="file:///usr/share/doc/libghc-comonad-doc/html/Control-Comonad.html#t:Comonad">Comonad</a></code>.</p></div></div><div id="synopsis"><p id="control.syn" class="caption expander" onclick="toggleSection('syn')">Synopsis</p><ul id="section.syn" class="hide" onclick="toggleSection('syn')"><li class="src short"><span class="keyword">data</span> <a href="#t:Density">Density</a> k a <span class="keyword">where</span><ul class="subs"><li><a href="#v:Density">Density</a> :: (k b -> a) -> k b -> <a href="Control-Comonad-Density.html#t:Density">Density</a> k a</li></ul></li><li class="src short"><a href="#v:liftDensity">liftDensity</a> :: <a href="file:///usr/share/doc/libghc-comonad-doc/html/Control-Comonad.html#t:Comonad">Comonad</a> w => w a -> <a href="Control-Comonad-Density.html#t:Density">Density</a> w a</li><li class="src short"><a href="#v:densityToAdjunction">densityToAdjunction</a> :: <a href="file:///usr/share/doc/libghc-adjunctions-doc/html/Data-Functor-Adjunction.html#t:Adjunction">Adjunction</a> f g => <a href="Control-Comonad-Density.html#t:Density">Density</a> f a -> f (g a)</li><li class="src short"><a href="#v:adjunctionToDensity">adjunctionToDensity</a> :: <a href="file:///usr/share/doc/libghc-adjunctions-doc/html/Data-Functor-Adjunction.html#t:Adjunction">Adjunction</a> f g => f (g a) -> <a href="Control-Comonad-Density.html#t:Density">Density</a> f a</li><li class="src short"><a href="#v:densityToLan">densityToLan</a> :: <a href="Control-Comonad-Density.html#t:Density">Density</a> f a -> <a href="Data-Functor-Kan-Lan.html#t:Lan">Lan</a> f f a</li><li class="src short"><a href="#v:lanToDensity">lanToDensity</a> :: <a href="Data-Functor-Kan-Lan.html#t:Lan">Lan</a> f f a -> <a href="Control-Comonad-Density.html#t:Density">Density</a> f a</li></ul></div><div id="interface"><h1>Documentation</h1><div class="top"><p class="src"><span class="keyword">data</span> <a name="t:Density" class="def">Density</a> k a <span class="keyword">where</span> <a href="src/Control-Comonad-Density.html#Density" class="link">Source</a></p><div class="subs constructors"><p class="caption">Constructors</p><table><tr><td class="src"><a name="v:Density" class="def">Density</a> :: (k b -> a) -> k b -> <a href="Control-Comonad-Density.html#t:Density">Density</a> k a</td><td class="doc empty"> </td></tr></table></div><div class="subs instances"><p id="control.i:Density" class="caption collapser" onclick="toggleSection('i:Density')">Instances</p><div id="section.i:Density" class="show"><table><tr><td class="src clearfix"><span class="inst-left"><a href="file:///usr/share/doc/libghc-comonad-doc/html/Control-Comonad-Trans-Class.html#t:ComonadTrans">ComonadTrans</a> <a href="Control-Comonad-Density.html#t:Density">Density</a></span> <a href="src/Control-Comonad-Density.html#line-57" class="link">Source</a></td><td class="doc empty"> </td></tr><tr><td class="src clearfix"><span class="inst-left"><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.8.2.0/Data-Functor.html#t:Functor">Functor</a> (<a href="Control-Comonad-Density.html#t:Density">Density</a> f)</span> <a href="src/Control-Comonad-Density.html#line-43" class="link">Source</a></td><td class="doc empty"> </td></tr><tr><td class="src clearfix"><span class="inst-left"><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.8.2.0/Control-Applicative.html#t:Applicative">Applicative</a> f => <a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.8.2.0/Control-Applicative.html#t:Applicative">Applicative</a> (<a href="Control-Comonad-Density.html#t:Density">Density</a> f)</span> <a href="src/Control-Comonad-Density.html#line-66" class="link">Source</a></td><td class="doc empty"> </td></tr><tr><td class="src clearfix"><span class="inst-left"><a href="file:///usr/share/doc/libghc-comonad-doc/html/Control-Comonad.html#t:Comonad">Comonad</a> (<a href="Control-Comonad-Density.html#t:Density">Density</a> f)</span> <a href="src/Control-Comonad-Density.html#line-51" class="link">Source</a></td><td class="doc empty"> </td></tr><tr><td class="src clearfix"><span class="inst-left"><a href="file:///usr/share/doc/libghc-semigroupoids-doc/html/Data-Functor-Apply.html#t:Apply">Apply</a> f => <a href="file:///usr/share/doc/libghc-semigroupoids-doc/html/Data-Functor-Apply.html#t:Apply">Apply</a> (<a href="Control-Comonad-Density.html#t:Density">Density</a> f)</span> <a href="src/Control-Comonad-Density.html#line-61" class="link">Source</a></td><td class="doc empty"> </td></tr><tr><td class="src clearfix"><span class="inst-left"><a href="file:///usr/share/doc/libghc-semigroupoids-doc/html/Data-Functor-Extend.html#t:Extend">Extend</a> (<a href="Control-Comonad-Density.html#t:Density">Density</a> f)</span> <a href="src/Control-Comonad-Density.html#line-47" class="link">Source</a></td><td class="doc empty"> </td></tr></table></div></div></div><div class="top"><p class="src"><a name="v:liftDensity" class="def">liftDensity</a> :: <a href="file:///usr/share/doc/libghc-comonad-doc/html/Control-Comonad.html#t:Comonad">Comonad</a> w => w a -> <a href="Control-Comonad-Density.html#t:Density">Density</a> w a <a href="src/Control-Comonad-Density.html#liftDensity" class="link">Source</a></p><div class="doc"><p>The natural transformation from a <code><code><a href="file:///usr/share/doc/libghc-comonad-doc/html/Control-Comonad.html#t:Comonad">Comonad</a></code> w</code> to the <code><a href="file:///usr/share/doc/libghc-comonad-doc/html/Control-Comonad.html#t:Comonad">Comonad</a></code> generated by <code>w</code> (forwards).</p><p>This is merely a right-inverse (section) of <code><a href="file:///usr/share/doc/libghc-comonad-doc/html/Control-Comonad-Trans-Class.html#v:lower">lower</a></code>, rather than a full inverse.</p><pre><code><a href="file:///usr/share/doc/libghc-comonad-doc/html/Control-Comonad-Trans-Class.html#v:lower">lower</a></code> . <code><a href="Control-Comonad-Density.html#v:liftDensity">liftDensity</a></code> ≡ <code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.8.2.0/Data-Function.html#v:id">id</a></code>
</pre></div></div><div class="top"><p class="src"><a name="v:densityToAdjunction" class="def">densityToAdjunction</a> :: <a href="file:///usr/share/doc/libghc-adjunctions-doc/html/Data-Functor-Adjunction.html#t:Adjunction">Adjunction</a> f g => <a href="Control-Comonad-Density.html#t:Density">Density</a> f a -> f (g a) <a href="src/Control-Comonad-Density.html#densityToAdjunction" class="link">Source</a></p><div class="doc"><p>The Density <code><a href="file:///usr/share/doc/libghc-comonad-doc/html/Control-Comonad.html#t:Comonad">Comonad</a></code> of a left adjoint is isomorphic to the <code><a href="file:///usr/share/doc/libghc-comonad-doc/html/Control-Comonad.html#t:Comonad">Comonad</a></code> formed by that <code><a href="file:///usr/share/doc/libghc-adjunctions-doc/html/Data-Functor-Adjunction.html#t:Adjunction">Adjunction</a></code>.</p><p>This isomorphism is witnessed by <code><a href="Control-Comonad-Density.html#v:densityToAdjunction">densityToAdjunction</a></code> and <code><a href="Control-Comonad-Density.html#v:adjunctionToDensity">adjunctionToDensity</a></code>.</p><pre><code><a href="Control-Comonad-Density.html#v:densityToAdjunction">densityToAdjunction</a></code> . <code><a href="Control-Comonad-Density.html#v:adjunctionToDensity">adjunctionToDensity</a></code> ≡ <code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.8.2.0/Data-Function.html#v:id">id</a></code>
<code><a href="Control-Comonad-Density.html#v:adjunctionToDensity">adjunctionToDensity</a></code> . <code><a href="Control-Comonad-Density.html#v:densityToAdjunction">densityToAdjunction</a></code> ≡ <code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.8.2.0/Data-Function.html#v:id">id</a></code>
</pre></div></div><div class="top"><p class="src"><a name="v:adjunctionToDensity" class="def">adjunctionToDensity</a> :: <a href="file:///usr/share/doc/libghc-adjunctions-doc/html/Data-Functor-Adjunction.html#t:Adjunction">Adjunction</a> f g => f (g a) -> <a href="Control-Comonad-Density.html#t:Density">Density</a> f a <a href="src/Control-Comonad-Density.html#adjunctionToDensity" class="link">Source</a></p></div><div class="top"><p class="src"><a name="v:densityToLan" class="def">densityToLan</a> :: <a href="Control-Comonad-Density.html#t:Density">Density</a> f a -> <a href="Data-Functor-Kan-Lan.html#t:Lan">Lan</a> f f a <a href="src/Control-Comonad-Density.html#densityToLan" class="link">Source</a></p></div><div class="top"><p class="src"><a name="v:lanToDensity" class="def">lanToDensity</a> :: <a href="Data-Functor-Kan-Lan.html#t:Lan">Lan</a> f f a -> <a href="Control-Comonad-Density.html#t:Density">Density</a> f a <a href="src/Control-Comonad-Density.html#lanToDensity" class="link">Source</a></p><div class="doc"><p>The <code><a href="Control-Comonad-Density.html#t:Density">Density</a></code> <code><a href="file:///usr/share/doc/libghc-comonad-doc/html/Control-Comonad.html#t:Comonad">Comonad</a></code> of a <code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.8.2.0/Data-Functor.html#t:Functor">Functor</a></code> <code>f</code> is obtained by taking the left Kan extension
(<code><a href="Data-Functor-Kan-Lan.html#t:Lan">Lan</a></code>) of <code>f</code> along itself. This isomorphism is witnessed by <code><a href="Control-Comonad-Density.html#v:lanToDensity">lanToDensity</a></code> and <code><a href="Control-Comonad-Density.html#v:densityToLan">densityToLan</a></code></p><pre><code><a href="Control-Comonad-Density.html#v:lanToDensity">lanToDensity</a></code> . <code><a href="Control-Comonad-Density.html#v:densityToLan">densityToLan</a></code> ≡ <code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.8.2.0/Data-Function.html#v:id">id</a></code>
<code><a href="Control-Comonad-Density.html#v:densityToLan">densityToLan</a></code> . <code><a href="Control-Comonad-Density.html#v:lanToDensity">lanToDensity</a></code> ≡ <code><a href="file:///usr/share/doc/ghc-doc/html/libraries/base-4.8.2.0/Data-Function.html#v:id">id</a></code>
</pre></div></div></div></div><div id="footer"><p>Produced by <a href="http://www.haskell.org/haddock/">Haddock</a> version 2.16.1</p></div></body></html>
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