<html><head></head><body><div><div><div style="display: none; border: 0px; width: 0px; height: 0px; overflow: hidden; visibility: hidden;"><img src="https://r.superhuman.com/x9v2nFL7pQhxMrXQelf-rQdiTRUu8G_LblOPVhXjFIc1pVwuwVaNlMuoYM692mrzohwnYtpLCCL06030MzfUR9k-Jz0q8LqamDG5ZiM8bPq_druaHdzOwEDQI3g0Qr6-g9tKWkRIwC_dDsBS63M5FKoel5kA9yafoYmNXEpG2Tk4_s6n6Aru6RY.gif" alt=" " width="1" height="0" style="display: none; border: 0px; width: 0px; height: 0px; overflow: hidden; visibility: hidden;"/><!--                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                --></div><div><div><div>@Asad that's a perfectly reasonable way to think about diagonal operations: as the data of a cartesian monoidal category, and the laws are correct in this case. I think we can get away with the additional abstraction to `Biapplicative` in this case, though.<br/></div><div><br/></div><blockquote><div style="text-decoration-color:initial;text-decoration-style:initial;text-decoration-line:none;" class="MessagePane-iframe-wrapper"><div style="text-decoration-skip-ink:initial;text-decoration-color:initial;text-decoration-style:initial;text-decoration-line:none;font-style:initial;" class="Shadowbox"><div><div class="oldSignatureStyling"><div class="ShadowBody"><div class=""><div class=""><div tabindex="-1" class=""><div class="unquoted-content"><p style="margin:0px;" class="">wouldn't the existence of appropriate `forall a. a -> t a a` and `forall a. x -> Unit t` functions pigeonhole it into being "the" cartesian monoidal structure on `->` (and thus naturally isomorphic to `(,)`)?<br/></p></div></div></div></div></div></div></div></div></div></blockquote><div><div><br/></div><div>Only if you chose that particular unit and that particular arrow. But there are other places where having a general `Biapplicative` contraint would make it useful. For example,  i'd like to use this in `smash` with `diag :: a → Smash a a`, defining the adjoining of a point to `a` and creating a diagonal in the subcategory of pointed spaces in Hask, so that I don't have to redefine this as `diag' = quotSmash . view _CanIso . diag . Just`. <br/></div><div><br/></div><div>Cheers,<br/></div><div>Emily</div><div><br/></div><div><br/></div><div><br/></div><div><br/></div></div><div><br/></div></div><br/><div class="gmail_signature"></div></div><br/><div><div class="gmail_quote">On Wed, Sep 16, 2020 at 6:35 PM, Asad Saeeduddin <span dir="ltr"><<a href="mailto:masaeedu@gmail.com" target="_blank">masaeedu@gmail.com</a>></span> wrote:<br/><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex"><div class="gmail_extra"><div class="gmail_quote sh-color-black sh-color" style="null" id="null"><p class="sh-color-black sh-color">Whoops, I just realized I've been responding to Carter
      specifically instead of to the list.<br/>
      <br/>
      I was having some trouble understanding the `unJoin` stuff but I
      read it a few more times and I think I understand it a little
      better now.<br/>
      <br/>
      In my personal experience the "abstracted version" of `x -> (x,
      x)` I use most often looks like this:<br/>
      <br/>
      ```<br/>
    </p>
    <pre class="sh-color-black sh-color">class SymmetricMonoidal t i p => CartesianMonoidal t i p</pre>
    <pre class="sh-color-black sh-color">  where</pre>
    <pre class="sh-color-black sh-color">  duplicate :: p x (x `t` x)</pre>
    <pre class="sh-color-black sh-color">  discard :: p x i

</pre>
    <pre class="sh-color-black sh-color">-- Laws:</pre>
    <pre class="sh-color-black sh-color">-- duplicate >>> first  discard = fwd lunit</pre>
    <pre class="sh-color-black sh-color">-- duplicate >>> second discard = fwd runit

</pre>
    <pre class="sh-color-black sh-color">-- where</pre>
    <pre class="sh-color-black sh-color">-- lunit :: Monoidal t i p => Iso p x (i `t` x)</pre>
    <pre class="sh-color-black sh-color">-- runit :: Monoidal t i p => Iso p x (x `t` i)</pre>
    <p class="sh-color-black sh-color">```<br/>
      <br/>
      i.e. adding a suitable duplicate and discard to some symmetric
      monoidal structure gives us a cartesian monoidal structure.<br/>
      <br/>
      This doesn't really seem to be relevant to what you folks are
      looking for, but it does bring up a question. If some `Bifunctor`
      `t` happens to form a monoidal structure on `->`, wouldn't the
      existence of appropriate `forall a. a -> t a a` and `forall a.
      x -> Unit t` functions pigeonhole it into being "the" cartesian
      monoidal structure on `->` (and thus naturally isomorphic to
      `(,)`)?</p>
    <div class="moz-cite-prefix sh-color-black sh-color">On 9/16/20 5:26 PM, Emily Pillmore
      wrote:<br/>
    </div>
    <blockquote cite="mid:kf5vrsny.e6040fe7-b608-4b4c-9b9a-cca791581d3e@we.are.superhuman.com" type="cite" class="sh-color-black sh-color">
      
      <div class="sh-color-black sh-color">
        <div class="sh-color-black sh-color">
          
          <div class="sh-color-black sh-color">
            <div class="sh-color-black sh-color">
              <div class="sh-color-black sh-color">Nice! <br/>
                <br/>
                That's kind of what I was going for with Carter earlier
                in the day, thanks Matthew. </div>
              <div class="sh-color-black sh-color"><br/>
              </div>
              <div class="sh-color-black sh-color">I think a diagonalization function and functor are
                both very sensible additions to `bifunctors` and
                `Data.Bifunctor`. The theory behind this is sound: The
                diagonalization functor Δ: Hask → Hask^Hask, forms the
                center of the adjoint triple `colim -| Δ -| lim : Hask →
                Hask^Hask`. <br/>
                <br/>
                Certainly the function `diag :: a → (a,a)` is something
                I've seen written in several libraries, and should be
                included in `Data.Tuple` as a `base` function. The clear
                generalization of this function is `diag ::
                Biapplicative f ⇒ a → f a a`. I'm in favor of both
                existing in their separate capacities. </div>
              <div class="sh-color-black sh-color"><br/>
              </div>
              <div class="sh-color-black sh-color">Thoughts? <br/>
              </div>
              <div class="sh-color-black sh-color"><br/>
              </div>
              <div class="sh-color-black sh-color">Emily</div>
            </div>
            <br/>
          </div>
          <br/>
          <div class="sh-color-black sh-color">
            <div class="gmail_quote sh-color-black sh-color">On Wed, Sep 16, 2020 at 3:49 PM,
              Carter Schonwald <span dir="ltr" class="sh-color-black sh-color"><<a href="mailto:carter.schonwald@gmail.com" target="_blank" rel="noopener noreferrer">carter.<wbr/>schonwald@<wbr/>gmail.<wbr/>com</a>></span>
              wrote:<br/>
              <blockquote style="margin:0 0 0
                .8ex;border-left:1px #ccc solid;padding-left:1ex" class="gmail_quote sh-color-black sh-color">
                <div class="gmail_extra sh-color-black sh-color">
                  <div id="null" style="null" class="gmail_quote sh-color-black sh-color">
                    <div class="sh-color-black sh-color" dir="auto">Is
                      the join bipure definition taking advantage of the
                      (a->) monad instance?  Slick!</div>
                    <div class="sh-color-black sh-color" dir="auto"><br/>
                    </div>
                    <div class="sh-color-black sh-color" dir="auto"><br/>
                    </div>
                    <div class="sh-color-black sh-color">
                      <div class="gmail_quote sh-color-black sh-color">
                        <div dir="ltr" class="gmail_attr sh-color-black sh-color">On Wed, Sep 16, 2020 at 3:39 PM
                          Matthew Farkas-Dyck <<a rel="noopener noreferrer" href="mailto:strake888@gmail.com" target="_blank">strake888@<wbr/>gmail.<wbr/>com</a>>
                          wrote:<br/>
                        </div>
                        <blockquote class="gmail_quote sh-color-black sh-color" style="margin:0 0 0
                          .8ex;border-left:1px #ccc
                          solid;padding-left:1ex">We also have<br/>
                          <br/>
                          <br/>
                          <br/>
                          diag = join bipure<br/>
                          <br/>
                          <br/>
                          <br/>
                          and (in pseudo-Haskell)<br/>
                          <br/>
                          <br/>
                          <br/>
                          diag = unJoin . pure<br/>
                          <br/>
                            where<br/>
                          <br/>
                              newtype Join f a = Join { unJoin :: f a a
                          } deriving (Functor)<br/>
                          <br/>
                              deriving instance Biapplicative f =>
                          Applicative (Join f)<br/>
                          <br/>
                          <br/>
                          <br/>
                          The latter seems on its face potentially
                          related to the instance for<br/>
                          <br/>
                          lists of fixed length, but i am not sure how
                          deep the connection may<br/>
                          <br/>
                          be.<br/>
                          <br/>
                        </blockquote>
                      </div>
                    </div>
                    <p class="sh-color-black sh-color">_______________________________________________
                      <br/>
                      Libraries mailing list
                      <br/>
                      <a href="mailto:Libraries@haskell.org" rel="noopener noreferrer" target="_blank">Libraries@<wbr/>haskell.<wbr/>org</a>
                      <br/>
                      <a href="http://mail.haskell.org/cgi-bin/mailman/listinfo/libraries" rel="noopener noreferrer" target="_blank">http:/<wbr/>/<wbr/>mail.<wbr/>haskell.<wbr/>org/<wbr/>cgi-bin/<wbr/>mailman/<wbr/>listinfo/<wbr/>libraries</a></p>
                  </div>
                </div>
              </blockquote>
            </div>
          </div>
          <br/>
        </div>
      </div>
      <br/>
      <fieldset class="mimeAttachmentHeader sh-color-black sh-color"></fieldset>
      <pre wrap="" class="moz-quote-pre sh-color-black sh-color">_______________________________________________
Libraries mailing list
<a href="mailto:Libraries@haskell.org" class="moz-txt-link-abbreviated" target="_blank" rel="noopener noreferrer">Libraries@<wbr/>haskell.<wbr/>org</a>
<a href="http://mail.haskell.org/cgi-bin/mailman/listinfo/libraries" class="moz-txt-link-freetext" target="_blank" rel="noopener noreferrer">http:/<wbr/>/<wbr/>mail.<wbr/>haskell.<wbr/>org/<wbr/>cgi-bin/<wbr/>mailman/<wbr/>listinfo/<wbr/>libraries</a>
</pre>
    </blockquote>
  


<p class="sh-color-black sh-color">_______________________________________________
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