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    <p>`Either a a -> a` is another very handy operation. That and `a
      -> (a, a)` together make up 90% of use cases for `duplicate ::
      p x (t x x)`.<br>
    </p>
    <div class="moz-cite-prefix">On 9/17/20 12:49 AM, Edward Kmett
      wrote:<br>
    </div>
    <blockquote type="cite"
cite="mid:CAJumaK8=fp8KN22QS4qggs2hWo_UuSTeiq5imsJh7isHCmq=xQ@mail.gmail.com">
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      <div dir="ltr">For what it's worth, I'd just like to see a
        no-nonsense 
        <div><br>
        </div>
        <div><font face="monospace">dup : a -> (a,a)</font></div>
        <div><font face="monospace">dup a = (a,a)</font></div>
        <div><br>
        </div>
        <div>in <font face="monospace">Data.Tuple</font>, where it is
          out of the way, but right where you'd expect it to be when
          looking for something for working with tuples.</div>
        <div><br>
        </div>
        <div>Yes, <font face="monospace">bipure</font> and <font
            face="monospace">id &&& id</font> exist, and
          generalize it on two incompatible axes, and if we had a proper
          cartesian category we'd be able to supply this in full
          generality, as a morphism to the diagonal functor, but all of
          these require a level of rocket science to figure out.</div>
        <div><br>
        </div>
        <div>I'd also happily support adding the equivalent in <font
            face="monospace">Data.Either</font> for <font
            face="monospace">Either a a -> a</font>, which invites
          bikeshedding names.</div>
        <div><br>
        </div>
        <div>-Edward</div>
      </div>
      <br>
      <div class="gmail_quote">
        <div dir="ltr" class="gmail_attr">On Wed, Sep 16, 2020 at 6:10
          PM Emily Pillmore <<a href="mailto:emilypi@cohomolo.gy"
            moz-do-not-send="true">emilypi@cohomolo.gy</a>> wrote:<br>
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                    <div>Just to clarify, that's not the "real" diagonal
                      in the space, but it is a super useful translation
                      that I'd like for free :)</div>
                  </div>
                  <br>
                </div>
                <br>
                <div>
                  <div class="gmail_quote">On Wed, Sep 16, 2020 at 9:08
                    PM, Emily Pillmore <span dir="ltr"><<a
                        href="mailto:emilypi@cohomolo.gy"
                        target="_blank" moz-do-not-send="true">emilypi@cohomolo.gy</a>></span>
                    wrote:<br>
                    <blockquote class="gmail_quote" style="margin:0px
                      0px 0px 0.8ex;border-left:1px solid
                      rgb(204,204,204);padding-left:1ex">
                      <div class="gmail_extra">
                        <div class="gmail_quote"
                          id="gmail-m_7220689800887981291null">
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                              <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>
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                                                    <p
                                                      style="margin:0px">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>
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                                </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>
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                              <div class="gmail_quote">On Wed, Sep 16,
                                2020 at 6:35 PM, Asad Saeeduddin <span
                                  dir="ltr"><<a
                                    href="mailto:masaeedu@gmail.com"
                                    rel="noopener noreferrer"
                                    target="_blank"
                                    moz-do-not-send="true">masaeedu@gmail.com</a>></span>
                                wrote:<br>
                                <blockquote style="margin:0px 0px 0px
                                  0.8ex;border-left:1px solid
                                  rgb(204,204,204);padding-left:1ex"
                                  class="gmail_quote">
                                  <div class="gmail_extra">
                                    <div
                                      id="gmail-m_7220689800887981291null"
                                      class="gmail_quote">
                                      <p>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 SymmetricMonoidal t i p => CartesianMonoidal t i p</pre>
                                      <pre>  where</pre>
                                      <pre>  duplicate :: p x (x `t` x)</pre>
                                      <pre>  discard :: p x i

</pre>
                                      <pre>-- Laws:</pre>
                                      <pre>-- duplicate >>> first  discard = fwd lunit</pre>
                                      <pre>-- duplicate >>> second discard = fwd runit

</pre>
                                      <pre>-- where</pre>
                                      <pre>-- lunit :: Monoidal t i p => Iso p x (i `t` x)</pre>
                                      <pre>-- runit :: Monoidal t i p => Iso p x (x `t` i)</pre>
                                      <p>```<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>On 9/16/20 5:26 PM, Emily
                                        Pillmore wrote:<br>
                                      </div>
                                      <blockquote type="cite">
                                        <div>
                                          <div>
                                            <div>
                                              <div>
                                                <div>Nice! <br>
                                                  <br>
                                                  That's kind of what I
                                                  was going for with
                                                  Carter earlier in the
                                                  day, thanks Matthew. </div>
                                                <div><br>
                                                </div>
                                                <div>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><br>
                                                </div>
                                                <div>Thoughts? <br>
                                                </div>
                                                <div><br>
                                                </div>
                                                <div>Emily</div>
                                              </div>
                                              <br>
                                            </div>
                                            <br>
                                            <div>
                                              <div class="gmail_quote">On
                                                Wed, Sep 16, 2020 at
                                                3:49 PM, Carter
                                                Schonwald <span
                                                  dir="ltr"><<a
                                                    rel="noopener
                                                    noreferrer"
                                                    href="mailto:carter.schonwald@gmail.com"
                                                    target="_blank"
                                                    moz-do-not-send="true">carter.schonwald@gmail.com</a>></span>
                                                wrote:<br>
                                                <blockquote
                                                  class="gmail_quote"
                                                  style="margin:0px 0px
                                                  0px
                                                  0.8ex;border-left:1px
                                                  solid
                                                  rgb(204,204,204);padding-left:1ex">
                                                  <div
                                                    class="gmail_extra">
                                                    <div
                                                      class="gmail_quote"
id="gmail-m_7220689800887981291null">
                                                      <div dir="auto">Is
                                                        the join bipure
                                                        definition
                                                        taking advantage
                                                        of the (a->)
                                                        monad instance? 
                                                        Slick!</div>
                                                      <div dir="auto"><br>
                                                      </div>
                                                      <div dir="auto"><br>
                                                      </div>
                                                      <div>
                                                        <div
                                                          class="gmail_quote">
                                                          <div
                                                          class="gmail_attr"
                                                          dir="ltr">On
                                                          Wed, Sep 16,
                                                          2020 at 3:39
                                                          PM Matthew
                                                          Farkas-Dyck
                                                          <<a
                                                          href="mailto:strake888@gmail.com"
                                                          rel="noopener
                                                          noreferrer"
                                                          target="_blank"
moz-do-not-send="true">strake888@gmail.com</a>> wrote:<br>
                                                          </div>
                                                          <blockquote
                                                          style="margin:0px
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class="gmail_quote">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>_______________________________________________
                                                        <br>
                                                        Libraries
                                                        mailing list <br>
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                                                          noreferrer"
                                                          href="mailto:Libraries@haskell.org"
target="_blank" moz-do-not-send="true">Libraries@haskell.org</a> <br>
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                                        <br>
                                        <fieldset></fieldset>
                                        <pre>_______________________________________________
Libraries mailing list
<a rel="noopener noreferrer" href="mailto:Libraries@haskell.org" target="_blank" moz-do-not-send="true">Libraries@haskell.org</a>
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</pre>
                                      </blockquote>
                                      <p>_______________________________________________
                                        <br>
                                        Libraries mailing list
                                        <br>
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                                          href="mailto:Libraries@haskell.org"
                                          rel="noopener noreferrer"
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                                          moz-do-not-send="true">Libraries@haskell.org</a>
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                                </blockquote>
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                  </div>
                </div>
                <br>
              </div>
            </div>
          </div>
          _______________________________________________<br>
          Libraries mailing list<br>
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            moz-do-not-send="true">Libraries@haskell.org</a><br>
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