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Where/when did the convention originate of marking pullback (and/or pushout) squares by that little right-angle symbol in the corner?

Pair of pullback squares; from Spivak, Category Theory for the Sciences; fair use for illustrative purposes of the use of this notation in the literature

The earliest instance I’ve been able to find is in Paul Taylor’s diagrams package, from ≤1994, as mentioned in e.g. the changelog notes for v3.81 at http://www.paultaylor.eu/diagrams/ . But it seems more likely that this was to meet the demand for a notation that was already established, rather than being the origin? But looking at various well-known category theory textbooks from before 2000 (Mac Lane Categories for the Working Mathematician; Mac Lane and Moerdijk Sheaves in Geometry and Logic; Borceux Handbook of Categorical Algebra; Johnstone Topos Theory), none of them seem to use it, as far as I can find.

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    $\begingroup$ @ToddTrimble I strongly suspect it has been invented by him. I've seen it in several papers of Peter Freyd, as well as in Cats and Allegators. The latter is full of notation I wish would become standard, like puncture sign for non-commuting diagrams. $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Mar 26, 2017 at 19:09
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    $\begingroup$ Indeed, the Freyd-Scedrov book from 1990 has the notation: books.google.com/… $\endgroup$
    – Todd Trimble
    Commented Mar 26, 2017 at 19:11
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    $\begingroup$ While we're at it maybe we can standardize this notation too? I've seen that right angle all over the place and pointing in several different directions. $\endgroup$
    – Jonathan Beardsley
    Commented Mar 27, 2017 at 4:49
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    $\begingroup$ In Freyd's notation the corner is actually attached to the arrows. He also seems to use a similar cross notation for products, and several other notations for (co)equalizers etc. $\endgroup$
    – Dmitri Pavlov
    Commented Mar 27, 2017 at 9:41
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    $\begingroup$ @მამუკაჯიბლაძე: That's one point of view (i.e., a corner is an arrowhead). Another point of view states that a corner is meant to represent the two newly constructed arrows; in the case of a pushout these two arrows give you ⌟. And a third point of view states that a corner is meant to represent the two old arrows, which would give you ⌜ for a pushout. $\endgroup$
    – Dmitri Pavlov
    Commented Mar 29, 2017 at 11:20

3 Answers 3

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In an email to me dated 17 February 1992, Peter Freyd said:

I was using a different notation in 1974 in lectures at Montreal. A high school teacher named Butler suggested the right-angle. It was an improvement. I have used it since.

When the diagram gets too complicated for the pullbacks to be rectangles, such as in the final chapter of my book Practical Foundations of Mathematics, I strongly recommend making them at least parallelograms. Then it is clear that pullback is acting as a functor that transforms one part of the diagram to another. In particular, in a type-theoretic setting pullback is substitution; whilst this has been known for a long time, Section 8.2 of the book actually proves it.

William Butler (had) proved some important results about monads, which you will find in "Toposes, Triples and Theories" by Barr and Wells (free TAC reprints copy).

Other very smart categorists who left academia to become schoolteachers include Christian Mikkelsen (who was the first to derive colimits from limits in an elementary topos) and Sjoerd Crans (who did weak higher dimensional category theory).

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If it is of any use, on page 251 of Taylor's "Practical Foundations of Mathematics", shortly after introducing pullbacks, he writes:

Pullbacks are often indicated with the right angle symbol, which was suggested by William Butler in 1974 and popularised by Peter Freyd.

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A small comment, that does not fit in its proper place. The department of Algebra in the University of Santiago de Compostela has studied categories since the second half of the sixties under the leadership of prof. Eduardo García-Rodeja. I made my studies later and the following notations were standard:

• Cartesian square:

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• coCartesian square:

enter image description here

So, I was not surprised to see Paul Taylor's notation. And for me, too, it was the first time I noticed a notation similar to the one it was used here in print. later, the main research areas where closed categories and homology theories, though after all these years, it has diversified a lot.

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    $\begingroup$ Very interesting, thankyou! What was your department then, and who were the people who you saw using this — either in terms of individuals, or in terms of what research area(s) they were in? $\endgroup$
    – Peter LeFanu Lumsdaine
    Commented Mar 27, 2017 at 11:13
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    $\begingroup$ I'va added some precisions. For me, personally, it was very useful to have a good background in categories to study cohomology in algebraic geometry. $\endgroup$
    – Leo Alonso
    Commented Mar 27, 2017 at 11:30

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