I've always struggled with the convention that if $f:X \rightarrow Y$ and $g:Y \rightarrow Z$, then $g \circ f : X \rightarrow Z$. Constantly reflecting back and forth is inefficient. Does anyone know of a category theory text that defines composition the other way? So that $f \circ g$ means what we would normally mean by $g \circ f$.
Category theory text that defines composition backwards?
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7If I might make a personal remark: I struggled with the same problem, and I actually found some text doing composition from left to right (sorry, forgot the name), but I came to appreciate that there is little point in speaking a language almost nobody else speaks. – 2012-12-14
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4To define $f \circ g$ backwards would be crazy and cause confusion (as the forward way is well established). Some texts define $(f\,;g)$ as the backward composition, but right now I don't recall any that would use it frequently. – 2012-12-14
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0@MichaelGreinecker Yes that's true. But the way I see it, you can learn things the simple way, even if in a publication you have to conform to certain standards which aren't quite as simple. – 2012-12-14
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1Here are some rough lecture notes using "postfix" notation: http://www.ii.uib.no/~wolter/teaching/v11-inf223/manuscript.pdf – 2012-12-14
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4Herstein (whom I met long ago, a very nice guy!) had a program at some point to write $f(x)$ as $(x)f$ to avoid this problem... but it was too late. In a way, our predicament can be productive, insofar as we are forced, for comprehension, to look beyond the (unfortunate) notation to see meaning. Another point is that _diagrams_ (the sine qua non of "categorical" modalities) avoid the goofiness of notation by physical demonstration of the composition of maps, etc. This does raise the _next_ issue, in effect, of our (collective) dependence on "temporal order". :) – 2012-12-14
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7I started reading the composition operator as "after" and it has really helped. You would read $g\circ f$ as $g$ after $f$ and then you always know which one comes first. – 2012-12-14
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0You might have some luck if you focus your attention on the 1960s or thereabouts. Left-to-right order seemed more fashionable in that era; for example, almost half of the papers in the 1966/67 _Seminar on triples and categorical homology theory_ were written with that convention. – 2012-12-14
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0My personal opinion is that if you find this a good enough to reason to seek a new category theory text, then category theory really isn't for you. – 2012-12-14
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0We almost always write a function like $f(x)$, not $(x)f$, for example $sin (x)$. So writing a composition as $(f\circ g)(x) = f(g(x))$ is natural. Since the category of sets is a typical category, the usual $f\circ g$ is also natural IMO. – 2012-12-18
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1It is difficult to judge which order of writing down composition is more natural. There are areas like group theory or category theory where the composition of (a lot more than just two) maps is used more often than mere application and e.g. $x^{fg} = (x^f)^g$ or $x(fg) = (xf)g$ looks natural enough, at least in languages where you read from left to right. For are more pointed opinion see http://www.iti.cs.tu-bs.de/TI-INFO/koslowj/RESEARCH/RPN . – 2012-12-20
3 Answers
I recall that the following textbooks on category theory have compositions written from left to right.
Freyd, Scedrov: "Categories, Allegories", North-Holland Publishing Co., 1990 .
Manes: "Algebraic Theories", GTM 26, Springer-Verlag, 1976.
Higgins: "Notes on Categories and Groupoids", Van Nostrand, 1971 (available as TAC Reprint No 7).
Other examples appear in group theory and ring theory, e.g.
- Lambek: "Lectures on rings and modules", Chelsea Publishing Co., 1976 (2ed).
or several books by P.M. Cohn.
But in order to avoid confusion, authors usually do not use the symbol $\circ$ for this. In particular when (as with noncommutative rings) it is helpful to have both readings available (so that module homomorphisms and scalars act on opposite sides). For instance, as far as I remember, Lambek uses $\ast$ instead.
There is, for example, the paper "Group Actions on Posets" by Babson and Kozlov where composition of morphisms is defined "reversed". Another approach which may be interesting to you is to reverse all diagrams (see e.g. "A Higher Category Approach to Twisted Actions on $C^*$-Algebras" by Buss, Meyer and Zhu). I for myself tend to use $f\bullet g := g\circ f$ to avoid any confusing.
Category Theory for the Sciences - Spivak