# Category theory text that defines composition backwards?

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 $g \circ f$.

• If 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. – Michael Greinecker Dec 14 '12 at 0:41
• To 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. – dtldarek Dec 14 '12 at 0:43
• Here are some rough lecture notes using "postfix" notation: ii.uib.no/~wolter/teaching/v11-inf223/manuscript.pdf – Nathan Grigg Dec 14 '12 at 0:45
• Herstein (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". :) – paul garrett Dec 14 '12 at 0:50
• I 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. – chris Dec 14 '12 at 0:58

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