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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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