Category theory text that defines composition backwards?

Solution 1:

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.

Solution 2:

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.