Back from holidays I am slowly working through the various interesting e-mails of Tom Leinster. The examples he presents in his e-mail on August 9 seem to lead to the following question: given on object A characterize the full subcategory L_A of all objects satisfying all "laws" that A satisfies. The algebraic case ("law" meaning equation) has two obvious generalizations: orthogonality and injectivity. For both of them the answer to the above question is nice and easy. INJECTIVITY: let H be the class of morphisms generated by {A} in the Galois connection "to be injective to". (That is, H consists of all morphisms to which A is injective.) The opposite class L_A = Inj H of all objects injective w.r.t. H consists of precisely all split subobjects of powers of A. This holds in every category with powers. Proof: injectivity classes are clearly closed under product and split subobject. Conversely, if B lies in Inj H, then it is a split subobject of the power of A to hom(B,A). In fact, the canonical morphism m from B to the power of A to hom(B,A) lies in H: given a morphism f: B -> A, then f factorizes through m via the projection of A^hom(B,A) corresponding to f. Consequently, B is injective w.r.t. m, and since B is the domain of m, this implies trivially that m is a split mono. ORTHOGONALITY: let K be the class of morphisms generated by {A} in the Galois connection "to be orthogonal to". The opposite class L_A = Ort K of all objects orthogonal to K is the closure of {A} under limits. This holds in every complete and cowellpowered category. Proof: orthogonality classes are clearly closed under limit, thus, Ort K contains the limit closure L of {A}. To prove the opposite inclusion observe that L is a reflective subcategory due to Freyd's SAFT: A is easily seen to be a cogenerator of L. For every object B in Ort K a reflection r: B -> B' in L lies in K (since A lies in L). Thus, B is orthogonal to r. This implies that r is a split mono. Now L contains B' and is closed under split subobjects, thus B lies in L. FINITARY LAWS The algebraic case has another feature: every equation, when translated as injectivity or orthogonality w.r.t. a morphism e:A-> B, has the property that both A and B are finitely presentable. We can thus decide to restrict our attention to finitary morphisms, i.e., morhisms with finitely presentable domains and codomains, as our "laws". If H is the class of all finitary morphisms to which A is injective, then the injectivity class Inj H is the closure of {A} under product, filtered colimit and pure subobject. This was proved by J. Rosicky, F. Borceux and myself in TAC 10 (2002), 148-161. If K is the class of all finitary morphisms to which A is orthogonal, then the orthogonality class Ort K is the closure of {A} under product, filtered colimit and A-pure subobject as proved by L. Sousa and myself in JPAA 276 (2004), 685-705. (The concept of A-pure subobject is a bit artificial, but unfortunately the above result is false if one substitutes it with pure morphism. Surprisingly, when generalizing finitary morphisms to k-ary morphisms for uncountable cardinals k, the corresponding result does hold with pure subobjects: see M. Hebert and J. Rosicky, Bull. London Math. Soc 33 (2001) 685-688.) xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx alternative e-mail address (in case reply key does not work): J.Adamek@tu-bs.de xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx