Subject: categories: Re: Laws

From: Jiri Adamek <[email-masked]>
Date: Fri, 25 Aug 2006 10:49:18 +0200 (CEST)

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.)

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