** ABSTRACT **

Let K be a full subcategory of a locally l-presentable category M (l any infinite regular cardinal). Then

- K is a l-injectivity class in M iff it is closed under products, l-filtered colimits and l-pure subobjects (iff it is axiomatizable by regular sentences in L(l,l))
- (l>
w) K is a
l-orthogonality class in M iff it is
closed under limits and l-filtered
colimits (iff it is axiomatizable by limit sentences in
L(l,l))
(This is known to be false for l
=w).
Here l-injective (l-orthogonal) means injective (orthogonal) w.r.t. a set of morphisms between l-presentable objects.

Other related results will be recalled.

Details and proofs can be found in the following manuscripts/papers available on request (or on the web where linked):

- [H] Hébert, M., Lambda*-injectivity + special compactness = lambda-injectivity, 1999.
- [H] Hébert, M., Purity and injectivity in accessible categories, JPAA 129 (1998), 143-147.
- [HR] Hébert, M., Rosicky, J., Uncountable orthogonality is a closure property, to appear in Bull. London Math. Soc.
- [HAR] Hébert, M., Adámek, J., Rosicky, J., More on orthogonality in locally presentable categories, to appear in Cahiers Top. Geom. Diff. Categ.