Injectivity and orthogonality classes: recent results

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.