# K-purity and orthogonality

## Michel Hebert

Adamek and Sousa recently solved the problem of characterizing the subcategories K of a locally $\lambda$-presentable category C which are $\lambda$-orthogonal in C, using their concept of K$\lambda$-pure morphism. We strengthen the latter definition, in order to obtain a characterization of the classes defined by orthogonality with respect to $\lambda$-presentable morphisms (where $f : A \rightarrow B is called$\lambda$-presentable if it is a$\lambda$-presentable object of the comma category A/C). Those classes are natural examples of reflective subcategories defined by proper classes of morphisms. Adamek and Sousa's result follows from ours. We also prove that$\lambda$-presentable morphisms are precisely the pushouts of morphisms between$\lambda\$-presentable objects of C.

Keywords: pure morphism, othogonality, injectivity, locally presentable categories, accessible categories

2000 MSC: 18A20, 18C35, 03C60, 18G05

Theory and Applications of Categories, Vol. 12, 2004, No. 12, pp 355-371.

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