# Algebraically closed and existentially closed substructures in categorical context

## Michel Hebert

We investigate categorical versions of algebraically closed (= pure) embeddings, existentially closed embeddings, and the like, in the context of locally presentable categories. The definitions of S. Fakir, as well as some of his results, are revisited and extended. Related preservation theorems are obtained, and a new proof of the main result of Rosicky, Adamek and Borceux, characterizing $\lambda$-injectivity classes in locally $\lambda$-presentable categories, is given.

Keywords: pure morphism, algebraically closed, existentially

2000 MSC: 18A20, 18C35, 03C60, 03C40

Theory and Applications of Categories, Vol. 12, 2004, No. 9, pp 269-298.

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