On the fullness of certain functors (Q584378)

The main result contained in the paper is the following: Let K: \({\mathcal C}\to {\mathcal D}\) be a continuous functor (i.e. \({\mathcal D}\) is weakly bounded by a regular cardinal \(\alpha\) and \({\mathcal C}\) has and K preserves \(\alpha\)-nice limits). (a) If K is full with respect to isomorphisms and \(K(C)=\{A|\) \(A\cong KX\) for some \({\mathcal C}\)-object \(X\}\) is closed in \({\mathcal D}\) under colimits of (S,\(\alpha)\)-sequences, then K is full. The converse is also true if \({\mathcal C}\) has and K preserves colimits of (S,\(\alpha)\)- sequences. (b) K is an equivalence iff it is essentially surjective on objects and full with respect to isomorphisms, and reflects isomorphisms. (c) K is an isomorphism iff it is essentially surjective on objects and full with respect to isomorphisms, and creates isomorphisms. The bounded categories with a generator, in the sense of Freyd and Kelly, are weakly bounded by some \(\alpha\)- in particular every locally presentable category is weakly bounded. Also the Cartesian closed topological categories and the generalized varieties are examples of weakly bounded categories.