Note on total categories

It is shown that, for a semi-topological functor T: A → X, the category A is total, that is, the Yoneda embedding of A has a left adjoint, if X is total. In particular, monadic categories over Set (possibly without rank) are total, and full reflective subcategories of total categories are total.

[1]  B. A. Rattray.  Adjoints to functors from categories of algebras , 1975 .

[2]  Jiří Adámek.  Colimits of algebras revisited , 1977, Bulletin of the Australian Mathematical Society.

[3]  Ross Street,et al.  Yoneda structures on 2-categories , 1978 .

[4]  Walter Tholen,et al.  Semi-topological functors I , 1979 .