The original motivation for the notion of separable functors [Năstăsescu, van den Bergh & van Oystaeyen (1989)] was the functorial form of Maschke's theorem over rings and its various cousins: namely the classical Maschke’s theorem about finite group rings over fields, generalizes to the statements that when the order of the group is invertible in the ground ring then the splitting of an exact sequence of $k G$-modules can be obtained functorially from its splitting as an exact sequence of underlying $k$-modules.
This “Maschke property” of the forgetful functor ${}_{k G} Mod\to {}_{k}Mod$ is what is generalized by the notion of separable functors.
A further example appears in the study of ring extensions $f \colon R\to S$: Such is separable iff the corresponding restriction of scalars-functor $\operatorname{Res}_R^S \colon {}_{S}Mod \to {}_{R}Mod$ is a separable functor.
Let $F \colon C\to D$ be a functor with hom-set components
Then $F$ is called separable [Năstăsescu, van den Bergh & van Oystaeyen (1989), p. 398] if each $F_{x,y}$ has a section which is “natural” in $x$ and $y$, in a suitable sense.
(Rafael’s theorem) Let $F\dashv G$ be a pair of adjoint functors. Then
$F$ is separable iff the adjunction unit $\eta \con id \to G F$ has a section (= a natural transformation $\nu$ which is its right inverse, $\eta\circ\nu = 1$),
$G$ is separable iff the adjunction counit $\epsilon \colon F G \to id$ has a retraction (i.e. a natural transformation $\zeta$ that is its left inverse, $\zeta\circ\epsilon =1$).
The latter condition is reminiscent of one of the many equivalent definitions of a separable algebra $A$ over a field, namely one for which multiplication, viewed as an $(A,A)$-bimodule map $A \otimes A^{\mathrm{op}} \to A$, has a right inverse.
Separable functors were defined in
Now there is a monograph available:
Other references
M. D. Rafael, Separable functors revisited, Comm. in Algebra 18 (1990), 1445-1459.
S. Caenepeel, B. Ion, G. Militaru, The structure of Frobenius algebras and separable algebras, $K$-Theory 19 (2000), no. 4, 365–402.
J. Gómez-Torrecillas, Separable functors in corings, Int. J. of Math. and Math. Sci. 30 (2002), 4, Pages 203-225, doi, pdf
A. Ardizzoni, Separable functors and formal smoothness, J. K-Theory 1 (2008), no. 3, 535–582, math.QA/0407095, doi, MR2009k:16069
A. Ardizzoni, G. Böhm, C. Menini, A Schneider type theorem for Hopf algebroids, J. Algebra 318 (2007), no. 1, 225–269 MR2008j:16103 doi math.QA/0612633 (arXiv version is unified, corrected); Corrigendum, J. Algebra 321:6 (2009) 1786-1796 MR2010b:16060 doi
The following article studies formal smoothness and generalizes the separable functors to the context of the so-called S-categories which are introduced therein:
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