There is a Beck's theorem for Karoubian triangulated categories, proposed by Konstevich and Rosenberg in July 2004, which is proved by using the Verdier's abelianization functor and graded monads; see page 36 of A. Rosenberg, Topics in noncommutative algebraic geometry, homological algebra and K-theory, preprint MPIM Bonn 2008-57, pdf.

In fact they got simulatenously (July 15, 2004) both versions: A-infinity and triangulated. The reference to the triangulated is above: while for A-infinity there is no write up, but Kontsevich gave a talk (I think Nov 2004, van den Bergh birthday conference) where he formulated and used the result; one of nice applications was to glue certain ordinary commutative schemes to get certain formal schemes. I remember very well the weeks preceding the result when we discussed possible shape of the result seeked at IHES. Later at the conference in Split, Kontsevich gave a talk where he gave some usages in noncommutative algebraic geometry.

I disagree with the statement: "Beck's theorem for comonad is equivalent to Grothendieck flat descent theory on scheme". Namely Grothendieck gave both the flat descent theory for quasicoherent sheaves (SGA I.8.1) which is a special case of Beck's theorem (though it has some symmetries which general noncommutative case does not have), but also the (stronger) flat descent theorem for affine schemes (SGA I.8.2) and for morphisms (cf. SGA I.8.5), which unlike the descent for quasicoherent sheaves, does not generalize to the noncommutative algebras and consequently to categorical setup either.

There is a Beck's theorem for Karoubian triangulated categories, proposed by Konstevich and Rosenberg in July 2004, which is proved by using the Verdier's abelianization functor and graded monads; see page 36 of A. Rosenberg, Topics in noncommutative algebraic geometry, homological algebra and K-theory, preprint MPIM Bonn 2008-57, pdf.

In fact they got simulatenously (July 15, 2004) both versions: A-infinity and triangulated. The reference to the triangulated is above: while for A-infinity there is no write up, but Kontsevich gave a talk (I think Nov 2004, van den Bergh birthday conference) where he formulated and used the result; one of nice applications was to glue certain ordinary commutative schemes to get certain formal schemes. I remember very well the weeks preceding the result when we discussed possible shape of the result seeked at IHES. Later at the conference in Split, Kontsevich gave a talk where he gave some usages in noncommutative algebraic geometry.

I disagree with the statement: "Beck's theorem for comonad is equivalent to Grothendieck flat descent theory on scheme". Namely Grothendieck gave both the flat descent theory for quasicoherent sheaves (SGA I.8.1) which is a special case of Beck's theorem (though it has some symmetries which general noncommutative case does not have), but also the (stronger) flat descent theorem for affine schemes (SGA I.8.2) and for morphisms (cf. SGA I.8.5), which unlike the descent for quasicoherent sheaves, does not generalize to the noncommutative algebras and consequently to categorical setup either.

There is a Beck's theorem for Karoubian triangulated categories, proposed by Konstevich and Rosenberg in July 2004, which is proved by using the Verdier's abelianization functor and graded monads; see page 36 of A. Rosenberg, Topics in noncommutative algebraic geometry, homological algebra and K-theory, preprint MPIM Bonn 2008-57, pdf.

In fact they got simulatenously (July 15, 2004) both versions: A-infinity and triangulated. The reference to the triangulated is above: while for A-infinity there is no write up, but Kontsevich gave a talk (I think Nov 2004, van den Bergh birthday conference) where he formulated and used the result; one of nice applications was to glue certain ordinary commutative schemes to get certain formal schemes. I remember very well the weeks preceding the result when we discussed possible shape of the result seeked at IHES. Later at the conference in Split, Kontsevich gave a talk where he gave some usages in noncommutative algebraic geometry.

There is a Beck's theorem for Karoubian triangulated categories, proposed by Konstevich and Rosenberg in July 2004, which is proved by using the Verdier's abelianization functor and graded monads; see page 36 of A. Rosenberg, Topics in noncommutative algebraic geometry, homological algebra and K-theory, preprint MPIM Bonn 2008-57, pdf.

In fact they got simulatenously (July 15, 2004) both versions: A-infinity and triangulated. The reference to the triangulated is above: while for A-infinity there is no write up, but Kontsevich gave a talk (I think Nov 2004, van den Bergh birthday conference) where he formulated and used the result; one of nice applications was to glue certain ordinary commutative schemes to get certain formal schemes. I remember very well the weeks preceding the result when we discussed possible shape of the result seeked at IHES. Later at the conference in Split, Kontsevich gave a talk where he gave some usages in noncommutative algebraic geometry.

I disagree with the statement: "Beck's theorem for comonad is equivalent to Grothendieck flat descent theory on scheme". Namely Grothendieck gave both the flat descent theory for quasicoherent sheaves (SGA I.8.1) which is a special case of Beck's theorem (though it has some symmetries which general noncommutative case does not have), but also the (stronger) flat descent theorem for affine schemes (SGA I.8.2) and for morphisms (cf. SGA I.8.5), which unlike the descent for quasicoherent sheaves, does not generalize to the noncommutative algebras and consequently to categorical setup either.