ABSTRACT
Let U : A ¯ → K ¯ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072621/6e3dda93-0297-4e21-9483-9ad6f30dbbc7/content/eq3143.tif"/> be a functor with left adjoint F, T ˜ = 〈 T , η , u 〉 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072621/6e3dda93-0297-4e21-9483-9ad6f30dbbc7/content/eq3144.tif"/> the monad it induces in K ¯ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072621/6e3dda93-0297-4e21-9483-9ad6f30dbbc7/content/eq3145.tif"/> and Φ : A ¯ → K ¯ T ˜ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072621/6e3dda93-0297-4e21-9483-9ad6f30dbbc7/content/eq3146.tif"/> the comparison functor. In l,l.l6(a,b), R.-E. Hoffmann raised the following questions:
Under what conditions will the restricted comparison functor Φ ¯ : Proj U → Proj U T ˜ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072621/6e3dda93-0297-4e21-9483-9ad6f30dbbc7/content/eq3147.tif"/> be an equivalence of categories?
Assuming that Φ ¯ : Proj U → Proj U T ˜ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072621/6e3dda93-0297-4e21-9483-9ad6f30dbbc7/content/eq3148.tif"/> is an equivalence of categories when is it possible to find a category C ¯ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072621/6e3dda93-0297-4e21-9483-9ad6f30dbbc7/content/eq3149.tif"/> and a functor V : C ¯ → K ¯ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072621/6e3dda93-0297-4e21-9483-9ad6f30dbbc7/content/eq3150.tif"/> , inducing the same monad T ˜ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072621/6e3dda93-0297-4e21-9483-9ad6f30dbbc7/content/eq3151.tif"/> in K ¯ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072621/6e3dda93-0297-4e21-9483-9ad6f30dbbc7/content/eq3152.tif"/> , such that:
A ¯ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072621/6e3dda93-0297-4e21-9483-9ad6f30dbbc7/content/eq3153.tif"/> is a full reflective subcategory of C ¯ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072621/6e3dda93-0297-4e21-9483-9ad6f30dbbc7/content/eq3154.tif"/> .
U = V ⋅ E https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072621/6e3dda93-0297-4e21-9483-9ad6f30dbbc7/content/eq3155.tif"/> , E being the embedding of A ¯ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072621/6e3dda93-0297-4e21-9483-9ad6f30dbbc7/content/eq3156.tif"/> in C ¯ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072621/6e3dda93-0297-4e21-9483-9ad6f30dbbc7/content/eq3157.tif"/> .
EF is left adjoint to V.
The comparison functor Φ ′ : C ¯ → K ¯ T ˜ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072621/6e3dda93-0297-4e21-9483-9ad6f30dbbc7/content/eq3158.tif"/> has a full and faithful left adjoint and Φ ′ ⋅ E = Φ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072621/6e3dda93-0297-4e21-9483-9ad6f30dbbc7/content/eq3159.tif"/> .
