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category_theory.monad.limits

Limits and colimits in the category of algebras #

This file shows that the forgetful functor forget T : algebra T ⥤ C for a monad T : C ⥤ C creates limits and creates any colimits which T preserves. This is used to show that algebra T has any limits which C has, and any colimits which C has and T preserves. This is generalised to the case of a monadic functor D ⥤ C.

TODO #

Dualise for the category of coalgebras and comonadic left adjoints.

(Impl) The natural transformation used to define the new cone

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(Impl) The natural transformation given by the algebra structure maps, used to construct a cocone c with apex colimit (D ⋙ forget T).

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(Impl) A cocone for the diagram (D ⋙ forget T) ⋙ T found by composing the natural transformation γ with the colimiting cocone for D ⋙ forget T.

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(Impl) Define the map λ : TL ⟶ L, which will serve as the structure of the coalgebra on L, and we will show is the colimiting object. We use the cocone constructed by c and the fact that T preserves colimits to produce this morphism.

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(Impl) Construct the colimiting algebra from the map λ : TL ⟶ L given by lambda. We are required to show it satisfies the two algebra laws, which follow from the algebra laws for the image of D and our commuting lemma.

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For D : J ⥤ algebra T, D ⋙ forget T has a colimit, then D has a colimit provided colimits of shape J are preserved by T.

If C has limits of shape J then any reflective subcategory has limits of shape J.

If C has limits then any reflective subcategory has limits.

If C has colimits of shape J then any reflective subcategory has colimits of shape J.

If C has colimits then any reflective subcategory has colimits.