Let be a set of morphisms in a category . An algebraically -injective object is an object equipped with the structure of, for every morphism in and every morphism , a specified morphism such that .
Assuming that is locally small and cocomplete (and is a small set), given an object , let be the following pushout:
where represents the copower of over Set. Then is a pointed endofunctor of , such that the (pointed) endofunctor algebras of are precisely the algebraically -injective objects.
When is locally small and cocomplete as before, if the algebraically-free monad on the pointed endofunctor exists, then by definition the algebraically -injective objects are its monad algebras. In particular, they are monadic over .
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John Bourke, Equipping weak equivalences with algebraic structure, 2017, arxiv
John Bourke, Iterated algebraic injectivity and the faithfulness conjecture, 2018, arxiv
Last revised on September 14, 2024 at 22:23:53. See the history of this page for a list of all contributions to it.