nLab algebraically injective object

Algebraically injective objects

Algebraically injective objects


Let JJ be a set of morphisms in a category CC. An algebraically JJ-injective object is an object XCX\in C equipped with the structure of, for every morphism i:ABi:A\to B in JJ and every morphism f:AXf:A\to X, a specified morphism g:BXg:B\to X such that gi=fg \circ i = f.


Algebras for a pointed endofunctor

Assuming that CC is locally small and cocomplete (and JJ is a small set), given an object XX, let F JXF_J X be the following pushout:

i:AB(C(A,X)A) X i:AB(C(A,X)B) F JX \array{ \coprod_{i:A\to B} (C(A,X) \cdot A) & \to & X \\ \downarrow & & \downarrow \\ \coprod_{i:A\to B} (C(A,X) \cdot B) & \to & F_J X }

where \cdot represents the copower of CC over Set. Then F JF_J is a pointed endofunctor of CC, such that the (pointed) endofunctor algebras of F JF_J are precisely the algebraically JJ-injective objects.


When CC is locally small and cocomplete as before, if the algebraically-free monad on the pointed endofunctor F JF_J exists, then by definition the algebraically JJ-injective objects are its monad algebras. In particular, they are monadic over CC.



  • John Bourke, Equipping weak equivalences with algebraic structure, 2017, arxiv

  • John Bourke, Iterated algebraic injectivity and the faithfulness conjecture, 2018, arxiv

Last revised on February 21, 2019 at 14:40:01. See the history of this page for a list of all contributions to it.