nLab algebraically injective object

Definition

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

Properties

Algebras for a pointed endofunctor

Assuming that $C$ is locally small and cocomplete (and $J$ is a small set), given an object $X$, let $F_J X$ be the following pushout:

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

where $\cdot$ represents the copower of $C$ over Set. Then $F_J$ is a pointed endofunctor of $C$, such that the (pointed) endofunctor algebras of $F_J$ are precisely the algebraically $J$-injective objects.

Monadicity

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

Solidity

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Related pages

• injective object, projective object
• algebraic small object argument
• algebraically fibrant object

References

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

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