nLab inert morphism

Definition

Definition

A morphism in $FinSet^{*/}$ is called an inert morphism if it is surjective and an injection on those elements that are not sent to the base point.

A morphism in $FinSet^{*/}$ is called an active morphism if only the basepoint goes to the basepoint.

(Lurie, def. 2.1.1.8, def. 2.1.2.1).

See at (infinity,1)-operad

Properties

Factorization system

$(Inert, Active)$ is an orthogonal factorization system on $FinSet^{*/}$.

(Lurie, remark 2.1.2.2)