nLab essentially injective functor

Idea

A functor $F\colon C \to D$ is essentially injective if it is injective on objects “up to isomorphism”, namely on isomorphism classes of objects.

Definition

$F\colon C \to D$ is essentially injective or essentially injective on objects if, for any two objects $c$ and $c'$ of $C$, if there is an isomorphism $F(c) \cong F(c')$ in $D$, then $c \cong c'$ in $C$. (The converse holds trivially, since functors preserve isomorphisms.) Equivalently, a functor is essentially injective if and only if it is injective on isomorphism classes.

Examples

• A functor between discrete categories (or, more generally, skeletal categories) is essentially injective iff it is a injective function between the classes of objects.

• An injective-on-objects functor is essentially injective if its image is skeletal, but not generally otherwise.

• A composite of any two essentially injective functors is essentially injective.

• If $g f$ is essentially injective, then $f$ is essentially injective.

• A conservative functor is essentially injective when it is full. More generally, any pseudomonic functor is essentially injective. More generally still, any fully faithful functor is essentially injective.

• An isocofibration is a functor that is (strictly) injective on objects, and is hence essentially injective.

Remark on terminology

Some sources call this property “isomorphism reflecting” or “isomorphism creating”. However, such terminology more accurately refers to conservative functors.

Related concepts

basic properties of…

• functors

  • faithful functor

  • full functor

  • full and faithful functor

  • injective-on-objects functor

  • essentially injective functor

  • embedding of categories

  • essentially surjective functor

    • surjective-on-objects functor

    • bijective-on-objects functor

    • dominant functor

  • split essentially surjective functor

  • essentially surjective and full functor

  • equivalence of categories

  • conservative functor

  • localization functor

  • pseudomonic functor

  • (co)continuous functor

  • final functor, dense functor

• 2-functors

  • full and faithful 2-functor

  • locally fully faithful 2-functor

  • equivalence of 2-categories

• (∞,1)-functors:

  • fully faithful (∞,1)-functor

  • essentially surjective (∞,1)-functor

  • equivalence of (∞,1)-categories

  • homotopy final functor, final (∞,1)-functor

References

• Jiří Rosický. Generalized Brown representability in homotopy categories. Theory and Applications of Categories 20 (2008): 18-24.