nLab essentially injective functor

Essentially injective functors

Essentially injective functors


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


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


Remark on terminology

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

basic properties of…


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

Last revised on December 4, 2023 at 20:57:56. See the history of this page for a list of all contributions to it.