nLab conservative functor





A functor F:CDF \colon C\to D is conservative if it is “isomorphism-reflecting”, i.e. if g:abg:a\to b is a morphism in CC such that F(g)F(g) is an isomorphism in DD, then gg is an isomorphism in CC.


Sometimes conservative functors are assumed to be faithful as well. If CC has, and FF preserves, equalizers, then conservativity implies faithfulness.

See conservative morphism for a generalization to an arbitrary 2-category.



Let 𝒞\mathcal{C} be a category with pullbacks. Given any morphism f:XYf \colon X \longrightarrow Y in 𝒞\mathcal{C} write

f *:𝒞 /Y𝒞 /X f^\ast \colon \mathcal{C}_{/Y} \longrightarrow \mathcal{C}_{/X}

for the functor of pullback along ff between slice categories (base change). If strong epimorphisms in 𝒞\mathcal{C} are preserved by pullback, then the following are equivalent:

  1. ff is a strong epimorphism;

  2. f *f^\ast is conservative.

(e.g. Johnstone, lemma A.1.3.2)


Every fully faithful functor is a conservative functor. An example of a functor that is conservative but not fully faithful is the inclusion of the groupoid core of a category into the category.


When CC and DD are pretoposes, a pretopos morphism F:CDF : C \to D is conservative if and only if for every object cCc \in C, the induced map between subobject lattices F (c):Sub(c)Sub(F(c))F^{(c)} : \operatorname{Sub}(c) \to \operatorname{Sub}(F(c)) is injective.


Every monadic functor is a conservative functor: for a TT-algebra homomorphism given by an invertible morphism f:ABf : A \to B, the inverse f 1:BAf^{-1} : B \to A is easily seen to also be a TT-algebra homomorphism.



A conservative functor F:CDF \colon C \to D reflects all limits and colimits that it preserves and which exist in the source category.


We discuss the case of limits (the argument for colimits is formally dual):

Let K:JCK \colon J \to C be a diagram in CC whose limit limK\lim K exists and such that lim(FK)F(limK)\lim (F\circ K) \,\simeq\, F (\lim K). Now if const cKconst_c \to K is a cone in CC that is sent to a limiting cone Fconst cF const_c in DD, then by the universal property of the limit in DD the morphism F(climK)F( c \to \lim K) is an isomorphism in DD, hence must have been an isomorphism in CC (by the assumption that FF is conservative), hence const cconst_c must have been a limiting cone in CC.


For an example of a conservative, but not faithful, functor f:ASetf: A\to Set having a left adjoint see Example 2.4 in:

Last revised on August 11, 2022 at 16:34:49. See the history of this page for a list of all contributions to it.