nLab conservative functor





A functor F:CDF \colon C\to D is conservative if it is “isomorphism-reflecting”, i.e. if g:abg \colon 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 (e.g. in the Elephant) 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.



Every fully faithful functor, and more generally any pseudomonic functor, is a conservative functor.

(Every fully faithful functor is pseudomonic.)

But further would-be converses of Exp. fail: not every conservative functor is full or faithful:


An example of a functor that is conservative but not full is the inclusion of the groupoid core Core(𝒞)𝒞Core(\mathcal{C}) \longrightarrow \mathcal{C} of a category 𝒞\mathcal{C} that is itself not a groupoid, into that category.


An example of a functor that is conservative but not faithful is the unique functor from any groupoid with two distinct isomorphisms f,g:xyf, g : x \to y to the terminal groupoid.


Every monadic functor is a conservative functor (see also at monadicity theorem):

For a T T -algebra homomorphism given by an invertible morphism f:ABf \colon A \to B, the inverse f 1:BAf^{-1} \colon B \to A is easily seen to also be a TT-algebra homomorphism.


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

f *:C /YC /X f^\ast \colon C_{/Y} \longrightarrow 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)


When CC and DD are pretoposes, a pretopos morphism F:CDF \colon 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.



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.


Formalization in cubical Agda:

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

Last revised on March 6, 2024 at 12:59:45. See the history of this page for a list of all contributions to it.