A functor is conservative if it is “isomorphism-reflecting”, i.e. if is a morphism in such that is an isomorphism in , then is an isomorphism in .
Sometimes (e.g. in the Elephant) conservative functors are assumed to be faithful as well. If has, and 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 of a category 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 to the terminal groupoid.
Every monadic functor is a conservative functor (see also at monadicity theorem):
For a -algebra homomorphism given by an invertible morphism , the inverse is easily seen to also be a -algebra homomorphism.
Let be a category with pullbacks. Given any morphism in write
for the functor of pullback along between slice categories (“base change”). If strong epimorphisms in are preserved by pullback, then the following are equivalent:
is a strong epimorphism;
is conservative.
(e.g. Johnstone, lemma A.1.3.2)
When and are pretoposes, a pretopos morphism is conservative if and only if for every object , the induced map between subobject lattices is injective.
A conservative functor 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 be a diagram in whose limit exists and such that . Now if is a cone in that is sent to a limiting cone in , then by the universal property of the limit in the morphism is an isomorphism in , hence must have been an isomorphism in (by the assumption that is conservative), hence must have been a limiting cone in .
Up to equivalence, every functor between small categories can be factored as an iterated localization followed by a conservative functor.
Geun Bin Im, Gregory Maxwell Kelly, Some remarks on conservative functors with left adjoints, J. Korean Math. Soc. 23 (1986), no. 1, 19–33, MR87i:18002b, pdf
Geun Bin Im, Gregory Maxwell Kelly, On classes of morphisms closed under limits, J. Korean Math. Soc. 23 (1986), no. 1, 1–18, pdf
Geun Bin Im, Gregory Maxwell Kelly, Adjoint-triangle theorems for conservative functors, Bull. Austral. Math. Soc. 36 (1987), no. 1, 133–136, MR88k:18005, doi
Formalization in cubical Agda:
For an example of a conservative, but not faithful, functor having a left adjoint:
Last revised on August 26, 2024 at 13:27:42. See the history of this page for a list of all contributions to it.