nLab conservative functor

Definition

Definition

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

Remark

Sometimes (e.g. in the Elephant) conservative functors are assumed to be faithful as well. If $C$ has, and $F$ preserves, equalizers, then conservativity implies faithfulness.

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

Examples

Example

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:

Example

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

Example

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

Example

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

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

Proposition

Let $C$ be a category with pullbacks. Given any morphism $f \colon X \longrightarrow Y$ in $C$ write

$$f^\ast \colon C_{/Y} \longrightarrow C_{/X}$$

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

1. $f$ is a strong epimorphism;

2. $f^\ast$ is conservative.

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

Example

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

Properties

Proposition

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

Proof

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

Let $K \colon J \to C$ be a diagram in $C$ whose limit $\lim K$ exists and such that $\lim (F\circ K) \,\simeq\, F (\lim K)$. Now if $const_c \to K$ is a cone in $C$ that is sent to a limiting cone $F const_c$ in $D$, then by the universal property of the limit in $D$ the morphism $F( c \to \lim K)$ is an isomorphism in $D$, hence must have been an isomorphism in $C$ (by the assumption that $F$ is conservative), hence $const_c$ must have been a limiting cone in $C$.

Proposition

Up to equivalence, every functor between small categories can be factored as an iterated localization followed by a conservative functor.

Related concepts

• monadicity theorem

• conservative (∞,1)-functor

• iterated localization

Literature

• 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

• Peter Johnstone, Sketches of an Elephant

Formalization in cubical Agda:

• 1lab, Functors

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

• Reinhard Börger, Walter Tholen, Example 2.4 in Strong regular and dense generators, Cahiers de Topologie et Géométrie Différentielle Catégoriques 32 3 (1991) 257-276 [numdam:CTGDC_1991__32_3_257_0, MR1158111]