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Definition

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

Examples

\begin{example}\label{FullyFaithfulFunctorIsConservative} Every fully faithful functor, and more generally any pseudomonic functor, is a conservative functor.

(Every fully faithful functor is pseudomonic.) \end{example}

But further would-be converses of Exp. \ref{FullyFaithfulFunctorIsConservative} fail: not every conservative functor is full or faithful:

\begin{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.
\end{example}

\begin{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. \end{example}

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

Properties

\begin{proposition} \label{ConservativeFunctorsReflectLimitsWhichTheyPreserve} A conservative functor $F \colon C \to D$ reflects all limits and colimits that it preserves and which exist in the source category. \end{proposition}

\begin{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 $C$. \end{proof}

Related concepts

• monadicity theorem

• conservative (∞,1)-functor

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; On classes of morphisms closed under limits, J. Korean Math. Soc. 23 (1986), no. 1, 1–18, Adjoint-triangle theorems for conservative functors, Bull. Austral. Math. Soc. 36 (1987), no. 1, 133–136, MR88k:18005, doi

• On classes of morphisms closed under limits, J. Korean Math. Soc. 23 (1986), no. 1, 1–18

• 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](http://www.numdam.org/item?id=CTGDC_1991__32_3_257_0), MR1158111]