nLab
conservative morphism

Definition

A morphism $f:A\to B$ in a 2-category $K$ is said to be (representably) conservative if for all objects $X$, the induced functor

$$K(X,A)\to K(X,B)$$K(X,A) \to K(X,B)

is conservative. In Cat, this is equivalent to $f$ being conservative in the usual sense.

Remarks

• Conservative morphisms often form the right class of a factorization system. In Cat, the left class consists of (possibly transfinitely) iterated localizations.

• Of course, any fully faithful morphism is also conservative, and in particular any inverter or equifier is conservative. Moreover, any inserter is also conservative (and faithful), though not generally fully-faithful.

• If $f$ is conservative in $K^{\mathrm{op}}$ (the 1-cell dual), then $f$ is said to be liberal. This joke is due to Carboni, Johnson, Street, and Verity; see codiscrete cofibration.

Related pages

• subcategory
• fully faithful morphism
• faithful morphism
• pseudomonic morphism
• discrete morphism