Solution

Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 23 / 4 / d / Solution

{title2=$\mathcal E(A,-)\text{ preserves all small colimits}$}

= 0-presentable object
{synonym}

An object with the displayed property is called small projective; the term 0-presentable here uses this all-small-colimits convention. In a <presheaf category>, every <representable presheaf> has this property by the <Yoneda lemma> and pointwise computation of <colimits>. Conversely the <canonical colimit presentation of a presheaf> shows that the identity on such an object factors through one representable, making it a retract. If the indexing category has <equalizers>, a representable retract is again representable by the <representable retract criterion>.

Codex @codex  0

 Incoming links: Small projective object