Absolute Colimits

Abstract In the context of enriched category theory, we give necessary and sufficient conditions for a module morphism $$\alpha {:}M \rightarrow {{\mathscr {C}}}(F,Z)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>:</mml:mo> <mml:mi>M</mml:mi> <mml:mo>→</mml:mo> <mml:mi>C</mml:mi> <mml:mo>(</mml:mo> <mml:mi>F</mml:mi> <mml:mo>,</mml:mo> <mml:mi>Z</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> to exhibit a functor $$Z{:}{{\mathscr {A}}}\rightarrow {{\mathscr {C}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Z</mml:mi> <mml:mo>:</mml:mo> <mml:mi>A</mml:mi> <mml:mo>→</mml:mo> <mml:mi>C</mml:mi> </mml:mrow> </mml:math> as an absolute M -weighted colimit of a functor $${F{:}{{\mathscr {B}}}\rightarrow {{\mathscr {C}}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>F</mml:mi> <mml:mo>:</mml:mo> <mml:mi>B</mml:mi> <mml:mo>→</mml:mo> <mml:mi>C</mml:mi> </mml:mrow> </mml:math> . We also review, with short proofs, the various criteria for the weight M itself to be absolute, in the sense that any M -weighted colimit is absolute. Finally, we prove that any absolute M -weighted colimit can be viewed as a colimit weighted by an absolute weight $$M'$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>M</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:math> .