In the context of enriched category theory, we give necessary and sufficient conditions for a module morphism <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\alpha {:}M \rightarrow {{\mathscr {C}}}(F,Z)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>:</mo> <mi>M</mi> <mo stretchy="false">→</mo> <mi mathvariant="script">C</mi> <mo stretchy="false">(</mo> <mi>F</mi> <mo>,</mo> <mi>Z</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> to exhibit a functor <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$Z{:}{{\mathscr {A}}}\rightarrow {{\mathscr {C}}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>Z</mi> <mo>:</mo> <mi mathvariant="script">A</mi> <mo stretchy="false">→</mo> <mi mathvariant="script">C</mi> </mrow> </math></EquationSource> </InlineEquation> as an absolute M-weighted colimit of a functor <InlineEquation ID="IEq3"> <EquationSource Format="TEX">${F{:}{{\mathscr {B}}}\rightarrow {{\mathscr {C}}}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>F</mi> <mo>:</mo> <mi mathvariant="script">B</mi> <mo stretchy="false">→</mo> <mi mathvariant="script">C</mi> </mrow> </math></EquationSource> </InlineEquation>. 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 <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$M'$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>M</mi> <mo>′</mo> </msup> </math></EquationSource> </InlineEquation>.

Absolute Colimits

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2026年7月17日

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Richard Garner,
Ross Street

摘要

In the context of enriched category theory, we give necessary and sufficient conditions for a module morphism $\alpha {:}M \rightarrow {{\mathscr {C}}}(F,Z)$ $α : M \to C (F, Z)$ to exhibit a functor $Z{:}{{\mathscr {A}}}\rightarrow {{\mathscr {C}}}$ $Z : A \to C$ as an absolute M-weighted colimit of a functor ${F{:}{{\mathscr {B}}}\rightarrow {{\mathscr {C}}}}$ $F : B \to C$ . 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 $M'$ $M^{'}$ .