A mathematical preprint sets out several equivalent ways to recognize an absolute weighted colimit, a construction in enriched category theory, and shows how the same colimit can be recast with a Cauchy weight after both the diagram and the weight are changed. Its central question is when a module morphism can exhibit a functor as an absolute weighted colimit.
The work is a formal analysis of categories, modules, functors and weighted colimits using enriched and bicategorical methods. It does not report an empirical sample or dataset. The document is identified as arXiv:2608.25607v1 and dated 26 August 2026.
Several routes to the same conclusion
Theorem 1.1 brings three descriptions of the same situation together. It links an absolute weighted colimit with explicit alpha and beta module-morphism conditions, identified in the paper as equations (1.4) and (1.5), and with representability of the composite module F∘M. In practical terms within the paper’s formal setting, the result lets the same question be approached through equivalent categorical tests.
Theorem 2.1 shifts the focus from one particular colimit to the weight M, the part of the construction that specifies how a diagram is combined. It gives equivalent criteria for M to be a left adjoint in the paper’s module category and for every M-weighted colimit to be absolute. The criteria include formulations using Yoneda, right-lifting and pointwise right adjoints.
The main reduction comes from a replacement construction. The setup uses a functor J that is identity on objects, a fully faithful functor F′ and a composite replacement module M′. Theorem 3.1 states that, after this specified replacement of the diagram and weight, an absolute weighted colimit has a Cauchy replacement weight M′.
Familiar constructions in the examples
The abstract framework is tied to a construction in the paper’s Set example. For a parallel pair of maps, the relevant M-weighted colimit of F is a coequaliser of u and v.
A separate example treats Kleisli objects in a 2-category. It gives an exact if-and-only-if statement: a Kleisli object is absolute precisely when it is the Kleisli object of a trivial monad.
The codescent example is more limited in scope. It supplies several sufficient conditions for a codescent cocone to be absolute, without presenting them as a complete general characterization.
What the later propositions add
The later results examine Cauchy weights under finite-presentability assumptions. Proposition 5.1 states that, under local finite-presentability assumptions, a finitely presentable or Cauchy module M takes values in Vfp, the paper’s class of finitely presentable objects. Proposition 5.2 adds that, in its directed cosieve setting, a finitely presentable M is the Kan extension of its restriction to some subcategory Ci, with the same conclusion noted for a Cauchy module.
The final structural result is narrower. In the stated vector-space and groupoid setting, M is Cauchy exactly when it is finite dimensional at every object and zero on all but finitely many connected components. That is a necessary-and-sufficient description for the specified setting, not a general characterization of Cauchy modules in every categorical environment.
The assumptions remain part of the result
The conclusions depend on the categorical frameworks in which they are stated. The absolute-weight criteria use the paper’s enriched and bicategorical assumptions, while the Cauchy reformulation requires the stated factorization and replacement setup. The codescent discussion gives selected sufficient conditions, and the final finite-dimensional characterization is confined to the specified vector-space and groupoid setting.
The declarations state that no data were generated for the manuscript. They report support from Discovery Grants DP160101519 and DP190102432 and report no competing interests.
Paper data and sources
Original title: Absolute colimits
Authors: Richard Garner, Ross Street
Journal/Repository: Absolute colimits, Applied Categorical Structures 34:43 (dedicated to Bob Paré, 2026)
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text