Codescent and bicolimits of pseudo-algebras. (English) Zbl 07833143
This paper establishes the bicocompleteness of 2-categories of pseudo-algebras of bifinitary pseudomonads.
The synopsis of the paper goes as follows.
The synopsis of the paper goes as follows.
- § 1
- shows that so-called \(\sigma\)-bicolimits can be reconstructed in any category from oplax bicolimits and bicoequalizers of codescent objects, which is a generalization of an observation that in \(\boldsymbol{Cat}\), \(\sigma\)-bicolimits are obtained as localizations of oplax colimit, which in turn can be obtained as bicoequalizers of codescent objects.
- § 2
- recalls some elements of pseudomonad theory [F. L. Nunes, “Pseudomonads and Descent, PhD Thesis (Chapter 1)”, Preprint, arXiv:1802.01767], particularly the 2-dimensional bar construction [I. J. Le Creurer et al., J. Pure Appl. Algebra 173, No. 3, 293–313 (2002; Zbl 1003.18007)].
- § 3
- establishes the 2-dimensional analogue of Linton theorem of monad theory [F. E. J. Linton, Lect. Notes Math. 80, 75–90 (1969; Zbl 0181.02902)] claiming that bicocompleteness of 2-categories of pseudo-algebras is tanamount to existence of bicoequalizers of codescent objects.
- § 4
- comes to the main result of the paper claiming the bicocompleteness of 2-categories of pseudo-algebras of bifinitary pseudomonads.
Reviewer: Hirokazu Nishimura (Tsukuba)
MSC:
| 18N15 | 2-dimensional monad theory |
| 18N10 | 2-categories, bicategories, double categories |
| 18C15 | Monads (= standard construction, triple or triad), algebras for monads, homology and derived functors for monads |
Keywords:
codescent; bicolimits; pseudo-algebras; pseudomonad; bifinitary; transfinite induction; bicoequalizer of codescent objectReferences:
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