×

Coequalizers and free triples. II. (English) Zbl 1422.18007

Let \(T\) be an endofunctor on a cocomplete category \(\mathcal{X}\) which has a factorization system (\(\mathcal{E}\),\(\mathcal{M}\)) for classes \(\mathcal{E}\) of epimorphisms and \(\mathcal{M}\) of monomorphisms. This paper examines the questions of when \(T\) freely generates a triple (or monad) on \(\mathcal{X}\), when an object \(X\) of \(\mathcal{X}\) freely generates a \(T\)-algebra (i.e., a morphism \(TY \to Y\)), and when the category of \(T\)-algebras has certain colimits. This article is actually a rewriting of an unpublished manuscript [M. Barr, Coequalizers and free triples, II. Manuscript (1979)] by the first author. This manuscript, which was itself a follow-up of his well-known article [M. Barr, Math. Z. 116, 307–322 (1970; Zbl 0194.01701)], was, in the words of the current authors, in need of a “clean-up”, for its“serious difficulties in the exposition” and “clumsiness”. A concept, called \(T\)-horn, is introduced to “confine the difficulties [of the manuscript] to one place and systematize [its] main construction”, which now consists of a transfinite sequence of successive approximations to the free \(T\)-algebras over an object \(X\) in \(\mathcal{X}\), or to coequalizers or other colimits in the category of \(T\)-algebras, and which stabilizes at them, whenever they exist, at least if \(\mathcal{X}\) is \(\mathcal{M}\)-well-powered. The authors point out the similarities and differences of this construction with others in various papers with related goals, in particular [V. Koubek and J. Reiterman, JPAA 14, 195–231 (1979; Zbl 0403.18002)]. The special case of \(\mathcal{X = Set}\) is treated in the last section.
(Note from the reviewer: The definition 2.3 of a factorization system (\(\mathcal{E}\),\(\mathcal{M}\)) is slightly incorrect where it appears, as it should require the unicity of the diagonal “filled in” map. This is of no consequence as the paper later requires the blanket assumption that \(\mathcal{E}\) and \(\mathcal{M}\) are made of epimorphisms and monomorphisms respectively, from which the unicity follows. Also, in 4.1 (resp. 4.2), it seems the phrase “between admissible maps \(X \to Z\) and maps \(JX \to K(Z,z)\) (resp. \(H_n \to K(Z,z))\)”) should rather be “between maps \(X \to Z\) and admissible maps \(JX \to K(Z,z)\)(resp. \(H_n \to K(Z,z))\)”)

MSC:

18C15 Monads (= standard construction, triple or triad), algebras for monads, homology and derived functors for monads
18A30 Limits and colimits (products, sums, directed limits, pushouts, fiber products, equalizers, kernels, ends and coends, etc.)
18A32 Factorization systems, substructures, quotient structures, congruences, amalgams
PDFBibTeX XMLCite
Full Text: Link

References:

[1] J. Ad´amek (1977), Colimits of algebras revisited. Bull Australian Math. Soc. 17, 433-450. · Zbl 0365.18007
[2] J. Ad´amek and V. Koubek (1980), Are colimits of algebras easy to construct? J. Algebra, 66, 226-250. · Zbl 0446.18003
[3] J. Ad´amek and V. Trnkov´a (2011), Initial algebras and terminal coalgebras in many-sorted sets. Math. Struct. Comp. Science 21, 481-509. · Zbl 1214.68227
[4] M. Barr (1970), Coequalizers and free triples. Math. Z. 116, 307-322. · Zbl 0194.01701
[5] M. Barr(1979), Coequalizers and free triples, II. Manuscript. · Zbl 0194.01701
[6] Camell Kachour (2013), Aspects of Globular Higher Category Theory. Doctoral thesis, Macquarie University. · Zbl 1276.18005
[7] G.M. Kelly (1980), A unified treatment of transfinite constructions of free algebras, colimits, associated sheaves, and so on. Bull Australian Math Soc, 22, 1-83. · Zbl 0437.18004
[8] V. Koubek and J. Reiterman (1979), Categorical construction of free algebras, colimits, and completions of partial algebras. J Pure Appl Algebra 14, 195-231. · Zbl 0403.18002
[9] H. Schubert (1972), Categories. Springer-Verlag. · Zbl 0253.18002
[10] Dept. Math. & Stats., McGill University, Montreal, QC
[11] Dept. Math., Clark University, Worcester, MA Dept. Math., Concordia University, Mon
[12] treal, QC
[13] Email: michael.barr@mcgill.ca
[14] jkennison@clarku.edu
[15] r.raphael@concordia.ca
[16] This article may be accessed at http://www.tac.mta.ca/tac/
[17] THEORY AND APPLICATIONS OF CATEGORIES will disseminate articles that significantly advance
[18] the study of categorical algebra or methods, or that make significant new contributions to mathematical
[19] science using categorical methods. The scope of the journal includes: all areas of pure category theory,
[20] including higher dimensional categories; applications of category theory to algebra, geometry and topology
[21] and other areas of mathematics; applications of category theory to computer science, physics and other
[22] mathematical sciences; contributions to scientific knowledge that make use of categorical methods.
[23] Articles appearing in the journal have been carefully and critically refereed under the responsibility of
[24] members of the Editorial Board. Only papers judged to be both significant and excellent are accepted
[25] for publication.
[26] Subscription informationIndividual subscribers receive abstracts of articles by e-mail as they
[27] are published. To subscribe, send e-mail to tac@mta.ca including a full name and postal address. Full
[28] text of the journal is freely available at http://www.tac.mta.ca/tac/.
[29] Information for authorsLATEX2e is required. Articles may be submitted in PDF by email
[30] directly to a Transmitting Editor following the author instructions at
[31] http://www.tac.mta.ca/tac/authinfo.html.
[32] Managing editor.Robert Rosebrugh, Mount Allison University: rrosebrugh@mta.ca
[33] TEXnical editor.Michael Barr, McGill University: michael.barr@mcgill.ca
[34] Assistant TEX editor.Gavin Seal, Ecole Polytechnique F´ed´erale de Lausanne:
[35] gavin seal@fastmail.fm
[36] Transmitting editors.
[37] Clemens Berger, Universit´e de Nice-Sophia Antipolis: cberger@math.unice.fr
[38] Julie Bergner, University of Virginia: jeb2md (at) virginia.edu
[39] Richard Blute, Universit´e d’ Ottawa: rblute@uottawa.ca
[40] Gabriella B¨ohm, Wigner Research Centre for Physics: bohm.gabriella (at) wigner.mta.hu
[41] Valeria de Paiva: Nuance Communications Inc: valeria.depaiva@gmail.com
[42] Richard Garner, Macquarie University: richard.garner@mq.edu.au
[43] Ezra Getzler, Northwestern University: getzler (at) northwestern(dot)edu
[44] Kathryn Hess, Ecole Polytechnique F´ed´erale de Lausanne: kathryn.hess@epfl.ch
[45] Dirk Hoffman, Universidade de Aveiro: dirk@ua.pt
[46] Pieter Hofstra, Universit´e d’ Ottawa: phofstra (at) uottawa.ca
[47] Anders Kock, University of Aarhus: kock@math.au.dk
[48] Joachim Kock, Universitat Aut‘onoma de Barcelona: kock (at) mat.uab.cat
[49] Stephen Lack, Macquarie University: steve.lack@mq.edu.au
[50] F. William Lawvere, State University of New York at Buffalo: wlawvere@buffalo.edu
[51] Tom Leinster, University of Edinburgh: Tom.Leinster@ed.ac.uk
[52] Matias Menni, Conicet and Universidad Nacional de La Plata, Argentina: matias.menni@gmail.com
[53] Ieke Moerdijk, Utrecht University: i.moerdijk@uu.nl
[54] Susan Niefield, Union College: niefiels@union.edu
[55] Robert Par´e, Dalhousie University: pare@mathstat.dal.ca
[56] Kate Ponto, University of Kentucky: kate.ponto (at) uky.edu
[57] Jiri Rosicky, Masaryk University: rosicky@math.muni.cz
[58] Giuseppe Rosolini, Universit‘a di Genova: rosolini@disi.unige.it
[59] Alex Simpson, University of Ljubljana: Alex.Simpson@fmf.uni-lj.si
[60] James Stasheff, University of North Carolina: jds@math.upenn.edu
[61] Ross Street, Macquarie University: ross.street@mq.edu.au
[62] Tim Van der Linden, Universit´e catholique de Louvain: tim.vanderlinden@uclouvain.be
[63] R. J. Wood, Dalhousie University: rjwood@mathstat.dal
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.