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Cauchy convergence in V-normed categories (with Dirk Hofmann and Walter Tholen)
Preprint 24-23, DMUC. arXiv 2404.09032

Abstract: Building on the notion of normed category as suggested by Lawvere, we introduce notions of Cauchy convergence and cocompleteness for such categories that differ from proposals in previous works. Key to our approach is to treat them consequentially as categories enriched in the monoidal-closed category of normed sets, i.e., of sets which come with a norm function. Our notions largely lead to the anticipated outcomes when considering individual metric spaces or normed groups as small normed categories (in fact, groupoids), but they can be challenging when trying to establish them for large categories, such as those of semi-normed or normed vector spaces and all linear maps as morphisms, not just because norms of vectors need to be allowed to have value infinity in order to guarantee the existence of colimits of (sufficiently many) infinite sequences. These categories, along with categories of generalized metric spaces, are the key example categories discussed in detail in this paper.
Working with a general commutative quantale V as a value recipient for norms, rather than only with Lawvere's quantale R+ of the extended real half-line, we observe that the categorically atypical, and perhaps even irritating, structure gap between objects and morphisms in the example categories is already present in the underlying normed category of the enriching category of V-normed sets. To show that this normed category and, in fact, all presheaf categories over it, are Cauchy cocomplete, we assume  the quantale V to satisfy a couple of light alternative extra properties which, however, are satisfied in all instances of interest to us.
Of utmost importance to the general theory is the fact that our notion of Cauchy convergence is subsumed by the notion of weighted colimit of enriched category theory. With this theory and, in particular, with results of Albert, Kelly and Schmitt, we are able to prove that all V-normed categories have correct-size Cauchy cocompletions, for V satisfying our light alternative assumptions.
We also show that our notions are suitable to prove a Banach Fixed Point Theorem for contractive endofunctors of Cauchy cocomplete normed categories.

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A variety of co-quasivarieties (with Carlos Fitas and Dirk Hofmann)
Preprint 24-21, DMUC. arXiv 2404.05017

Abstract: It is shown that the duals of several categories of topological flavour, like the categories of ordered sets, generalised metric spaces, probabilistic metric spaces, topological spaces, approach spaces, are quasivarieties, presenting a common proof for all such results.

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On toposes, algebraic theories, semi-abelian categories and compact Hausdorff spaces (with Francis Borceux)
Preprint  24-12, DMUC.

Abstract: In this paper we study the categories (C*)^op and ((C*)^op)^T of T-models in (C*)^op for an arbitrary algebraic theory T, when C is a topos or the category CHaus of compact Hausdorff spaces. It is well-known that, when C is a topos, (C*)^op is semi-abelian. We show that (CHaus*)^op is semi-abelian, as well as ((CHaus*)^op)^T, and that, when C is a topos having locales of subobjects, ((C*)^op)^T is also semi-abelian. In addition, we prove the representability of actions in (CHaus*)^op.
 

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Effective descent morphisms of filtered preorders (with George Janelidze)
Preprint  23-37, DMUC. arXiv 2312.14315

Abstract: We characterize effective descent morphisms of what we call filtered preorders, and apply these results to slightly improve a known result, due to the first author and F. Lucatelli Nunes, on the effective descent morphisms in lax comma categories of preorders. A filtered preorder, over a fixed preorder X, is defined as a preorder A equipped with a profunctor X-> A and, equivalently, as a set A equipped with a family (A_x)_{x\in X} of upclosed subsets of A with x' \leq x => A_x\subseteq A_x'.

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M.M. Clementino, F. Lucatelli Nunes, R. Prezado, Lax comma categories: cartesian closedness, extensivity, topologicity, and descent. Theory Appl. Categ. 41 (16) (2024). Preprint 24-26, DMUC. arXiv 2405.03773

M.M. Clementino, D. Rodelo, A note on Mal'tsev objects. Port. Math. (to appear). Preprint  23-36, DMUC. arXiv 2311.17628

F. Borceux, M. M. Clementino, On coherent systems of subobjects with application to torsion theory. Theory Appl. Categ. 40 (2024), 249-277. Preprint  22-33, DMUC. arXiv 2210.14337

M.M. Clementino, F. Lucatelli Nunes, Lax comma 2-categories and admissible 2-functors. Theory Appl. Categ. 40 (2024), 180-226. Preprint  20-06, DMUC. arXiv 2002.03132

M.M. Clementino, F. Lucatelli Nunes, Lax comma categories of ordered sets. Quaestiones Mathematicae 46 (S1) (2023) 145-159. Preprint  22-49, DMUC. arXiv 2212.13541

M.M. Clementino, C. Ruivo, On split extensions of preordered groups. Port. Math. 80 (2023) 327-341. Preprint  22-40, DMUC. arXiv 2212.07064

M.M. Clementino, A. Montoli, D. Rodelo, On lax protomodularity for Ord-enriched categories. J. Pure Appl. Algebra 227, Issue 8 (2023) 107348. Preprint  22-22, DMUC. arXiv 2210.14332

M.M. Clementino, C. Fitas,  On presheaf submonads of quantale enriched categories. Quaestiones Mathematicae 46 (10) (2023) 2079-2108. Preprint  22-02, DMUC. arXiv 2202.06805

M.M. Clementino, An invitation to topological semi-abelian algebras. In: New perspectives in Algebra, Topology and Categories, Coimbra Mathematical Texts, Vol. 1, Springer Nature and University of Coimbra (2021), pp. 27-66.
 
M.M. Clementino, (T,V)-Cat is extensive. Theory Appl. Categ. 36, nº 14 (2021), 368-378. Preprint 21-18, DMUC.

M.M. Clementino, A. Montoli, On the categorical behaviour of V-groups. J. Pure Appl. Algebra 225 (2021) 106550. Preprint  20-18, DMUC. arXiv 2005.07738


M.M. Clementino, I. López Franco, Lax orthogonal factorisations in ordered structures. Theory Appl. Categ. 35, nº 36 (2020) 1379-1423. Preprint  17-06, DMUC. arXiv 1702.02602

M.M. Clementino, D. Hofmann, W. Ribeiro, Cartesian closed exact completions in topology. J. Pure Appl. Algebra 224 (2020) 610-629. Preprint  18-46, DMUC. arXiv 1811.03993. DOI: 10.1016/j.jpaa.2019.06.003

M.M. Clementino, G. Janelidze, Another note on effective descent morphisms of topological spaces and relational algebras. Topology Appl. 273 (2020), 106961. Preprint  18-32, DMUC.

F. Cagliari, M.M. Clementino, Topological groups have representable actions. Bulletin of the Belgium Math. Society -- Simon Stevin 26 (2019), 519-526. Preprint  18-29, DMUC.

M.M. Clementino, N. Martins-Ferreira, A. Montoli, On the categorical behaviour of preordered groups. J. Pure Appl. Algebra 223 (2019), 4226-4245. Preprint  18-22, DMUC. DOI: 10.1016/j.jpaa.2019.01.006

M.M. Clementino, J. Picado, A. Pultr, The other closure and complete sublocales. Appl. Categ. Structures 26 (2018), 891-906. Preprint  17-40, DMUC. DOI: 10.1007/s10485-018-9516-4

M.M. Clementino, M. Gran, G. Janelidze, Some remarks on protolocalizations and proto-additive reflections. J. Algebra Appl., Volume 17, Number 11 (2018), 1850207. DOI: 10.1142/S0219498818502079. Preprint  17-13, DMUC. arXiv 1702.08822

M.M. Clementino, Ignacio López Franco, Lax orthogonal factorisations in quantale-monad-enriched categories. Logical Methods in Computer Science Vol. 13 (3:32) 2017, pp. 1–16. Preprint  17-03, DMUC. arXiv 1701.05510

M.M. Clementino, D. Hofmann, The rise and fall of V-functors, Fuzzy Sets and Systems 321 (2017) 29-49. Preprint  16-06, DMUC.

M.M. Clementino, Ignacio López Franco, Lax orthogonal factorisation systems. Adv. Math. 302 (2016) 458-528. Preprint  15-09, DMUC; arXiv 1503.06469

M.M. Clementino, D. Hofmann, G. Janelidze, On exponentiable morphisms in classical algebras. Appl. Categ. Structures 24 (2016) 733-742. Preprint  15-56, DMUC

F. Cagliari, M.M. Clementino, S. Mantovani, Fibrewise injectivity in order and topology. Topology Appl. 200 (2016) 61-78. Preprint  14-20, DMUC

D. Chikhladze, M.M. Clementino, D. Hofmann, Representable (T,V)-categories. Appl. Categ. Structures 23 (2015) 829-858. Preprint  12-47, DMUC

M.M. Clementino, A. Montoli, L. Sousa, Semidirect products of (topological) semi-abelian algebras. J. Pure Appl. Algebra 219 (2015) 183-197. Preprint 13-47, DMUC

F. Borceux, M.M. Clementino, A. Montoli, On the representability of actions for topological algebras. In: Categorical Methods in Algebra and Topology, Textos de Matemática, DMUC, Vol. 46 (2014) 41-66. Preprint  14-19, DMUC

M.M. Clementino, W. Tholen, From lax monad extensions to Topological theories. In: Categorical Methods in Algebra and Topology, Textos de Matemática, DMUC, Vol. 46 (2014) 99-123. Preprint  14-39, DMUC

M.M. Clementino, D. Hofmann, A. Montoli, Covering morphisms in categories of relational algebras. Appl. Categ. Structures 22 (2014) 767-788. Preprint 12-45, DMUC

M.M. Clementino, E. Colebunders, W. Tholen, Lax Algebras as Spaces. In: Monoidal Topology. A Categorical Approach to Order, Metric and Topology, Encyclopedia Math. Appl. 153, Cambridge Univ. Press (2014), pp. 375-465: contents

M.M. Clementino, D. Hofmann, G. Janelidze, The monads of classical algebra are seldom weakly cartesian. J. Homotopy Relat. Struct. 9 (2014) 175-197. Preprint 12-46, DMUC

F. Cagliari, M.M. Clementino, S. Mantovani, Fibrewise complete posets and continuous lattices. In: Recent Progress for Topology, Computer Science, Fuzzy Mathematics and Economics, Univ. of the Basque Country, Bilbao (2013), pp. 69-75.

M.M. Clementino, D. Hofmann, G. Janelidze, On exponentiability of étale algebraic homomorphisms. J. Pure Appl. Algebra 217 (2103) 1195-1207. Preprint 11-35, DMUC.

M.M. Clementino, W. Tholen, Proper maps for lax algebras and the Kuratowski-Mrówka Theorem. Theory Appl. Categ. 27 (2012) 327-346. Preprint 12-16, DMUC

F. Cagliari, M.M. Clementino, S. Mantovani, Fibrewise injectivity and Kock-Z"oberlein monads. J. Pure Appl. Algebra 216 (2012) 2411-2424. Preprint 11-32, DMUC

M.M. Clementino, D. Hofmann, Descent morphisms and a van Kampen Theorem in categories of lax algebras. Topology Appl. 159 (2012) 2310-2319. Preprint 11-01, DMUC

M.M. Clementino, G. Janelidze, A note on effective descent morphisms of topological spaces and relational algebras. Topology Appl. 158 (2011) 2431-2436. Preprint 10-55, DMUC

M.M. Clementino, D. Hofmann, On the completion monad via the Yoneda embedding in quasi-uniform spaces. Topology Appl. 158 (2011) 2423-2430. Preprint 10-46, DMUC

M.M. Clementino, A note on the categorical van Kampen Theorem. Topology Appl. 158 (2011) 926-929. Preprint 10-45, DMUC

M.M. Clementino, G. Gutierres, On regular and homological closure operators. Cah. Topol. Géom. Différ. Catég. LI (2010) 127-142. Preprint 09-49, DMUC

A. Akhvlediani, M.M. Clementino, W. Tholen, On the categorical meaning of Hausdorff and Gromov distances, I. Topology Appl. 157 (2010) 1275-1295. Preprint 09-01, DMUC    

M.M. Clementino, D. Hofmann, Lawvere completeness in topology. Appl. Categ. Structures 17 (2009) 175-210.

M.M. Clementino, D. Hofmann, I. Stubbe, Exponentiable functors between quantaloid-enriched categories. Appl. Categ. Structures 17 (2009) 91-101. Preprint 06-50, DMUC

M.M. Clementino, D. Hofmann, Relative injectivity as cocompleteness for a class of distributors. Theory Appl. Categ. 21 (2008) 210-230.

F. Borceux, M.M. Clementino, M. Gran, L. Sousa, Protolocalisations of homological categories. J. Pure Appl. Algebra 212 (2008) 1898–1927. Preprint 06-60, DMUC

M.M. Clementino, D. Hofmann, On some special classes of continuous maps. Chapter 39, Part 4, of  Open Problems in Topology 2, pages 367-376, edited by Elliott Pearl, Elsevier 2007. Preprint 06-37, DMUC

M.M. Clementino, Towards categorical behaviour of groups, CIM Bulletin 23 (2007), 7-12.

M.M. Clementino, D. Dikranjan, W. Tholen, Torsion theories and radicals  in normal categories. J. Algebra 305 (2006) 98-129. Preprint 05-19, DMUC

F. Borceux, M.M. Clementino, Topological protomodular algebras. Topology Appl. 153 (2006) 3085-3100. Preprint 04-10, DMUC

M.M. Clementino, D. Hofmann, Exponentiation in V-categories. Topology Appl. 153 (2006) 3113-3128.

F. Borceux, M.M. Clementino, Topological semi-abelian algebras. Adv. Math. 190 (2005) 425-453.

M.M. Clementino, D. Hofmann, G. Janelidze, Local homeomorphisms via ultrafilter convergence. Proc. Amer. Math. Soc. 133 (2005) 917-922.

M.M. Clementino, D. Hofmann, Effective descent morphisms in categories of lax algebras. Appl. Categ. Structures 12 (2004) 413-425. Preprint 02-20, DMUC

M.M. Clementino, D. Hofmann, On extensions of lax monads. Theory Appl. Categ. 13 (2004) 41-60.

M.M. Clementino, D. Hofmann, Walter Tholen, One setting for all: metric, topology, uniformity, approach structure. Appl. Categ. Structures 12 (2004) 127-154.

M.M. Clementino, E. Giuli, W. Tholen, A Functional Approach to General Topology. In: Categorical Foundations. Special Topics in Order, Topology, Algebra, and Sheaf Theory, Encyclopedia Math. Appl. 97, CUP (2004), pp. 103-163.

M.M. Clementino, D. Hofmann, Walter Tholen, Exponentiability in categories of lax algebras. Theory Appl. Categ. 11 (2003) 337-352.

M.M. Clementino, W. Tholen, Metric, Topology and Multicategory - A Common Approach. J. Pure Appl. Algebra 179 (2003) 13-47.

M.M. Clementino, D. Hofmann, Topological features of lax algebras. Appl. Categ. Structures 11 (2003) 267-286.

M.M. Clementino, D. Hofmann, Walter Tholen, The convergence approach to exponentiable maps. Port. Math. (N.S.) 60 (2003) 139-160.

M.M. Clementino, D. Hofmann, On limit stability of special classes of continuous maps. Topology Appl. 125 (2002) 471-488.

M.M. Clementino, D. Hofmann, Triquotient maps via ultrafilter convergence. Proc. Amer. Math. Soc. 130 (2002) 3423-3431.

M.M. Clementino, On finite triquotient maps. J. Pure Appl. Algebra 168 (2002) 387-389.

M.M. Clementino, W. Tholen, A note on local compactness. Rend. Ist. Mat. Univ. Trieste, Suppl. 2, XXXII (2001) 17-31.

M.M. Clementino, On connectedness via closure operators. Appl. Categ. Structures 9 (2001) 539-556.

M.M. Clementino, E. Giuli, W. Tholen, What is a quotient map with respect to a closure operator? Appl. Categ. Structures 9 (2001) 139-151.

M.M. Clementino, W. Tholen, Separated and connected maps. Appl. Categ. Structures 6 (1998) 373-401.

M.M. Clementino, W. Tholen, A Characterization of the Vietoris Topology. Topology Proc. 22 (1997) 71-95.

M.M. Clementino, W. Tholen, Separation versus connectedness. Topol. Appl. 75 (1997) 143-181.

M.M. Clementino, W. Tholen, Tychonoff's Theorem in a Category. Proc. Amer. Math. Soc. 124 (1996) 3311-3314.

M.M. Clementino, E. Giuli, W. Tholen, Topology in a Category: Compactness. Port. Math. (N.S.) 53 (1996) 397-433.

M.M. Clementino, On categorical notions of compact object. Appl. Categ. Structures 4 (1996) 15-29.

M.M. Clementino, Constant morphisms and constant subcategories. Appl. Categ. Structures 3 (1995) 119-137.

M.M. Clementino, On connectedness and disconnectedness. In: Topology with Applications (Szekszard, Hungary, 1993), Math. Studies 4 (Bolyai Soc.), pgs. 71-82.

M.M. Clementino, Weakly hereditary regular closure operators. Topology Appl. 49 (1993) 129-139.

M.M. Clementino, Hausdorff separation in categories. Appl. Categ. Structures 1 (1993) 285-295.

M.M. Clementino, T0- and T1-reflections. Comment. Math. Univ. Carolin. 33 (1992) 493-504.

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Editions
                                Applied Categorical Structures
                               Cahiers de Topologie et Géométrie Différentielle Catégoriques                              
                               Categories and General Algebraic Structures with Applications
                      Portugaliae Mathematica
                          Theory and Applications of Categories

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