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Cogeneration of algebras in regular categories

Jiří Adámek
1976 Bulletin of the Australian Mathematical Society  
Relations in a category are introduced and investigated by Gri I let [S] . By a relation i?  ...  In this case, the colimit of X is just the cointersection of X. . (i < a) ; more precisely, if V. : X. •*• A (i < a) III. Relations III.l.  ... 
doi:10.1017/s0004972700022796 fatcat:dn2fkayulnh6rj2ejcerrmbdzm

Abstract elementary classes and accessible categories [article]

Tibor Beke, Jiri Rosicky
2011 arXiv   pre-print
We compare abstract elementary classes of Shelah with accessible categories having directed colimits.  ...  Our aim is to relate these two approaches. In Section 5, we introduce a hierarchy of accessible categories with directed colimits.  ...  Accessible categories In order to define accessible categories one just needs the concept of a λ-directed colimit where λ is a regular cardinal number.  ... 
arXiv:1005.2910v2 fatcat:ewjdxud2nnbu5ei6efbjutbz7y

Limits of abstract elementary classes [article]

M. Lieberman, J. Rosický
2015 arXiv   pre-print
Moreover, we provide a broader view of this closure phenomenon, considering a variety of categories of accessible categories with additional structure, and relaxing the assumption that the morphisms be  ...  We show that the category of abstract elementary classes (AECs) and concrete functors is closed under constructions of "limit type," which generalizes the approach of Mariano, Zambrano and Villaveces away  ...  Introduction One of the main virtues of accessible categories is that they are closed under constructions of limit type ( [9] ).  ... 
arXiv:1504.04683v2 fatcat:c6sij4km4vegznvsy42lezeepy

Abstract elementary classes and accessible categories

T. Beke, J. Rosický
2012 Annals of Pure and Applied Logic  
Proposition 4.8 and Claim 4.9 in [8]. Accessible categories In order to define accessible categories one just needs the concept of a λ-directed colimit where λ is a regular cardinal number.  ...  We also investigate ranks of objects in accessible categories, and the effect of accessible functors on ranks.  ...  Our aim is to relate these two approaches. In Section 5, we introduce a hierarchy of accessible categories with directed colimits.  ... 
doi:10.1016/j.apal.2012.06.003 fatcat:3q74a4w3h5hxtc7q63gw4u4umi

A characterization of quasi-toposes

Francis Borceux, Maria Cristina Pedicchio
1991 Journal of Algebra  
A locally a-presentable category 8 can thus be presented as the category of cc-left exact presheaves on a small category $9 with a-colimits.  ...  In this paper, we focus our attention on some semantic conditions.  ...  Consider first the category d of sets provided with a reflexive binary relation, and mappings preserving that relation. Limits in & are computed in the obvious way.  ... 
doi:10.1016/0021-8693(91)90306-s fatcat:alp4vwu4ibeshipcckjueqps7a

Notes on commutation of limits and colimits [article]

Marie Bjerrum, Peter Johnstone, Tom Leinster, William F. Sawin
2015 arXiv   pre-print
On the other hand, if the corresponding class of limits contains either pullbacks or equalizers, then the class of colimits is contained in that of pseudo-filtered colimits.  ...  We show that there are infinitely many distinct closed classes of colimits (in the sense of the Galois connection induced by commutation of limits and colimits in Set) which are intermediate between the  ...  Introduction The fact that colimits over filtered categories commute with finite limits in the category of sets (and in many related categories) is very well known, and most students encounter it as part  ... 
arXiv:1409.7860v2 fatcat:v4wt5isd5va3zarotsyxzneru4

A Theory of Recursive Domains with Applications to Concurrency

Gian Luca Cattani, Marcelo P. Fiore, Glynn Winskel
1997 BRICS Report Series  
<br />In particular, we generalise the traditional approach based on order-theoretic structures to category-theoretic ones.  ...  A motivation for this development is the need of a domain theory for concurrency, with an account of bisimulation.  ...  A relational structure R on an ωCat 0 -category K induces a ωCat 0 -category of relations {K | R} with objects {C | R} given by an object C of K together with a relation R on it, maps are required to be  ... 
doi:10.7146/brics.v4i35.18961 fatcat:ob4e7tmiofe3xaaxact4ub7aby

On sifted colimits in the presence of pullbacks [article]

Ruiyuan Chen
2021 arXiv   pre-print
We show that in a category with pullbacks, arbitrary sifted colimits may be constructed as filtered colimits of reflexive coequalizers.  ...  We also prove generalizations of these results for κ-small sifted and filtered colimits, and their interaction with λ-small limits in place of finite ones, generalizing Garner's characterization of algebraic  ...  of the presheaf category [C op , Set] on the closure of the representables under said colimits.  ... 
arXiv:2109.12708v2 fatcat:5gvg6kcy4ranrl4djmbkjxoq7u

Regular and effective regular categories of locales [article]

Panagis Karazeris, Konstantinos Tsamis
2020 arXiv   pre-print
We do not appeal to the existence of points (which would render the two results trivial) but rely on the treatment of the subject by methods that are valid in the internal logic of a topos.  ...  colimits.  ...  the context of the project "Strengthening Human Resources Research Potential via Doctorate Research" (MIS-5000432).  ... 
arXiv:2004.04404v2 fatcat:nyhh5cthifa4zbdtikcyjnayji

Note on filtered colimits of Hilbert spaces [article]

Branko Nikolić, Alessandra Di Pierro
2019 arXiv   pre-print
The category of Hilbert spaces and contractions has filtered colimits, and tensoring preserves them. We also discuss (problems with) bounded maps.  ...  Introduction While working on formally adding certain directed colimits to the fusion category of Fibonacci anyons, we noticed a gap in the literature in the formal examination of directed colimits in  ...  Explicitly, vectors of L " colimF are equivalence classes on disjoint union of underlying sets of F A, formed by relating Proposition 3.1.  ... 
arXiv:1911.10060v2 fatcat:z46ry2itqfhwdeucvrhgvl3fae

Are colimits of algebras simple to construct?

Jir̆í Adámek, Václav Koubek
1980 Journal of Algebra  
Then we present a construction of colimits of B-algebras, which is based on iteration of pushouts in the base category X.  ...  Generalizing Schubert's result above, he proves the reflexivity of the category fi in case & preserves colimits of n,-chains (or weaker colimits of this type).  ...  With the above description of 6 and k, it is easy to see that (since b, is not in the image of 6): cw + c&J with t,aEAcA+A,. v> Now we can readily verify that the coequalizer off, 1, is not simple.  ... 
doi:10.1016/0021-8693(80)90122-2 fatcat:hp5yj5mvp5e5tjxk25bm6s37gy

Remarks on exactness notions pertaining to pushouts [article]

Richard Garner
2012 arXiv   pre-print
We prove three results relating to diexact categories: firstly, that a category is a pretopos if and only if it is diexact with a strict initial object; secondly, that a category is diexact if and only  ...  if it is Barr-exact, and every pair of monomorphisms admits a pushout which is stable and a pullback; and thirdly, that a small category with finite limits and pushouts of Mal'cev spans is diexact if  ...  Introduction Amongst the first facts that a category theorist will learn about limits and colimits is that certain limit or colimit types suffice for the construction of other ones.  ... 
arXiv:1201.0805v1 fatcat:uspwbrblnzhsrosfqpocokite4

Canonical Selection of Colimits [chapter]

Till Mossakowski, Florian Rabe, Mihai Codescu
2017 Lecture Notes in Computer Science  
Colimits are a powerful tool for the combination of objects in a category.  ...  Therefore, we study how to usefully choose one representative of the isomorphism class of all colimits of a given diagram. We develop criteria that colimit selections should meet.  ...  In most practically relevant signature categories, the construction of a colimit can be reduced to the construction of the colimit in Set of the corresponding sets of symbols.  ... 
doi:10.1007/978-3-319-72044-9_12 fatcat:awjv67h3tndfdot4w4x7efq4f4

Canonical Selection of Colimits [article]

Till Mossakowski, Florian Rabe, Mihai Codescu
2017 arXiv   pre-print
Colimits are a powerful tool for the combination of objects in a category.  ...  In the context of modeling and specification, they are used in the institution-independent semantics (1) of instantiations of parameterised specifications (e.g. in the specification language CASL), and  ...  In most practically relevant signature categories, the construction of a colimit can be reduced to the construction of the colimit in Set of the corresponding sets of symbols.  ... 
arXiv:1705.09363v1 fatcat:eun4yw6h2jc4fkfepxd5ddiayi

Algebraic lattices and locally finitely presentable categories

Hans-E. Porst
2011 Algebra Universalis  
We show that subobjects and quotients respectively of any object K in a locally finitely presentable category form an algebraic lattice. The same holds for the internal equivalence relations on K.  ...  In fact, these results turn out to be-at least in the case of subobjects-nothing but simple consequences of well known closure properties of the classes of locally finitely presentable categories and accessible  ...  We will answer both questions in the style of the previous sections by showing that both, the category Kerp(K) of kernel pairs on K and the category EqRel(K) of internal equivalence relations on K are  ... 
doi:10.1007/s00012-011-0129-0 fatcat:ry2zbmprxjesvfjwcavvmjtqfe
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