Finitary sketches and finitely accessible categories

1995 ◽  
Vol 5 (3) ◽  
pp. 315-322 ◽  
Author(s):  
Jiří Adámek ◽  
Jiří Rosický
Keyword(s):  
Accessible Categories ◽  
Accessible Category

Every accessible category is proved to be sketchable by a sketch with finite colimits. In contrast, a finitely accessible category is presented that cannot be sketched by a finitary sketch, i.e., a sketch with finite limits and finite colimits. Also, a category sketchable by a finitary sketch is found that is not finitely accessible.

scholarly journals WEAK SATURATION AND WEAK AMALGAMATION PROPERTY

2019 ◽  
Vol 84 (3) ◽  
pp. 929-936
Author(s):  
IVAN DI LIBERTI
Keyword(s):  
Model Theory ◽  
Existence And Uniqueness ◽  
Classical Model ◽  
Sufficient Conditions ◽  
Amalgamation Property ◽  
Accessible Categories ◽  
Image Position ◽  
Accessible Category ◽  
Weak Amalgamation

AbstractWe study the two model-theoretic concepts of weak saturation and weak amalgamation property in the context of accessible categories. We relate these two concepts providing sufficient conditions for existence and uniqueness of weakly saturated objects of an accessible category ${\cal K}$. We discuss the implications of this fact in classical model theory.

Final coalgebras in accessible categories

2011 ◽  
Vol 21 (5) ◽  
pp. 1067-1108 ◽  
Author(s):  
PANAGIS KARAZERIS ◽  
APOSTOLOS MATZARIS ◽  
JIŘÍ VELEBIL
Keyword(s):  
The Other ◽  
Linear Orders ◽  
Other Hand ◽  
Accessible Categories ◽  
Self Similar ◽  
Accessible Category

We propose a construction of the final coalgebra for a finitary endofunctor of a finitely accessible category and study conditions under which this construction is available. Our conditions always apply when the accessible category is cocomplete, and is thus a locally finitely presentable (l.f.p.) category, and we give an explicit and uniform construction of the final coalgebra in this case. On the other hand, our results also apply to some interesting examples of final coalgebras beyond the realm of l.f.p. categories. In particular, we construct the final coalgebra for every finitary endofunctor on the category of linear orders, and analyse Freyd's coalgebraic characterisation of the closed unit as an instance of this construction. We use and extend results of Tom Leinster, developed for his study of self-similar objects in topology, relying heavily on his formalism of modules (corresponding to endofunctors) and complexes for a module.

PURE SEMISIMPLE FINITELY ACCESSIBLE CATEGORIES AND HERZOG'S CRITERION

2007 ◽  
Vol 06 (06) ◽  
pp. 1001-1025 ◽  
Author(s):  
A. I. CÁRCELES ◽  
J. L. GARCÍA
Keyword(s):  
Sufficient Conditions ◽  
Representation Type ◽  
Locally Finite ◽  
Finite Representation ◽  
Finite Representation Type ◽  
Artinian Rings ◽  
Semisimple Ring ◽  
Accessible Categories ◽  
Accessible Category ◽  
Necessary And Sufficient

Let [Formula: see text] be a finitely accessible category with products, and assume that its symmetric category [Formula: see text] is also finitely accessible and pure semisimple. We study necessary and sufficient conditions in both categories for [Formula: see text] (and hence [Formula: see text]) to be of locally finite representation type. In particular, we obtain a generalization of Herzog's criterion for finite representation type of left pure semisimple and right artinian rings. As an application, we prove that a left pure semisimple ring R with enough idempotents which has a self-duality is of locally finite representation type if and only if it is left locally finite.

scholarly journals Enriched accessible categories

1996 ◽  
Vol 54 (3) ◽  
pp. 489-501 ◽  
Author(s):  
Francis Borceux ◽  
Carmen Quinteriro
Keyword(s):  
Category Theory ◽  
Closed Category ◽  
Cauchy Completion ◽  
Accessible Categories ◽  
Symmetric Monoidal Closed Category ◽  
Monoidal Closed Category ◽  
Accessible Category

We consider category theory enriched in a locally finitely presentable symmetric monoidal closed category ν. We define the ν-filtered colimits as those colimits weighted by a ν-flat presheaf and consider the corresponding notion of ν-accessible category. We prove that ν-accessible categories coincide with the categories of ν-flat presheaves and also with the categories of ν-points of the categories of ν-presheaves. Moreover, the ν-locally finitely presentable categories are exactly the ν-cocomplete finitely accessible ones. To prove this last result, we show that the Cauchy completion of a small ν-category Cis equivalent to the category of ν-finitely presentable ν-flat presheaves on C.

scholarly journals Obstruction theory in action accessible categories

2013 ◽  
Vol 385 ◽  
pp. 27-46 ◽  
Author(s):  
Alan S. Cigoli ◽  
Giuseppe Metere ◽  
Andrea Montoli
Keyword(s):  
Obstruction Theory ◽  
Accessible Categories

scholarly journals A classification of accessible categories

2002 ◽  
Vol 175 (1-3) ◽  
pp. 7-30 ◽  
Author(s):  
Jiřı́ Adámek ◽  
Francis Borceux ◽  
Stephen Lack ◽  
Jiřı́ Rosický
Keyword(s):  
Accessible Categories

scholarly journals PURE-INJECTIVES RELATIVE TO A COTORSION PAIR: APPLICATIONS

2012 ◽  
Vol 55 (1) ◽  
pp. 59-68
Author(s):  
SERGIO ESTRADA ◽  
PEDRO A. GUIL ASENSIO
Keyword(s):  
Classical Theory ◽  
General Class ◽  
Cotorsion Pair ◽  
Coherent Sheaves ◽  
Accessible Categories

AbstractFinitely accessible categories naturally arise in the context of the classical theory of purity. In this paper we generalise the notion of purity for a more general class and introduce techniques to study such classes in terms of indecomposable pure injectives related to a new notion of purity. We apply our results in the study of the class of flat quasi-coherent sheaves on an arbitrary scheme.

Sign in / Sign up

Export Citation Format

Share Document