The nLab says a categorry $\mathcal can be written as a colimit of a λ \lambda-filtered small diagram whose objects are in S S. Essential localizations of locally finitely presentable categories and presheaf categories are fully described.Author: John Baez Format: MarkdownItex Why is Emily Riehl's definition of "locally presentable" category in *Categories in Context* simpler than the nLab definition? Are they equivalent? Rosický (to appear) as an equational hull of the 2-category LFP (of locally finitely presentable categories), are characterized by the above properties (2) and (3). Analogously, precontinuous categories, introduced in J. This can be viewed as a nonadditive generalization of the classical Roos Theorem characterizing essential localizations of categories of modules. Concrete versions of the characterizations of locally -presentable and -generated categories are given as. presentable and -generated objects are also investigated from this viewpoint. (2) commutativity of filtered colimits with finite limits, (3) distributivity of filtered colimits over arbitrary products, and (4) product-stability of regular epimorphisms. By concrete we mean that starting with a category of -structures, the theories obtained are extensions of the original ones, and the equivalences of categories are concrete isomorphisms.
Locally presentable category generator#
We will show that algebraically exact categories with a regular generator are precisely the essential localizations of varieties and that, in this case, algebraic exactness is equivalent to (1) exactness. Rosický (to appear), as an equational hull of the 2-category VAR of all varieties of finitary algebras. American plain imitation voile ( mercerised plain cloth of quite presentable. This result is new, even in the case where is empty and -filtered colimits are just arbitrary (small) colimits.Īlgebraically exact categories have been introduced in J. those under the other four categories show a considerable fallingoff.
Locally presentable category free#
When consists of the finite discrete categories, these are the finitary varieties.As a by-product of this theory, we prove that the free completion under -filtered colimits distributes over the free completion under limits. When consists of the finite categories, these are precisely the locally finitely presentable categories of Gabriel and Ulmer.
A surprising number of the main results from the theory of accessible categories remain valid in the -accessible context.The locally -presentable categories are defined as the cocomplete -accessible categories. Every -accessible category is accessible thus the choice of different sound provides a classification of accessible categories, as referred to in the title. The -accessible categories are then the categories with -filtered colimits and a small set of -presentable objects which is “dense with respect to -filtered colimits”.We suppose always that satisfies a technical condition called “soundness”: this is the “suitable” case mentioned above. An object of a category is called -presentable when the corresponding representable functor preserves -filtered colimits. A small category is called -filtered when -colimits commute with -limits in the category of sets. Makkai and Pitts Every iso-full subcategory of a locally nitely presentable category closed under limits and ltered colimits ( 0) is re. For a suitable collection of small categories, we define the -accessible categories, generalizing the λ-accessible categories of Lair, Makkai, and Paré here the λ-accessible categories are seen as the -accessible categories where consists of the λ-small categories. ON REFLECTIVE SUBCATEGORIES OF LOCALLY PRESENTABLE CATEGORIES 1307 about iso-full subcategories of L, i.e., those containing every isomorphism of L with domain and codomain in the subcategory: Theorem.