The cocomplete categories in which every object is a λ-directed colimit of λ-presentable objects of a certain set (Definition 1.17) We will see that there are many equivalent ways in which locally λ-presentable categories can be introduced: they are precisely A category is locally A-presentable iff it has colimits and is generated (in some strong sense) by a set of λ-presentable objects. More generally, an object A is λ-presentable (for a cardinal λ) if every morphism from A to a λ-directed colimit colim i∈ I D i factorizes (essentially uniquely) through some D i.

The basic concept, a finitely presentable object, can be regarded as a generalization of the concept of a finite (or compact) element in a Scott domain, i.e., an element a such that for each directed set every morphism A → colim i∈ I D i factorizes (essentially uniquely) through D i for some i ∈ I. The category of simplicial sets is finitely accessible.

The category -Mod of (left) -modules is locally finitely presentable for any ring. Besides, locally presentable categories are closed under a number of categorical constructions (limits, comma-categories), see also Chapter 2. The category Set of all sets and functions is locally finitely presentable, since every set is the direct limit of its finite subsets, and finite sets are finitely presentable. The idea of “adding on many copies of something” translates into the notion of cell complex, which has long played an important role in algebraic topology.The first chapter is devoted to an important class of categories, the locally presentable categories, which is broad enough to encompass a great deal of mathematical life: varieties of algebras, implicational classes of relational structures, interesting cases of posets (domains, lattices), etc., and yet restricted enough to guarantee a number of completeness and smallness properties. In turn, these authors were formalizing arguments which, for the basic cases derived from the category of simplicial sets, were due to Quillen, Bousfield and Kan, Jardine, Hirschhorn and others. Smith's notion of combinatorial model category (see Beke, Dugger and Rosický ), slightly modified by Barwick with his notion of tractable model category. Stack Exchange Network Stack Exchange network consists of 181 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. We show that every small subobject in X is contained in a small pure subobject. The applicability of this theory, in its abstract form, to model categories came out with J. I am trying proving that this category is combinatorial, I have seen some reference for the cofibran. Let x be a locally-presentable monoidal category. Refer there for historical remarks about these notions. Our discussion is based in large part on the book of Adamek and Rosický about locally presentable and accessible categories. We therefore start with a review of that part of category theory. This kind of discussion usually makes essential use of arguments about cardinality, and the most convenient categorical setting in which to do that is the notion of locally presentable category. A locally finitely presentable category & is equivalent to the category Lex 6 of left exact contravariant functors on a small cocomplete category 8. The properties of these categories are studied extensively, in particular their close relationship with other types of categories. In a nutshell, this says that by adding in enough copies of pushouts along arrows from a fixed set, we can ensure the right lifting property with respect to those arrows. The notions of a locally -presentable and locally -generated categories are introduced, where is a regular cardinal. The theory of model categories is most effective when used in conjunction with the small object argument.

Locally presentable category