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\(\lambda\)-presentable morphisms, injectivity and (weak) factorization systems. (English) Zbl 1111.18002

A morphism \(f \colon A \to B\) in a category \({\mathcal C}\) is called \(\lambda\)-presentable if it is a \(\lambda \)-presentable object of the comma category \((A \downarrow {\mathcal C})\). The author shows that every \(\lambda_m\)-injectivity class (i.e., the class of all the objects injective with respect to some class of \(\lambda\)-presentable morphisms) is a weakly reflective subcategory determined by a functorial weak factorization system cofibrantly generated by the class \(K\) of all the \(\lambda\)-presentable morphisms with respect to which it is injective. This means that the left-hand side of the weak factorization system can be obtained as retractions of transfinite compositions of elements of \(K\).
This was essentially known for small-injectivity classes since at least P. Gabriel and M. Zisman [“Calculus of fractions and homotopy theory”, Ergeb. Math. Grenzgeb. 35 (1967; Zbl 0186.56802)], and referred to as the “small object argument”. An analogous result is obtained for orthogonality classes and factorization systems, where the \(\lambda\)-filtered colimits play the role of the transfinite compositions in the injectivity case. \(\lambda\)-presentable morphisms are also used to organize and clarify some related results (and their proofs), in particular on the existence of enough injectives (resp., pure-injectives) in categories. Finally, locally \(\lambda\)-presentable categories are shown to have all their morphisms as a transfinite composition of \(\lambda\)-presentable ones, which implies that they are cellularly generated (by the set \(C_\lambda\) of their morphisms between \(\lambda\)-presentable objects).

MSC:

18A32 Factorization systems, substructures, quotient structures, congruences, amalgams
18A20 Epimorphisms, monomorphisms, special classes of morphisms, null morphisms
13B22 Integral closure of commutative rings and ideals
13C11 Injective and flat modules and ideals in commutative rings
55U35 Abstract and axiomatic homotopy theory in algebraic topology
18A40 Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.)

Citations:

Zbl 0186.56802
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References:

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