nLab algebraic small object argument

The algebraic small object argument

The algebraic small object argument

Idea

The ordinary small object argument is a way of constructing (often combinatorial) weak factorization systems from a set of “generators”. While powerful and useful, it has several defects, such as:

• Its result is not uniquely determined. In particular, it does not converge: we just go on until we’ve gone on long enough, but going on longer would produce a different result.

• Relatedly, it has no universal property. This makes it hard to deal with category-theoretically.

• It does not suffice to construct every weak factorization system, not even every accessible one.

The algebraic small object argument is a refinement of the small object argument, due to Garner, that remedies these defects.

Properties

• The result of the algebraic small object argument is an algebraic weak factorization system, which is “freely generated” by the input data in an appropriate sense.

• If the input data is a set of arrows (rather than a category or double category), then the algebraic right class consists of the algebraically injective objects for the generating class of arrows.

• If the underlying category is locally presentable, then this awfs is in particular an accessible weak factorization system. Conversely, any accessible wfs can be generated by the algebraic small object argument; see Rosicky.

• Not every accessible algebraic wfs can be generated by the algebraic small object argument as above: every accessible wfs admits some algebraic realization that’s generated by the algebraic SOA above, but it could admit other algebraic realizations that are not. However, there is a further refinement of the algebraic SOA, due to Bourke and Garner, that takes as input a double category, and does suffice to generate all accessible algebraic wfs.

Algebraic model structures: Quillen model structures, mainly on locally presentable categories, and their constituent categories with weak equivalences and weak factorization systems, that can be equipped with further algebraic structure and “freely generated” by small data.

structuresmall-set-generatedsmall-category-generatedalgebraicized
weak factorization systemcombinatorial wfsaccessible wfsalgebraic wfs
model categorycombinatorial model categoryaccessible model categoryalgebraic model category
construction methodsmall object argumentsame as $\to$algebraic small object argument