Holmstrom Small object argument

nLab

A tool for constructing functorial factorizations in categories. See Hovey, Section 2.1.

Without explaining what the individual terms mean, the statement is as follows:

Theorem: Suppose $C$ is a category containing all small colimits, and $I$ is a set of maps in $C$. Suppose the domains of the maps of $I$ are small relative to $I$-cell. Then there is a functorial factorization $(\gamma, \delta)$ on $C$ such that, for all morphisms $f$ in $C$, the map $\gamma(f)$ is in $I$-cell and the map $\delta(f)$ is in $I$-inj.

See also Dundas, pp26.

Goerss and Schemmerhorn formulates it as follows: If a model category is cofibrantly generated, then factorizations can be chosen to be natural. The proof is also presented.

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