## Small object argument for enriched factorisation systems

### Mathieu Anel – 10 March 2021

If C is a category enriched over S, it makes sense for a factorization system (L,R) on C to be enriched over a factorization system (A,B) on S. This subsumes the classical notion of weak/functorial/unique factorization systems and more. I shall present a version of the small object argument in this context which produces the enriched factorization system generated by a diagram of maps in C. The construction specializes to Quillen's original argument and Kelly’s version for unique factorization system. (jww Simon Henry)

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