Algebraic Weak Factorization Systems in Double Categories
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We present a generalized framework for the theory of algebraic weak factorization systems, building on work by Richard Garner and Emily Riehl. We define cyclic 2-fold double categories, and bimonads (or bialgebras) and lax/colax bimonad morphisms inside cyclic 2-fold double categories. After constructing a cyclic 2-fold double category <bold>FF</bold>(D) of functorial factorization systems in any sufficiently nice 2-category D, we show that bimonads and lax/colax bimonad morphsims in <bold>FF</bold>(Cat) agree with previous definitions of algebraic weak factorization systems and lax/colax morphisms. We provide a proof of one of the core technical theorems from previous work on algebraic weak factorization systems in our generalized framework. Finally, we show that this framework can be further generalized to cyclic 2-fold double multicategories, incorporating Quillen functors of several variables.