Abstract. Algebraic weak factorisation systems (awfs) refine weak factorisation systems by requiring that the assignations sending a map to its first and second factors should underlie an interacting comonad-monad pair on the arrow category. We provide a comprehensive treatment of the basic theory of awfs-drawing on work of previous authors-and complete the theory with two main new results. The first provides a characterisation of awfs and their morphisms in terms of their double categories of left or right maps. The second concerns a notion of cofibrant generation of an awfs by a small double category; it states that, over a locally presentable base, any small double category cofibrantly generates an awfs, and that the awfs so arising are precisely those with accessible monad and comonad. Besides the general theory, numerous applications of awfs are developed, emphasising particularly those aspects which go beyond the non-algebraic situation.

Given a locally presentable enriched category E together with a small dense full subcategory A of arities, we study the relationship between monads on E and identity-on-objects functors out of A, which we call Apretheories. We show that the natural constructions relating these two kinds of structure form an adjoint pair. The fixpoints of the adjunction are characterised as the A-nervous monads-those for which the conclusions of Weber's nerve theorem hold-and the A-theories, which we introduce here.The resulting equivalence between A-nervous monads and A-theories is best possible in a precise sense, and extends almost all previously known monad-theory correspondences. It also establishes some completely new correspondences, including one which captures the globular theories defining Grothendieck weak ω-groupoids.Besides establishing our general correspondence and illustrating its reach, we study good properties of A-nervous monads and A-theories that allow us to recognise and construct them with ease. We also compare them with the monads with arities and theories with arities introduced and studied by Berger, Melliès and Weber. arXiv:1805.04346v1 [math.CT] 11 May 2018 K G G J b b D Explicitly, J is given on objects by Ja = Ha, and on homs by B(a, b)

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We describe a perfect correspondence between skew monoidal categories and certain generalised multicategories, called skew multicategories, that arise in nature.

We investigate the extent to which the weak equivalences in a model category can be equipped with algebraic structure. We prove, for instance, that there exists a monad T such that a morphism of topological spaces admits T -algebra structure if and only it is a weak homotopy equivalence. Likewise for quasi-isomorphisms and many other examples. The basic trick is to consider injectivity in arrow categories. Using algebraic injectivity and cone injectivity we obtain general results about the extent to which the weak equivalences in a combinatorial model category can be equipped with algebraic structure.