The small object argument is a transfinite construction which, starting from a set of maps in a category, generates a weak factorisation system on that category. As useful as it is, the small object argument has some problematic aspects: it possesses no universal property; it does not converge; and it does not seem to be related to other transfinite constructions occurring in categorical algebra. In this paper, we give an "algebraic" refinement of the small object argument, cast in terms of Grandis and Tholen's natural weak factorisation systems, which rectifies each of these three deficiencies.
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.
We show that the classifying category C(T) of a dependent type theory T with axioms for identity types admits a non-trivial weak factorisation system. We provide an explicit characterisation of the elements of both the left class and the right class of the weak factorisation system. This characterisation is applied to relate identity types and the homotopy theory of groupoids
The Vietoris monad on the category of compact Hausdorff spaces is a topological analogue of the power-set monad on the category of sets. Exploiting Manes' characterisation of the compact Hausdorff spaces as algebras for the ultrafilter monad on sets, we give precise form to the above analogy by exhibiting the Vietoris monad as induced by a weak distributive law, in the sense of Böhm, of the power-set monad over the ultrafilter monad. arXiv:1811.00214v1 [math.CT] 1 Nov 2018
Many kinds of categorical structure require the existence of finite limits,
of colimits of some specified type, and of "exactness" conditions between the
finite limits and the specified colimits. Some examples are the notions of
regular, or Barr-exact, or lextensive, or coherent, or adhesive category. We
introduce a general notion of exactness, of which each of the structures listed
above, and others besides, are particular instances. The notion can be
understood as a form of cocompleteness "in the lex world" -- more precisely, in
the 2-category of finitely complete categories and finite-limit preserving
functors.Comment: 38 pages; v2: final journal version, various minor changes and new
Section 5.9 dealing with filtered colimit
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