Eilenberg-Moore categories and Kan-injectivity

Lurdes Sousa
Monday, July 22, 2013 - from 17:00 to 17:25
In a poset enriched category $\cal X$, an object $X$ is said to be left Kan-injective w.r.t. a morphism $h: A\to B$, if for every $f:A\to X$ the left Kan-extension $f/h$ of $f$ along $h$ exists and fulfils $f=(f/h)h$. In the category of $T_0$ topological spaces the left Kan-injective spaces w.r.t. subspace inclusions are precisely the continuous lattices; and w.r.t. all dense subspace inclusions they are the continuous Scott domains. In the category of locales, the left Kan-injective objects w.r.t. sublocale inclusions closed under finite suprema are the stably locally compact locales. In these examples, and many others, the left Kan-injective objects are just the Eilenberg-Moore algebras of a Kock-Z\"oberlein (KZ) monad [2], as observed by Mart\'{\i}n Escard\'{o} and others in a number of papers in the late 90's. In [1] we defined left Kan-injectivity also for morphisms: suppose $X$ and $Y$ are left Kan-injective w.r.t. $h:A\to B$. Then the morphism $g:X\to Y$ is called left Kan-injective w.r.t. $h$ provided that the equality $(gf)/h=g(f/h)$ holds for all $f:A\to X$. We thus have, for every class $\mathcal {H}$ of morphisms, a (non-full, in general) subcategory $\,\mbox{LInj}\, \mathcal {H}$ of all objects and morphisms that are left Kan-injective w.r.t. every member of $\mathcal {H}$. This way, we obtain a Galois connection between classes of morphisms and subcategories enjoying good properties on limits, colimits and reflectivity, resembling the one between orthogonality and full reflectivity. In particular, every Eilenberg-Moore category of a KZ-monad is a left Kan-injective subcategory in the same manner as every Eilenberg-Moore category of an idempotent monad is an orthogonal (full) subcategory. And, similarly to the classical Orthogonal Subcategory Problem of Peter Freyd and Max Kelly, this leads us to a ``Left Kan-injective Problem": When is a subcategory of $\cal X$ of the form $\,\mbox{LInj}\, \mathcal {H}$ a Eilenberg-Moore category of a KZ-monad over $\cal X$? An answer to this problem, obtained in recent joint work with Ji\v r\'{\i} Ad\'amek and Ji\v r\'{\i} Velebil, will be presented in this talk.