CABool is monadic over almost all categories

Strict Ω-categories Are Monadic over Polygraphs

We give a direct proof that the category of strict ω-categories is monadic over the category of polygraphs.

On the Characterization of Monadic Categories over Set

M-completeness Is Seldom Monadic over Graphs

For a set M of graphs the category CatM of all M-complete categories and all strictly M-continuous functors is known to be monadic over Cat. The question of monadicity of CatM over the category of graphs is known to have an affirmative answer when M specifies either (i) all finite limits, or (ii) all finite products, or (iii) equalizers and terminal objects, or (iv) just terminal objects. We pr...

Kz-monadic Categories and Their Logic

Given an order-enriched category, it is known that all its KZ-monadic subcategories can be described by Kan-injectivity with respect to a collection of morphisms. We prove the analogous result for Kan-injectivity with respect to a collection H of commutative squares. A square is called a Kan-injective consequence of H if by adding it to H Kan-injectivity is not changed. We present a sound logic...

The Monadic Quantiier Alternation Hierarchy over Graphs Is Innnite

We show that the monadic second-order quantiier alternation hierarchy over nite directed graphs and over nite two-dimensional grids is innnite. For this purpose we investigate sets of grids where the width is a function in the height. The innniteness of the hierarchy is then witnessed by n-fold exponential functions for increasing integers n.