reference: Bill Lawvere, [[axiomatic cohesion]], [pdf](http://www.tac.mta.ca/tac/volumes/19/3/19-03.pdf) ## II. Cohesion versus non-cohesion; quality types He interprets geometric morphisms as "Contrasts" between cohesion and non-cohesion and between variation and non-variation. There is also a distinction between cohesion and variation. +-- {: .num_defn} ###### Definition A full and faithful functor $f^*: S\to F$ between [[extensive category|extensive categories]] which is a [[nLab:Frobenius functor]] in that it is reflective and coreflective by the same functor by definition *exhibits $F$ as a **quality type** over $S$. =-- +-- {: .num_defn} ###### Definition (category of cohesion) =-- ## III. Extensive quality; intensive quality in its rarefied and condensed aspects; the canonical qualities form and substance +-- {: .num_defn} ###### Definition An **extensive quality** quality on a category of cohesion $p:E\to S$ is defined to be a functor $h$ such that * $h$ preserves finite coproducts * the codomain of $h$ is a quality type $q:F\to S$ * $q_! h=p_!$ $$\array{ E&\stackrel{p_!}{\to}&S\\ \downarrow^h&\nearrow^{q_!}\\ F }$$ i.e. an extensive quality of $X$ has the same number of connected pieces as $X$. =--