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 (i.e. there is a $g$ s.t. $(g\dashv f^*\dashv g)$ ) by definition *exhibits $F$ as a **quality type** over $S$. Short: A quality type is a full and faithful Frobenius pair/triple. =-- +-- {: .num_defn} ###### Definition (category of cohesion) Let $E,S$ be cartesian-closed extensive categories. $E$ is called a **category of cohesion relative to $S$ if $$(p_!\dashv p^*\dashv p_*\dashv p^!):E\stackrel{p_*}{\to} S$$ is an [[nLab:adjoint quadruple]] of functors such that (a) $p_!$ preserves finite products and $p^!$ is full and faithful. Thus for toposes this would be phrased as $p$ is "connected surjective" and "local", $p^!$ is a subtopos, and $p^*$ is an exponential ideal. (b) $p_!$ preserves $S$-parametrized powers in that $$p_![p^* W,X]\simeq [W,p_! X]$$ is a natural isomorphism for all $X\in E$ and all $W\in S$. This "continuity" property follows from (a) if all hom sets in $S$ are finite; it also holds if the contrast with $S$ is determined as in IV below ba an infinitely divisible interval in $E$. (c) The canonical map $p_*\to p_!$ in $S$ is epimorphic (Schreiber calls this "pieces have points"). This holds iff the other canonical map $p^*\to p^!$ in $E$ is monomorphic (Schreiber calls this "discrete objects are concrete). Lawvere calls this property the "Nullstellensatz". =-- +-- {: .un_lemma} ###### Comments 1. The functors directed "downwards" (i.e. $p_!$ and $p_*$) express the opposition between "points" and "pieces". 1. The functors directed "upwards" (i.e. $p^*$ and $p^!$) express the opposition between "pure anti-cohesion" (discreteness) and "pure cohesion" (codiscreteness). 1. (c) implies that $p_!(X\times Y)\to p_!(X)\times p_!(Y)$ "mapping pieces of a product to the product of pieces" is an epimorphism. 1. If (c) is an isomorphism, this implies (a) and (b). In particular a cartesian closed quality type is a category of cohesion (in an extreme sense). =-- ## 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 $q:F\to S$ * $q_! h=p_!$ $$\array{ E&\stackrel{p_!}{\to}&S\\ \downarrow^h&\mathrlap{\nearrow}^{q_!}\\ F }$$ i.e. an extensive quality of $X$ has the same number of connected pieces as $X$. =-- +-- {: .num_theorem} ###### Theorem (Hurewicz) Any category of cohesion $p$ has a canonical extensive quality defined by $F(X,Y)=p_![Y,X]$ where $h$ is the identity on objects and preserves finite products and exponentiation. =-- +-- {: .num_defn} ###### Definition An **intensive quality** on a category of cohesion $p:E\to S$ is a functor $s_*$ from $p$ to a quality type $q:L\to S$ such that * $s_*$ preserves finite products and finite coproducts. * $q_*s_*=p_*$ $$\array{ E&\stackrel{p}{\to}&S\\ \downarrow^{s_*}&\mathrlap{\nearrow}^{q}\\ L }$$ =--