The Continuity axiom over Sets

Matias Menni
Tuesday, July 23, 2013 - from 16:00 to 16:25
Following Lawvere's terminology in his TAC 2007 paper we say that a geometric morphism ${p:\calE \rightarrow \calS}$ makes $\calE$ a {\em pre-cohesive} topos over $\calS$ if the pair ${ p^* \dashv p_*}$ extends to a string of adjoints ${p_! \dashv p^* \dashv p_* \dashv p^!}$ such that ${p_!:\calE \rightarrow \calS}$ preserves finite products, ${p^*:\calS \rightarrow \calE}$ is full and faithful and the induced natural transformation ${\theta:p_* \rightarrow p_!}$ is epi. Such a pre-cohesive topos is called {\em Sufficiently Cohesive} if the subobject classifier $\Omega$ of $\calE$ is `connected' in the sense that ${p_! \Omega = 1}$. On the other hand, the {\em Continuity} axiom is said to hold if the canonical natural transformation ${ p_! (X^{p^* S}) \rightarrow (p_! X)^S}$ is an iso for every $X$ in $\calE$ and $S$ in $\calS$. During CT2011, Lawvere explained that this axiom isolates a positive property that `combinatorial' examples of pre-cohesive toposes do not satisfy. Motivated by this we show that if ${\calS = \Sets}$ and $\calE$ is a presheaf topos then $p$ satisfies Continuity if and only if $\theta$ is an iso. It follows that Continuity and Sufficient Cohesion are incompatible for pre-cohesive presheaf toposes. We also build examples of pre-cohesive Grothendieck toposes satisfying both Continuity and Sufficient Cohesion.