Preservation of Meets and Joins

definition

A Monotone Map $f:P\to Q$ preserves meets if $f(a\wedge b)\cong f(a)\wedge f(b)$ for all $a,b\in P$, and preserves joins if $f(a\vee b)\cong f(a)\vee f(b)$ for all $a,b$. (More generally, for all subsets: $f(\bigwedge A)\cong \bigwedge f(A)$.)

Sources: 7 Sketches Definition 1.92, Proposition 1.111, Theorem 1.115, Exercise 1.94; DaoFP §10.7 (adjoints preserve (co)limits).

• Failure to preserve joins is a Generative Effect (Definition 1.93). Adam’s thesis restricts to maps that preserve meets but not joins: they “behave well when restricting to subsystems, but throw up surprises when joining systems”.
• For any monotone map, $f(a)\vee f(b)\le f(a\vee b)$ and $f(a\wedge b)\le f(a)\wedge f(b)$ automatically (7S Exercise 1.94); preservation is the reverse inequality.
• Right Adjoints Preserve Meets and left adjoints preserve joins; conversely, when the domain has all meets/joins, preservation characterizes adjointness (Adjoint Functor Theorem for Preorders).
• Example 1.113: right adjoints need not preserve joins.
• Categorical generalization: Right Adjoints Preserve Limits (DaoFP §10.7), and a Functor preserving colimits is “co-continuous”.