[002D] Theorem 5·c.
  • $\vf$ is an absolute extension if and only if $\vf$ is a pointwise extension and can be computed as an absolute colimit if and only if $Y_Z\circ\vf$ is an extension.
  • Following §1.14 (with 1.1), $\vf$ is an absolute extension if and only if $\vf : \esq FJIR$ is exact, i.e. $\widetilde{\vf}$ is an isomorphism (in such a case, the zig-zag criterion and all local criteria take special forms, see (Guitart, 1977)).
  • $\vf$ is an absolute extension if and only if, taking the comma square $\esq{d_0}{d_1}UF$, the induced functor $J\circ d_1 \downarrow Y \to Z \downarrow R$ is opaque and surjective on objects.
  • As in §2.1, $F : A \to B$ is opaque if and only if the square is exact, i.e. (cf. §1.1) if and only if $F\otimes \widetilde{1_F} : F^\text{o}\otimes F \otimes F^\text{o} \to F^\text{o}$ is an isomorphism, or equivalently if and only if the pro-comonad on $ B$ associated to $F$ (see (Harting, 1977) (Thiébaud, 1971)) is idempotent.
  • If $ B_F$ is the full subcategory of $ B$ on the image of $F$, i.e. on objects of the form $FA$ for $A\in A_o$, $F$ is opaque if and only if the triangle is an absolute left extension.
  • $F$ is opaque if and only if it can be factored as a composition $T\circ Q$ with $T$ fully faithful and $Q$ co-fully faithful.
  • Thanks to Isbell’s zig-zag theorem, if $F$ is bijective on objects, then $F$ is opaque if and only if $F$ is an epimorphism of $\Cat$.