The main motivation for our definition and our study of exact squares, as exposed through sections 1 to 4 was the study of Paré’s thesis about absolute limits (Paré, 1969) and the characterizations of absolute extensions given in (Guitart, 1977) (partially continuing on the line of some results in Thiébaud (Thiébaud, 1971) and Harting (Harting, 1977)).

We call absolute left extension a diagram in $\Cat$ of the form such that for each $H: Z\to K$ the composite $H\vf$ is a left extension (i.e. $HR\cong \Lan_J HF$).

A functor $F : A \to B$ is said to be opaque if for each $a,a\tick\in A_o$ and $b : Fa\to Fa\tick$ in $ B$ there is a zig-zag in $A$, the image of which under $F$ fits into a lantern diagram

• $\vf$ 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 $\Cat$.

A functor $F : A \to B$ is called consistent if it satisfies Frei’s “C condition” (Frei, 1976), which following (Hilton, 1966) p. 241 is equivalent to the following condition: for all $A\in A_o$, $FA \cong \varprojlim \big( FA\downarrow F \to A \xto{F} B \big)$, and which, following Linton (Linton, 1969) is equivalent to the fact that $\exp_F^\op$ is fully faithful on the pairs $(B,FA)$.

A functor $F : A \to B$ is called very rich if for all $A,A\tick\in A_o$, and $b : FA\to FA\tick$, there exists a span $A\xleftarrow{r} V \xto{r\tick} A\tick$ in $ A$ such that $Fr$ is invertible and $b = Fr\tick\circ (Fr)^{-1}$.

• For every functor we have the implications

$F$ is very rich $\To$ $F$ is rich (Hilton) $\To$ $F$ is opaque $\To$ $F$ is consistent (Frei)

and each converse implication is false in general.

• An opaque functor is consistent and co-consistent (i.e. $F^\op$ is consistent).
• If $L\dashv U$ is an adjunction, $U$ is opaque if and only if $L$ is opaque.
• If $L\dashv U$ is an adjunction, $U$ is very rich if and only if it is rich, if and only if it is opaque, if and only if it is consistent.

Another important use of opaque functors is the following: Mac Donald (MacDonald, 1976) shows that Kan extensions of cohomological theories can be taken along rich functors; in fact, it can also be taken along opaque functors (and more in general with exact squares).