A 2-category is a good context for doing a lot of category theory (including internal category theory, enriched category theory, and so on), but there are some things that are hard to do there, such as to talk about weighted limits and colimits. This leads one to introduce the notion of a 2-category equipped with proarrows, which is a 2-category along with extra data that plays the role of profunctors, allowing the definition of weighted limits and other aspects of category theory.
However, it would also be nice if the extra data in a proarrow equipment were somehow determined by the 2-category we started with. This is especially so when talking about functors between equipments, since functors between 2-categories are often easier to construct. It turns out that in many cases, including the most common ones, this is the case: we can construct the proarrows in terms of the underlying 2-category. This was originally realized by Ross Street.
The idea is to identify a profunctor with its collage, aka its cograph, which is a special sort of cospan in $Cat$ (or $V Cat$, or whatever other 2-category one wants to start with). One then simply has to characterize, in 2-categorical terms, which cospans are collages, and how to do things like compose them. It turns out that in most cases the characterization is precisely that they are the two-sided codiscrete cofibrations — i.e. the two-sided discrete fibrations in the opposite 2-category.
Suppose that $K$ is a 2-category with finite 2-colimits, and $A,C\in K$. A cofibration from $A$ to $C$ is a cospan $A\to B \leftarrow C$ which is an internal two-sided fibration in $K^{op}$. As remarked at fibration in a 2-category, there is a 2-monad on $Span_{K^{op}}(A,C)$ whose algebras are such fibrations. In other words, there is a 2-comonad on $Cospan_K(A,C)$ whose coalgebras are such fibrations. This 2-comonad is defined by
where $I$ is the interval category $(0\to 1)$ and $(-\times I)$ denotes the copower with $I$. In the pushouts, the map $A\to A\times I$ is the inclusion at $0$ and $C\to C\times I$ is the inclusion at $1$.
A cospan $A\to B \leftarrow C$ in a 2-category $K$ is codiscrete if it is codiscrete in the 2-category $Cospan_K(A,C)\simeq (A+C)/K$. This means that for any $X\in Cospan(A,C)$, the hom-category $Cospan(A,C)(B,X)$ is equivalent to a discrete category. Explicitly, it means that given any two morphisms $B \;\rightrightarrows\; X$ of cospans, if there exists a 2-cell from one to the other in $Cospan(A,C)$, then it is unique and invertible.
A codiscrete cofibration is a two-sided cofibration which is codiscrete as a cospan.
We sketch a characterization of cofibrations in $V Cat$, where $V$ is any Bénabou cosmos. Let $A\overset{f}{\to} B \overset{g}{\leftarrow} C$ be a cospan and let $D = (A\times I) +_A B +_C (C\times I)$. We claim that $D$ has the following description.
Its objects are the disjoint union of those of $A$, $B$, and $C$, i.e. $ob(D) = ob(A) \sqcup ob(B) \sqcup ob(C)$.
$A$ and $B$ and $C$ are (disjoint) full subcategories of $D$.
There are no morphisms in $D$ from $A$ to $B$, or from $B$ to $C$, or from $A$ to $C$. That is, for $a\in A$, $b\in B$, and $c\in C$ we have $D(a,b) = D(b,c) = D(a,c) = \emptyset$.
If $a\in A$, $b\in B$, and $c\in C$, we have $D(b,a) = B(b,f a)$, $D(c,b) = B(g c, b)$, and $D(c,a) = B(g c, f a)$.
That $D$ is a $V$-category is immediate, and it is easy to check the universal property. We write $A \overset{i}{\to} D \overset{j}{\leftarrow} C$ for the inclusions.
Now suppose that $B$ is a coalgebra for the 2-comonad in question. Therefore, in particular we have a map $h\colon B\to D$ in $Cospan(A,C)$, so that $h f = i$ and $h g = j$ (or perhaps only isomorphic; it really makes no difference here). Moreover, the counit of the comonad is the obvious map $k\colon D\to B$, so we must have $k h = 1_B$.
Since $i$ and $j$ are injective on objects and have disjoint images, so must be $f$ and $g$. And since $i$ and $j$ are fully faithful $V$-functors, the action of $f$ and $g$ on homs must be split monic in $V$, and the action of $h$ on homs in $A$ and $B$ must be split epic. But since $h k = 1_B$, the action of $h$ on homs must also be split monic, hence an isomorphism, and hence so must that of $f$ and $g$ be. Therefore, $f$ and $g$ are fully faithful inclusions with disjoint images.
Clearly $h$ must take the images of $f$ and $g$ to the images of $i$ and $j$, respectively. Because $k h = 1_B$, it must be that $h$ takes the rest of $B$ to itself, sitting in the canonical copy of $B$ inside $D$. This uniquely defines $h$, as long as $B$ satisfies the condition that
It is then easy to check that if $f$ and $g$ are fully faithful with disjoint images and this condition holds, then $B$ is in fact a coalgebra for the comonad in question, i.e. a two-sided cofibration from $A$ to $C$.
Note that such a cofibration from $A$ to $C$ can be identified with the following data: a category $B' = B\setminus (A\cup C)$, profunctors $m\colon A ⇸ B$, $n\colon B ⇸ C$, and $p\colon A ⇸ C$, and a morphism $n m \to p$ of profunctors. Such a thing is sometimes called a gamut from $A$ to $C$.
Now a 2-cell in $Cospan(A,C)$ is simply a natural transformation between functors $B \;\rightrightarrows\; X$ whose components on the images of $A$ and $C$ are isomorphisms. Thus, if $B$ is a cofibration as above with the property that $B \setminus (A\cup C)$ is empty, then it must be codiscrete. The converse is easy to check, taking $X$ to be the ordinal $4 = (0\le 1 \le 2\le 3)$ as a category. But a gamut with $B'=\emptyset$ is nothing but a profunctor $A ⇸C$; hence codiscrete cofibrations in $V Cat$ can be precisely identified with the collages of profunctors.
A codiscrete cofibration in the 2-category $Topoi$ of topoi can be identified with a left exact functor.
Codiscrete cofibrations in the 2-category $Dbl$ of double categories, double functors, and horizontal transformations can be identified with double profunctors.
The examples of profunctors suggest that given any 2-category $K$ with finite 2-colimits, we may try to canonically equip it with proarrows by defining the proarrows $A ⇸C$ to be the codiscrete cofibrations. The sticky point is then how to define units and composition of such proarrows in order to obtain an equipment.
The unit is obvious: we should take the unit proarrow of $A$ to be the cospan $A\to A\times I \leftarrow A$, which is always a codiscrete cofibration.
Binary composition is subtler. The obvious way to compose codiscrete cofibrations $A \to B \leftarrow C$ and $C\to D \leftarrow E$, of course, is to take a pushout $B +_C D$. It is not hard to show (see references):
In any 2-category with finite 2-colimits, if $B$ and $D$ are cofibrations, then so is $B +_C D$.
However, $B +_C D$ will not be codiscrete even if $B$ and $D$ are. In $V Cat$, if $B$ and $D$ are collages of profunctors $m$ and $n$, then $B +_C D$ represents the gamut consisting of $m$, $n$, and the composite profunctor $n m$, with the middle category being $C$. Thus, in order to obtain the correct composite, we need to forget about $C$ somehow. The best way to do this seems to be using a factorization system in a 2-category, akin the way in which we construct the bicategory of relations from any regular category.
We are looking for a 2-categorical factorization system $(\mathcal{E},\mathcal{M})$ in $K$ such that if we have a two-sided cofibration $A\to C\leftarrow B$ and we factor $A+B \to C$ into an $\mathcal{E}$-map and an $\mathcal{M}$-map, then the resulting cospan $A\to E \leftarrow B$ is a codiscrete cofibration. Codiscreteness means in particular that the $\mathcal{E}$-map $A+B\to E$ should be codiscrete, i.e. representably cofaithful and co-conservative. Moreover, if $A\to C\leftarrow B$ was already a codiscrete cofibration, then $A+B\to C$ should already be in $\mathcal{E}$. This suggests the following definition.
A 2-category with finite 2-limits and 2-colimits is pre-equippable if it has a factorization system $(\mathcal{E},\mathcal{M})$ such that
It is equippable if in addition it satisfies:
Co-conservative morphisms are also called liberal. Recall that by definition of codiscreteness, if $A\to C \leftarrow B$ is a codiscrete cofibration, then $A+B\to C$ is cofaithful and liberal; thus the first two conditions are compatible.
The example to keep in mind is $V Cat$, for any suitable $V$, where $\mathcal{E}$ is the class of essentially surjective $V$-functors and $\mathcal{M}$ is the class of $V$-fully-faithful functors.
Any morphism which is right orthogonal to codiscrete cofibrations is representably fully faithful. In particular, if $K$ is pre-equippable, then every morphism in $\mathcal{M}$ is representably fully faithful.
For any $X$ in $K$, we have a codiscrete cofibration $X\to X \times I \leftarrow X$, and thus $X+X \to X\times I$ is in $\mathcal{E}$. But orthogonality with respect to all such morphisms is precisely representable fully-faithfulness.
Any representably fully faithful morphism is right orthogonal to any cocomma object?. In particular, $K$ is pre-equippable and every codiscrete cofibration is a cocomma object, then $\mathcal{M}$ is precisely the class of representably fully faithful morphisms.
Maps out of a cocomma object are in canonical correspondence with 2-cells in $K$. But representable fully-faithfulness means that 2-cells lift uniquely along such a map. Hence so do maps out of a cocomma object, and hence any representably fully faithful map is right orthogonal to all cocomma cospans.
If $K$ is pre-equippable, then any inverter or equifier is in $\mathcal{M}$, and every morphism in $\mathcal{E}$ is cofaithful and liberal.
Any inverter is always right orthogonal to any liberal morphism, and any equifier is always right orthogonal to any cofaithful morphism.
In an equippable 2-category, we can compose cofibrations in the desired way.
If $K$ is equippable, $A\to E \leftarrow B$ is a two-sided cofibration, and $A+B \to F \to E$ is an $(\mathcal{E},\mathcal{M})$-factorization, then $A\to F \leftarrow B$ is a codiscrete cofibration. In particular, the category $CodCofib(A,B)$ is coreflective in the 2-category $Cofib(A,B)$.
Since $\mathcal{E}$-morphisms are cofaithful and liberal, $A\to F \leftarrow B$ is certainly codiscrete. That it is a cofibration is proven as in (MB, 4.18). Coreflectivity follows by orthogonality for the factorization system $(\mathcal{E},\mathcal{M})$, since all codiscrete cofibrations are in $\mathcal{E}$ by assumption.
Therefore, in an equippable 2-category, we can define the composite of codiscrete cofibrations $A\to B\leftarrow C$ and $C\to D\leftarrow E$ to be the codiscrete coreflection of the cofibration $A \to B +_C D\leftarrow E$.
If $K$ is equippable, there is a 2-category $CodCofib(K)$, with the same objects as $K$, and with codiscrete cofibrations as 1-morphisms. Moreover, there is a locally fully faithful identity-on-objects (pseudo) 2-functor $(-)_* K\to CodCofib(K)$ such that each 1-morphism $f_*$ has a right adjoint. Therefore, $K$ is canonically a 2-category equipped with proarrows (hence the term “equippable”).
This is essentially (MB, 4.20).
One can then impose additional axioms on $K$ to get good behavior of this equipment, and try to characterize the equipments arising in this way; see (MB, section 5) and (PC).
Note that since coreflections are determined by a universal property, the composite of codiscrete cofibrations is independent of the chosen factorization system $(\mathcal{E},\mathcal{M})$. In fact, there are two different “extreme” ways that we might try to define an equippable factorization system; we could either
In the second case we mean that $\mathcal{M}$ is the class of all morphisms right orthogonal to the morphisms $A+B\to C$ such that $A\to C \leftarrow B$ is a codiscrete cofibration, and then $\mathcal{E}$ is the class of all morphisms left orthogonal to $\mathcal{M}$. This implies, of course, that $\mathcal{E}$ contains the codiscrete cofibrations.
Neither of the above choices is guaranteed to produce a factorization system (since the factorizations may not exist), but if either one does, then that factorization system is automatically pre-equippable. In the first case this is obvious, since all codiscrete cofibrations are cofaithful and liberal, while in the second case, it follows since inverters and equifiers are then necessarily in $\mathcal{M}$, and anything left orthogonal to inverters and equifiers must be cofaithful and liberal. Thus, a 2-category is equippable if either of these two choices produces a factorization system for which $\mathcal{M}$ is closed under pushout and tensors with $I$.
The (essentially surjective, $V$-fully faithful) factorization system is generated by the codiscrete cofibrations, and is equippable.
It suffices to show that a $V$-functor $f\colon A\to B$ is right orthogonal to codiscrete cofibrations if and only if it is $V$-fully faithful, i.e. each morphism $A(a,a') \to B(f a, f a')$ is an isomorphism in $V$. For “if”, it suffices to observe that $V$-fully faithful functors are right orthogonal to all essentially surjective ones, and any codiscrete cofibration is essentially surjective. For “only if,” suppose given $a,a'\in A$, let $X=Y=I$ be the unit $V$-category, consider the object $B(f a,f a')\in V$ as a $V$-profunctor $X \to Y$, and let $E$ be its collage. Then we have a square
where the bottom arrow is the identity on the nontrivial hom-object $B(f a,f a')$. A lifting in this square supplies a section of $A(a,a') \to B(f a, f a')$, and uniqueness of lifting against the collage of $A(a,a')$ (also as a profunctor $I\to I$) shows that it is an inverse isomorphism; hence $f$ is $V$-fully faithful.
Finally, it is straightforward to verify that $V$-fully-faithful functors are closed under pushout and tensors with $I$.
In $V Cat$, every liberal is automatically cofaithful, and there is a pre-equippable factorization system in which $\mathcal{E}$ is the class of liberal morphisms. However, it is not equippable, even when $V=Set$.
This is essentially (MB, 3.4). In this case $\mathcal{M}$ consists of the $V$-fully faithful morphisms which are additionally closed under absolute colimits, while $\mathcal{E}$ consists of the functors which are surjective up to absolute colimits (“Cauchy dense” functors). When $V=Set$, all absolute colimits are generated by retracts, and it is easy to construct an example of a fully faithful functor closed under retracts and a pushout of it which is no longer closed under retracts.
An equippable 2-category with $\mathcal{E} =$ liberal cofaithfuls = liberals is called faithfully co-conservational in (MB). This is the only case considered there, but the proofs generalize directly to any equippable 2-category. Note that $V Cat$ is not faithfully co-conservational, since the above factorization system is only pre-equippable: $\mathcal{M}$ is not closed under pushout. Its sub-2-category $V Cat_{cc}$ of Cauchy complete $V$-categories is faithfully co-conservational, but this is arguably just because when restricted to $V Cat_{cc}$, the above factorization coincides with the other, better one. Thus, it seems that perhaps in general it is better to consider the factorization system generated by the codiscrete cofibrations.
Ross Street, “Fibrations in bicategories”, MR, and correction.
Aurelio Carboni and Scott Johnson? and Ross Street and Dominic Verity, “Modulated bicategories” (MB) MR.
Bob Rosebrugh and Richard Wood?, “Proarrows and cofibrations” (PC), MR