Profunctors are categories

Here we use the context of Definition 1. (or, equivalently, of Definition 2.) of page ‘Many Faces of a Profunctor’, that is, we identify a profunctor F:\ct A^{op}\times\ct B\to \Set (according to Definition 3.) with the category \ct F that disjointly contains (isomorphic copies of) \ct A and \ct B and for all objects a\in Ob\ct A,\ b\in Ob\ct B, the set F(a,b) is considered as the homset \ct F(a\ig b) and its elements are called heteromorphisms. Their compositions with arrows of \ct A (from the left) and arrows of \ct B (from the right) are given by the action of the functor F on the arrows.
This setup naturally induces a functor p:\ct F\to\bf 2 where \bf 2 is the category of two objects 0,1 and a single nonidentity arrow 0\to 1 whose preimage at p is exactly the collection of all heteromorphisms in \ct F.
In this setting we write \ct F:\ct A\ag\ct B to express that \ct F is a profunctor from category \ct A to category \ct B.

Definition. Let \ct F and \ct G be profunctors \ct A\ag\ct B. By a profunctor morphism \teta:\ct F\to\ct G we simply mean a functor \ct F\to\ct G such that both of its restrictions, to \ct A and to \ct B, are the identity functors.

Note that such a functor \teta indeed straightly corresponds to a natural transformation F\to G where F,G:\ct A^{op}\times\ct B\to \Set are F(a,b):=\ct F(a\ig b),\ G(a,b):=\ct G(a\ig b).

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Yet another forms of Yoneda lemma

Given categories \ct A,\ \ct B and objects a\in Ob\ct A,\ b\in Ob\ct B, we let M_{a,b}:\ct A\ag\ct B denote the category where we freely add an arrow m:a\to b to the disjoint union of \ct A and \ct B.
This can be achieved e.g. by taking the colimit of the following \sf M-shaped diagram:

    \[\dia@R=1pc{{\bf 1}\ar[rd]^0 \ar[dd]_a && {\bf 1} \ar[ld]_1 \ar[dd]^b \\ & {\bf 2} & \\ \ct A && \ct B}\]

where \bf 1 is the terminal category and \bf 2 contains two objects, denoted by 0 and 1, and a single nonidentity arrow 0\to 1, whose image in the colimit will be the freely adjoined arrow m.
Alternatively, we can consider the profunctor induced by the span \ct A\overset{a}\ot 1\overset{b}\to\ct B (which is just the profunctor composition a^*\,b_*:\ct A\ag\ct B).

It’s straightforward to check that M_{a,b}(x\ig y)=\{\alpha m \beta\mid \alpha:x\to a,\ \beta:b\to y\}\cong \ct A(x\ig a)\times\ct B(b\ig y).

Based on the construction, for any profunctor \ct F:\ct A\ag\ct B, the profunctor morphisms M_{a,b}\to\ct F (functors that are identical on both \ct A and \ct B) are uniquely determined by where they take the freely added arrow m, and that can be any heteromorphism a\to b in \ct F.

Theorem. (Yoneda lemma for profunctors.) For any profunctor \ct F:\ct A\ag\ct B and objects a\in Ob\ct A,\ b\in Ob\ct B we have a bijection

    \[\ct Prof(M_{a,b}\ig \ct F)\,\cong\,\ct F(a\ig b)\]

natural in both a and b.

It’s also straightforward to verify that, for profunctors \ct F,\ct G:\ct A\ag\ct B, a functor \ct F\to\ct G that acts as the identity on both \ct A and \ct B, naturally corresponds to a natural transformation F\tto G where F,G are the hom functors of \ct F,\,\ct G restricted to heteromorphisms (i.e. \ct A^{op}\times\ct B\to\ct Set), and vice-versa.

This way, involving our previous observation about M_{a,b}, we can reformulate the above theorem as, for any functor F:\ct A^{op}\times\ct B\to\ct Set we have

    \[Nat(\ct A(-,a)\times\ct B(b,-)\ig F)\,\cong\, F(a,b)\]

naturally in a,b.

That clearly implies both the covariant and the contravariant versions of the original Yoneda lemma when applying it either with \ct A=1 or with \ct B=1.

Presheaves as simple extensions

A presheaf on a category \ct A is a functor \ct A^{op}\to\ct Set, so using \ct A^{op}\cong \ct A^{op}\times 1 we can even consider it as a profunctor \ct A\ag 1, which can be viewed as a left \ct A-module, or… as a simple right extension of \ct A by a single object (the object of 1).
Similarly, a presheaf on the opposite category \ct A^{op} can be though of as a right \ct A-module, or as a simple left extension.

Definition. A category \ct B with object b\in\ct B is a simple right extension of category \ct A by a new object b, if \ct A is a full(y embedded) subcategory of \ct B, \ b\notin Ob\ct A, \ Ob\ct B=Ob\ct A\cup\{b\}\, and b is not the domain of any morphism besides its identity 1_b.
Examples. For any object a\in Ob\ct A we obtain a simple right extension by joining a new object b and copying all the arrows \alpha:x\to a as \alpha_b:x\to b where x\in Ob\ct A. This extension category will be denoted by \ct A\>a.
Observe that by construction, \ct A\>a(x\ig b)\cong\ct A(x\ig a), so it’s just the hom functor \ct A(-\ig a):\ct A^{op}\to\ct Set seen as a profunctor \ct A\ag 1 where 1 is embedded into \ct A\>a as the new object b.

We can also define the empty and the unique simple right extensions by prescribing that each \ct B(a\ig b) contains exactly 0 or 1 elements, respectively.

If \ct B and \ct C are simple right extensions of \ct A, with new objects b,\, c, respectively, then a functor \ct B\to\ct C is said to be a morphism of extensions whenever it takes b to c and fixes all objects and arrows in \ct A.

Theorem. (Yoneda lemma for simple right extensions.)
For every simple right extension \ct B of \ct A, the morphisms of extensions \,(\ct A\>a)\to \ct B\, correspond to the new arrows a\to b in B, naturally in a\in\, Ob A.

Proof. Such a morphism is uniquely determined by where it takes the copy of 1_a:a\to a in (\ct A\>a) and that can be any new arrow a\to b.

Theorem. (co-Yoneda lemma for simple right extensions.)
Every simple right extension \ct B of \ct A is a colimit of simple extensions \ct A\>a.

Proof. Let a simple right extension \ct B of \ct A with new object b be fixed, and consider the following diagram D on the slice category \ct A / b: an object of that is a new arrow f:a\to b and the diagram assigns the simple extension \ct A\>a to it, whereas to an arrow \alpha:f\to g (with \alpha:a\to a_1\in\ct A,\ f\,=\,\alpha g) it assigns \fii\mapsto \fii\alpha.

Now, we get a cocone over D to \ct B by defining f^D\to\ct B to be the morphism of extensions (\ct A\>a)\to\ct B that corresponds to f:a\to b, and actually any cocone over D, with vertex \ct C determines a morphism of extensions \ct B\to\ct C by sending any new arrow f:a\to b in \ct B to the image of (the copy of) 1_a under the corresponding leg map (\ct A\>a)\to\ct C of the cocone.

A Bridge Construction

A category \ct H is a bridge between categories \ct A and \ct B if these are disjoint full subcategories of \ct H and \ct H has no more objects. In notation: \ct H:\ct A \rightleftharpoons \ct B.

In other words, if \ct Iso denotes the category of 2 objects (name them 0,1) and 2 nonidentity arrows, {\bf i}:0\to 1,\ {\bf j}:1\to 0 which are inverses of each other, then a  bridge  is a category over \ct Iso, i.e. a category \ct H equipped with a functor H:\ct H\to\ct Iso. The full subcategories \ct A:=H^{-1}(0) and \ct B:=H^{-1}(1) are called banks of the bridge.
The arrows in the H-preimage of the two arrows {\bf i} and {\bf j} in \ct Iso are referred to as through arrows or heteromorphisms in the bridge.

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Co/lax functors without apriori comparison cells

A profunctor \ct F:\ct A\ag\ct B is functorial iff every object of \ct A has a reflection in \ct B within \ct F. Fixing (arbitrarily) one reflection arrow for each object \ct A yields a functor F:\ct A\to\ct B such that F_*\cong \ct F.
Dually, \ct F is co-functorial iff every object of \ct B has a coreflection in \ct A, yielding a functor G:\ct B\to\ct A such that G^*\cong\ct F.
If both are satisfied, then F above is left adjoint to G.

This analogy continues in dimension 2 (for bicategories or double categories), yielding the colax functors as double profunctors with the reflection property and the lax functors as double profunctors with the coreflection property. If a double profunctor has both property, it determines a colax/lax adjunction.

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