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:

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

where 1 is the terminal category and 2 contains two objects and a single nonidentity arrow 0\to 1.
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 can even condider 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 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\ct 1 where \ct 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.

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

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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Plenty faces of a profunctor

What is a profunctor?

Recall that a category is a directed graph equipped with an associative and unital composition of its edges. In the categorical context, the nodes of the graph are also called objects and edges are called arrows or morphisms. The unitality requirement says that each object a has a unit arrow a\to a, called identity, which doesn’t affect the result of the compositions.

A functor is a composition and identity preserving graph morphism from a category to a category.

A profunctor, on the other hand, connects up two categories by (potentially) adding more morphisms in between them ‘from the outer world’. In other perspective, we can describe this setting as one of the two categories acting from the left and the other one acting from the right on the newly added morphisms. This can also be grasped as simply being a two variable functor to the category of sets, contravariant in the first argument.
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Functor as a fibration

In topology, any continuous function f:A\to B can be considered as a fibration over space B, with total space A, where the fibres f^{-1}(b) are glued together according to the topology of A.

Likewise, any functor F:\ct A\to \ct B can be regarded as a kind of fibration over \ct B. Each object b\in Ob\ct B determines a fibre category F^{-1}(b) with arrows which are mapped to 1_b.

In this generality, the fibres connect up by means of profunctors between them and then F and the original composition of \ct A are encapsulated in the arising mapping \ct B\to \ct Prof.
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On the two definitions of adjunctions

(Note that we write compositions from left to right and accordingly apply most maps on the right.)

A functor F:\ct A\to\ct B  is called a left adjoint to a functor U:\ct B\to\ct A  if there is a bijection between the homsets:

    \[\ct B(A^F\ig B) \simeq \ct A(A\ig B^U)\]

natural in both A and B. In this case U is called a right adjoint to F, and we write F\adj U.

A functor F:\ct A\to\ct B is called a left adjoint to a functor U:\ct B\to\ct A if there are natural transformations \eta:1_{\ct A}\tto FU and \eps:UF\tto 1_{\ct B} satisfying the zig-zag identities:

    \[\matrix{\eta F\cdot F\eps = 1_F &\sep U\eta\cdot \eps U=1_U}\]

So, why are these two definitions equivalent?
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Morita equivalence by categorical bridges

Let \mathcal Iso denote the category of 2 objects (name them 0,1) and 2 nonidentity arrows, f:0\to 1,\ g:1\to 0 which are inverses of each other.

We define a  bridge  as a category over \mathcal Iso, i.e. a category \mathcal H equipped with a functor B:\mathcal H\to\mathcal Iso. The full subcategories \mathcal A:=B^{-1}(0) and \mathcal B:=B^{-1}(1) are called banks of the bridge, and we write \mathcal H:\mathcal A \rightleftharpoons \mathcal B.
The arrows in the B-preimage of the two arrows in \mathcal Iso are referred to as through arrows or heteromorphisms in the bridge.
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