## Yet another forms of Yoneda lemma

Given categories and objects , we let denote the category where we freely add an arrow to the disjoint union of and .
This can be achieved e.g. by taking the colimit of the following -shaped diagram:

where is the terminal category and contains two objects and a single nonidentity arrow .
Alternatively, we can consider the profunctor induced by the span (which is just the profunctor composition ).

It’s straightforward to check that .

Based on the construction, for any profunctor , the profunctor morphisms (functors that are identical on both and ) are uniquely determined by where they take the freely added arrow , and that can be any heteromorphism in .

Theorem. (Yoneda lemma for profunctors.) For any profunctor and objects we have a bijection

natural in both and .

It’s also straightforward to verify that, for profunctors , a functor that acts as the identity on both and , naturally corresponds to a natural transformation where are the hom functors of restricted to heteromorphisms (i.e. ), and vice-versa.

This way, involving our previous observation about , we can reformulate the above theorem as, for any functor we have

naturally in .

That clearly implies both the covariant and the contravariant versions of the original Yoneda lemma when applying it either with or with .

## Presheaves as simple extensions

A presheaf on a category is a functor , so using can even condider it as a profunctor , which can be viewed as a left -module, or… as a simple right extension of by a single object (the object of ).
Similarly, a presheaf on the opposite category can be though of as a right -module, or as a simple left extension.

Definition. A category with object is a simple right extension of category if is a full(y embedded) subcategory of , , and is not the domain of any morphism besides its identity .
Examples. For any object we obtain a simple right extension by joining a new object and copying all the arrows as where . This extension category will be denoted by .
Observe that by construction, , so it’s just the hom functor seen as a profunctor where is embedded into as the new object .

We can also define the empty and the unique simple right extensions by prescribing that each contains exactly 0 or 1 elements, respectively.

Theorem. (Yoneda lemma for simple right extensions.)
Every simple right extension of is a colimit of simple extensions . (…..)

## A Bridge Construction

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

In other words, if denotes the category of 2 objects (name them ) and 2 nonidentity arrows, which are inverses of each other, then a  bridge  is a category over , i.e. a category equipped with a functor . The full subcategories and are called banks of the bridge.
The arrows in the -preimage of the two arrows and in are referred to as through arrows or heteromorphisms in the bridge.

## Co/lax functors without apriori comparison cells

A profunctor is functorial iff every object of has a reflection in within . Fixing (arbitrarily) one reflection arrow for each object yields a functor such that .
Dually, is co-functorial iff every object of has a coreflection in , yielding a functor such that .
If both are satisfied, then above is left adjoint to .

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.

## 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 has a unit arrow , 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.
Continue reading “Plenty faces of a profunctor”

## Functor as a fibration

In topology, any continuous function can be considered as a fibration over space , with total space , where the fibres are glued together according to the topology of .

Likewise, any functor can be regarded as a kind of fibration over . Each object determines a fibre category with arrows which are mapped to .

In this generality, the fibres connect up by means of profunctors between them and then and the original composition of are encapsulated in the arising mapping .
Continue reading “Functor as a fibration”

## On the two definitions of adjunctions

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

Def.1
A functor   is called a left adjoint to a functor   if there is a bijection between the homsets:

natural in both and . In this case is called a right adjoint to , and we write .

Def.2
A functor is called a left adjoint to a functor if there are natural transformations and satisfying the zig-zag identities:

So, why are these two definitions equivalent?

## Morita equivalence by categorical bridges

Let denote the category of 2 objects (name them ) and 2 nonidentity arrows, which are inverses of each other.

We define a  bridge  as a category over , i.e. a category equipped with a functor . The full subcategories and are called banks of the bridge, and we write .
The arrows in the -preimage of the two arrows in are referred to as through arrows or heteromorphisms in the bridge.
Continue reading “Morita equivalence by categorical bridges”

## Situation-Tree Logic on a Profunctor

Let be a given profunctor. We regard it as a category, fully containing both and , with additional heteromorphisms .
Objects of will be regarded as abstract situations (we are given this and that perhaps with certain elementary conditions), objects of will be called models, and a heteromorphism is regarded as an interpretation of situation in model .
Continue reading “Situation-Tree Logic on a Profunctor”

## An abstract infinitary presevervation theorem

We generalize the following results of infinitary first order logic:
A class of models is axiomatizable by universal / existential / positive / negative formulas iff it is closed under submodels / extensions / homomorphic images / surjective homomorphic preimages.