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.
Throughout the note, letters denote objects of
and letters
denote objects of
.
Note that if there are no heteromorphisms of the form , or if we delete them, then we receive a profunctor
that we denote by
. We define
similarly.
Also note that a bridge is thus built up by these two profunctors and
, connected by a so called Morita context which is exactly the structure that is needed to provide composition of consecutive zigzag heteromorphisms, i.e. those of the form
or
.
Since both banks are assumed to be full subcategories, these zigzag compositions must exist purely within the bank or
.
Examples
- As mentioned, any profunctor
or
is in particular, a bridge, where either
or
empty.
- Any two full subcategories
of a category
determines a bridge by taking disjoint copies of
and
and adding heteromorphisms from
.
To give a specific example, we obtain a bridgebetween groups and commutative semigroups with semigroup homomorphisms as heteromorphisms.
- In particular, every category
has an identity bridge
where every object is duplicated and every morphism is present in 4 instances. Note that this, as a category, is equivalent to
itself.
The construction
Given an arbitrary profunctor , we can formally invert those heteromorphisms which are both reflection and coreflection arrows at once.
So, we can define a bridge for any
by adding heteromorphisms of the form
, which would represent the formal composition
, where we need to regard the formal compositions
and
equal whenever
and
. and we need to consider every consequences of these equations.
In other words, we quotient out by the smallest equivalence relation that is closed under compositions and generated by all pairs that satisfy
.
Assume and
are heteromorphisms, then the following compositions are defined by means of the unique arrow in
[resp. in
] that exists by the coreflection [resp. reflection] property of
to make the diagrams commutative:
Examples
4. Consider the category whose objects are natural numbers and arrows
are
matrices over a given field
, composition is matrix multiplication. We can define a natural profunctor
to the category of vector spaces over
, by setting a through arrow
to be an
-tuple of elements of
, i.e.
.
Compositions are defined straightforwardly. A through arrow will be a reflection and coreflection arrow at once iff it is a basis of
.
Applying the bridge construction, we receive a category where the object is effectively isomorphic to every
dimensional vector space.
5. Let be the profunctor with heteromorphisms
the element-like relations
, i.e. those that satisfy
for the maximal element
of the Boolean algebra
if not
then
if
and
then
for all elements and
.
We compose by defining


