Let be the category of objects
where
is a set,
is a unary function, and
is a nominated element and morphisms
which are functions
preserving the structure such that
and
.
Verify that is a category
on
is indeed a function which preserves the structure (
,
) and thus a morphism
Given three objects and two functions
and
the function
is an arrow in
, that is to say, it preserves structure:
and
Show that is a
-object
is indeed a Set,
is indeed an unary function and
is indeed an element of
.
Show that for each -object
there is an unique arrow
and describe the behaviour of the carrying function.
We construct a carrying function recursively:
is indeed a morphism and thus an arrow.
The morphism maps to the transitive closure of
under
.
Consider objects of form where
is a set and
. For two such objects a morphism
is a function
that respects the selected subsets:
Show that such objects and morphisms form a category
on
is indeed a function which respects the distinguished subset (
) and thus a morphism
Given three objects and two functions
and
the function
is an arrow in
, that is to say,
:
Consider pairs where
is a set and
is a binary relation on
.
Show that these pairs are objects of a category finding a sensible notion of morphism.
For two such objects a morphism shall be a function
that respects the binary relation:
We call the category comprised of such objects and arrows
.
For every there exists
on
is indeed a function which respects the binary relation (
) and thus a morphism
There exists an associative partial binary operation on the arrows of
Topological spaces , where
is a Set and
, and continuous maps
, that is
, form a Category
.
on
is indeed a continuous map and thus a morphism
Given three spaces and two continuous maps
and
the map
is an arrow in
, that is to say, it is contiuous:
Show that is a monoid under composition, where
is an object of a category
.
is associative and total on
by the definition of category and
is obviously closed under
.
is required to exist by the definition of category and indeed an identity of
on
.
Each monoid is a category (again).
We construct a category with exactly one object and associate to every element
an arrow
. We further define
.
Show that in
is associative.
Consider the composition of two partial functions
and
:
where all restricted versions of
,
, and
are total.
Extending the above to three partial functions ,
, and
:
Consider the category of pointed sets.
Show that and
are “essentially the same” category.
The idea is to identify the undefined parts of a partial functions image with :
where
is total.
Verify that for every monoid both
and
are categories of structured sets.
Since the two proofs are perfectly analogous we cover only the case .
For every -Set
there exists
on
is indeed a structure preserving function (
).
There exists an associative partial binary operation on the arrows of
Given three R-Sets ,
, and
and two continuous maps
and
the map
is an arrow in
, that is to say
:
The format of the proof was chosen to demonstrate that is indeed a structured set.