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.Given two morphisms and we show that they are pointwise identical by induction over :
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 ofGiven three objects and two functions and the function is an arrow in , that is to say, :
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.