functor

For a language with state, we can work in a functor category.

Then as we can take the free group functor.

G is not a functor, but nevertheless it carries important information.

A functor is an operation on spaces and functions between them.

Such a variable cannot take a term made of a functor applied to other terms as a value.

Here, has domain 1 and is an identity functor.

This allows the entire chain complex to be treated as a functor.

In the language of representable functor one can state the above result as follows.

A continuous functor is one that preserves all small limits.

The part consists of expressions that are evaluated when the functor is called.

There are for such that has a basis in , so is a free functor.

Another common work around is using a functor.

Finally there is a functor which takes the product of two topological spaces.

For a second example, consider the same functor 1+Nx(-) as before.

Recall that we could also express a sheaf as a special kind of functor.

If Y is a point, then the direct image equals the global sections functor.

Then the fact is that the functor is an equivalence.

Every flat resolution is acyclic with respect to this functor.

In category theory, a correspondence from to is a functor .

For these objects, a commonly considered forgetful functor is as follows.

It is useful to regard the dual group functor.

An alternative approach to the dual point of view is to use the functor of points.

A direct system is then just a covariant functor .

A particularly common type of functor is the predicate.

This functor is left exact but not necessarily right exact.