the terminal broker
An arrow r from A to B indicates that B is in relation with A via the channel r.
Assume that our category is concrete. (i.e. each object is a structured set) Let the size of the set Hom(A,B) indicate the depth of the relationship B is in with A.
A broker has relationships with everyone. However, each of these relationships is as superficial as possible. (After all he only cares about business and has only a finite amount of time in his hands. Let us not forget that time is money!) Of course, client A may be delusional and thinking that his relationship with his broker B has some depth to it. In that case, the broker has done a very good job! (Is not the propagation of such a delusion the dream of every public relations person?)
Most successful broker is a person 1 such that Hom(A,1) is one element set and Hom(1,A) is bijective to A. In other words, he knows everyone "minimally" and is liked by everyone "maximally". Such a broker is not only a terminal object of our category, but also represents the forgetful functor from our category to the category of sets.
Such objects exist in categories like Set and Top, but algebraic categories have no tolerance for them. In algebraic categories, the terminal object is often the one-element algebra and the representing object for the forgetful functor is the free algebra generated on the one-element set. So these two objects differ... In some sense, algebraic categories have separation of powers: If you know everybody minimally, then you are not allowed to be liked by everyone maximally (and vice versa).
A jerk on the other hand is 1 such that both Hom(A,1) and Hom(1,A) are one-element sets. In other words, he knows everyone, but nobody really likes him. They avoid him as much as possible. Such a broker is called a zero object, a very suiting name indeed...
Assume that our category is concrete. (i.e. each object is a structured set) Let the size of the set Hom(A,B) indicate the depth of the relationship B is in with A.
A broker has relationships with everyone. However, each of these relationships is as superficial as possible. (After all he only cares about business and has only a finite amount of time in his hands. Let us not forget that time is money!) Of course, client A may be delusional and thinking that his relationship with his broker B has some depth to it. In that case, the broker has done a very good job! (Is not the propagation of such a delusion the dream of every public relations person?)
Most successful broker is a person 1 such that Hom(A,1) is one element set and Hom(1,A) is bijective to A. In other words, he knows everyone "minimally" and is liked by everyone "maximally". Such a broker is not only a terminal object of our category, but also represents the forgetful functor from our category to the category of sets.
Such objects exist in categories like Set and Top, but algebraic categories have no tolerance for them. In algebraic categories, the terminal object is often the one-element algebra and the representing object for the forgetful functor is the free algebra generated on the one-element set. So these two objects differ... In some sense, algebraic categories have separation of powers: If you know everybody minimally, then you are not allowed to be liked by everyone maximally (and vice versa).
A jerk on the other hand is 1 such that both Hom(A,1) and Hom(1,A) are one-element sets. In other words, he knows everyone, but nobody really likes him. They avoid him as much as possible. Such a broker is called a zero object, a very suiting name indeed...
