Comma category

A comma category is a structure in the mathematical category theory, which was introduced in 1963 by FW Lawvere. The name is derived from the notation originally used by Lawvere.

Definition

For the general construction of the comma category you look at two functors. Typically, one is defined by two on the terminal category: many category theoretical representations only consider this case.

Be, and categories, T and S functors. The comma category is defined as follows:

  • The objects are triples with object, object and arrow is.
  • The arrows from to are pairs, where each and arrows are in and so that the following diagram commutes:

Special cases

Category of objects under A

The first special case occurs when terminally and S identical functor (that is ). ( Then, for a fixed object in and the single arrow in ). The relevant point category is called under the category of objects written. The objects can be listed briefly, since the establishment of the indication of redundant power; note we briefly as - often also called to it to identify as an injection. Similarly, we can reduce the appearance of an arrow, as is always elected. The following diagram commutes:

Is an initial object of. There is already a beginning of object, it must be isomorphic.

Examples:

  • The category of the punctured topological spaces is isomorphic to the category of topological spaces below a selected one-point space.
  • The category of unitary algebras for a body is isomorphic to the category of unitary circles under.

Category of objects over A

Analogously, we can choose identical and terminal. We then obtain the category of objects over (where A is the selected object by S of is ). This comma category we write as; in algebraic geometry, the name is common. It is under the dual concept to objects. The objects are pairs; point is on for projection. An arrow in the comma category with the source and target is given by a mapping that makes the following diagram commute:

Is a final object of. Is a final object of this, it should be isomorphic.

  • Category theory
William Lawvere Initial and terminal objects Algebra over a field
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