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Concept
Comma category
Summary
In mathematics, a comma category (a special case being a slice category) is a construction in . It provides another way of looking at morphisms: instead of simply relating objects of a to one another, morphisms become objects in their own right. This notion was introduced in 1963 by F. W. Lawvere (Lawvere, 1963 p. 36), although the technique did not become generally known until many years later. Several mathematical concepts can be treated as comma categories. Comma categories also guarantee the existence of some s and colimits. The name comes from the notation originally used by Lawvere, which involved the comma punctuation mark. The name persists even though standard notation has changed, since the use of a comma as an operator is potentially confusing, and even Lawvere dislikes the uninformative term "comma category" (Lawvere, 1963 p. 13).
The most general comma category construction involves two functors with the same codomain. Often one of these will have domain 1 (the one-object one-morphism category). Some accounts of category theory consider only these special cases, but the term comma category is actually much more general.
Suppose that , , and are categories, and and (for source and target) are functors:
We can form the comma category as follows:
The objects are all triples with an object in , an object in , and a morphism in .
The morphisms from to are all pairs where and are morphisms in and respectively, such that the following diagram commutes:
Morphisms are composed by taking to be , whenever the latter expression is defined. The identity morphism on an object is .
Overcategory
The first special case occurs when , the functor is the identity functor, and (the category with one object and one morphism). Then for some object in .
In this case, the comma category is written , and is often called the slice category over or the category of objects over . The objects can be simplified to pairs , where . Sometimes, is denoted by .
Official source
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