Inserter category

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In category theory, a branch of mathematics, the inserter category is a variation of the comma category where the two functors are required to have the same domain category.

Category theory logic and mathematics

Category theory formalizes mathematical structure and its concepts in terms of a labeled directed graph called a category, whose nodes are called objects, and whose labelled directed edges are called arrows. A category has two basic properties: the ability to compose the arrows associatively, and the existence of an identity arrow for each object. The language of category theory has been used to formalize concepts of other high-level abstractions such as sets, rings, and groups. Informally, category theory is a general theory of functions.

Mathematics Field of study concerning quantity, patterns and change

Mathematics includes the study of such topics as quantity, structure (algebra), space (geometry), and change. It has no generally accepted definition.

In mathematics, a comma category is a construction in category theory. It provides another way of looking at morphisms: instead of simply relating objects of a category to one another, morphisms become objects in their own right. This notion was introduced in 1963 by F. W. Lawvere, 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 limits 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".

Contents

Definition

If C and D are two categories and F and G are two functors from C to D, the inserter category Ins(F, G) is the category whose objects are pairs (X, f) where X is an object of C and f is a morphism in D from F(X) to G(X) and whose morphisms from (X, f) to (Y, g) are morphisms h in C from X to Y such that . [1]

Properties

If C and D are locally presentable, F and G are functors from C to D, and either F is cocontinuous or G is continuous; then the inserter category Ins(F, G) is also locally presentable. [2]

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References

  1. Seely, R. A. G. (1992). Category Theory 1991: Proceedings of an International Summer Category Theory Meeting, Held June 23-30, 1991. American Mathematical Society. ISBN   0821860186 . Retrieved 11 February 2017.
  2. Adámek, J.; Rosický, J. (10 March 1994). Locally Presentable and Accessible Categories. Cambridge University Press. ISBN   0521422612 . Retrieved 11 February 2017.