Inserter category explained

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.

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

G(h) \circ f = g \circ F(h) .^[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]

Notes and References

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