Inserter category

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]

References

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.
Adámek, J.; Rosický, J. (10 March 1994). Locally Presentable and Accessible Categories. Cambridge University Press. ISBN 0521422612. Retrieved 11 February 2017.

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[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.