Isomorphism (category theory)

From Maths
Jump to: navigation, search
(Unknown grade)
This page is a stub
This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:
Needs proof-reading and checking, created to reduce the clutter in the "wanted pages" list


Given a pair of arrows, [ilmath]A\mathop{\longrightarrow}^fB[/ilmath] and [ilmath]B\mathop{\longrightarrow}^g A[/ilmath] in a category [ilmath]\mathscr{C} [/ilmath], we say that [ilmath]f[/ilmath] and [ilmath]g[/ilmath] are isomorphisms[1] if:

  • [ilmath]g\circ f=\text{Id}_A[/ilmath] and [ilmath]f\circ g=\text{Id}_B[/ilmath]

We may also call [ilmath]f[/ilmath] and [ilmath]g[/ilmath] an inverse pair of isomorphisms. For clarity I say again: both [ilmath]f[/ilmath] and [ilmath]g[/ilmath] are themselves isomorphisms

See also


  1. An Introduction to Category Theory - Harold Simmons - 1st September 2010 edition