A collection of all the objects isomorphic to
a given object. Talking about the isomorphism class (of a
poset, say) ensures that we will only consider its
properties as a poset, and will not consider other incidental
properties it happens to have.
For a category F of finitely generated left F-modules, the Grothendieck group G(F) is the abelian group generated by symbols [M], one for every isomorphism class of modules M in F and relations [M] = [L] + [N] for any short exact sequence 0 [right arrow] L [right arrow] M [right arrow] N [right arrow] 0 in F.