Thus, there exist the categories of sets (with empty structure) and mappings, of groups and group-homomorphisms, of rings and ring-homomorphisms, of vector spaces and linear transformations, of topological spaces and continuous mappings, and so on.
We need to mention the objects and morphisms in the categories of sets and of classes, and also to spell out the functors of interest on them.
We also evaluated differences in head injury incidence rates between male and female players across categories of setting, contact, and competition level.
Examination of outbreaks within categories of setting, identification of the index case, and clinical characteristics provided additional insights.
Unadjusted rate ratios (RR), compared incidence rates between categories of sex, injury mechanism, setting and competition level.
But what is the category of sets if not the distinguished Boolean local topos being sought?
Though he did not abandon this idea, later he proposed a generalization of the category of sets (and mappings) instead.
Is it possible that this universe can be constructed syntactically, like the free topos, without reference to any previously given category of sets, or by a universal property?
This is defined as the union of a class of sets containing the empty set (the initial object in the category of sets) and closed under the power-set operation and under transfinite unions thus, as a subcategory of the category of sets.
