category
Think of a category as a mathematical world together with the maps allowed to move between its inhabitants. The inhabitants are the “objects” (sets, groups, spaces, modules), and the arrows are the “morphisms” (functions, homomorphisms, continuous maps). Category theory makes a deliberate bet: most of what matters about an object is encoded not in its internal guts but in how it relates to everything else through arrows. The discipline of speaking only about objects and arrows turns out to capture constructions across all of mathematics at once.
Formally, a category C consists of a collection of objects, and for each ordered pair of objects (A, B) a collection of morphisms Hom(A, B). There is a composition operation that, given f in Hom(A, B) and g in Hom(B, C), produces g ∘ f in Hom(A, C). Two axioms are required: composition is associative, h ∘ (g ∘ f) = (h ∘ g) ∘ f whenever defined; and each object A has an identity morphism id_A in Hom(A, A) that acts as a two-sided unit, f ∘ id_A = f and id_B ∘ f = f.
A subtlety worth flagging: the “collection” of objects may be too big to be a set (the objects of all sets do not form a set). A category is called small if its objects and morphisms each form a set, and locally small if every Hom(A, B) is a set even when the objects are not. Most working categories — Set, Grp, Ring, Top, Vect — are locally small but not small, which is why size distinctions appear in careful statements like the Yoneda lemma.
In the category Grp, objects are groups and morphisms are group homomorphisms; composition is ordinary function composition and id_G is the identity map. In a poset viewed as a category, objects are the elements and there is exactly one arrow a -> b precisely when a ≤ b, with composition forced by transitivity.
A familiar category and a degenerate one — both satisfy the same two axioms.
Every category C has an opposite category C^op with the same objects but all arrows reversed; statements about C^op are exactly the dual statements about C, which is the engine behind every “co-” notion (coproduct, colimit, cohomology).