Cones, pullbacks, equalizers, and the limit zoo
Limits generalize products, equalizers, and pullbacks into a single concept: the universal cone over a diagram. Every limit is defined by a universal property, and together they capture an enormous range of mathematical constructions.
A cone over a diagram D consists of an apex object N and a family of morphisms from N to each object in D, such that every triangle commutes. Select a diagram shape, add your apex, then click objects to draw cone legs.
Key insight: A limit of a diagram D is a universal cone — a cone through which every other cone factors uniquely. The shape of the diagram determines the type of limit: a discrete pair gives a product, a parallel pair gives an equalizer, and a cospan gives a pullback.
The pullback (or fiber product) P = A ×ᶜ B consists of all pairs (a, b) where f(a) = g(b). It is the limit of the cospan A → C ← B. Below, see the pullback square and the concrete fiber product in Set.
Key insight: The pullback "synchronizes" two maps into a common target. In Set, it is the set of compatible pairs. Pullbacks generalize intersections, inverse images, and fiber products. They are limits of cospan diagrams (A → C ← B).
Given two parallel morphisms f, g : A → B, the equalizer Eq(f, g) is the subobject of A on which f and g agree: Eq(f, g) = { a ∈ A | f(a) = g(a) }. It is the limit of the parallel pair diagram.
Key insight: The equalizer is the largest subobject on which two parallel maps agree. In algebra, equalizers correspond to solution sets of equations; in topology, to closed subspaces where continuous maps coincide.
Every limit arises from a diagram of a particular shape (the index category). Explore the four fundamental limit types, their index categories, and what they look like in Set.
Key insight: All limits are the same concept applied to different diagram shapes. Terminal objects, products, equalizers, and pullbacks are not four separate ideas — they are four instances of the single idea of a universal cone. A category with all finite limits is called finitely complete.