Cones, pullbacks, equalizers, and the limit zoo
Products, pullbacks, and equalizers look like different ideas. They are all the same one: the best possible cone over a diagram.
Choose a diagram shape, add an apex, then click objects to draw cone legs. Every triangle has to commute.
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.
Press Show Challenger X and watch a rival cone factor through the pullback. Hover over pairs to see which ones match.
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).
Toggle the equalizer on and hover over elements to find where f and g agree.
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 subspaces where continuous maps coincide (closed subspaces when the codomain is Hausdorff).
Switch between the four tabs to see each limit's diagram shape and what it looks like in Set.
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.
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