the projections of the product. In the category of sets, for instance, the products are given by Cartesian products and the projections are just the natural projections onto the various factors. Powers. A special case of a product is when the diagram F is a constant functor to an object X of C. The limit of this diagram is called the Jth power of X and denoted XJ. Equalizers. If J is a category with two objects and two parallel morphisms from one object to the other, then a diagram of shape J is a pair of parallel morphisms in C. The limit L of such a diagram is called an equalizer of those morphisms. Kernels. A kernel is a special case of an equalizer where one of the morphisms is a zero morphism. Pullbacks. Let F be a diagram that picks out three objects X, Y, and Z in C, where the only non-identity morphisms are f : X → Z and g : Y → Z. The limit L of F is called a pullback or a fiber product. It can nicely be visualized as a commutative square: Inverse limits. Let J be a directed set (considered as a small category by adding arrows i → j if and only if i ≥ j) and let F : Jop → C be a diagram. The limit of F is called (confusingly) an inverse limit or projective limit. If J = 1, the category with a single object and morphism, then a diagram of shape J is essentially just an object X of C. A cone to an object X is just a morphism with codomain X. A morphism f : Y → X is a limit of the diagram X if and only if f is an isomorphism. More generally, if J is any category with an initial object i, then any diagram of shape J has a limit, namely any object isomorphic to F(i). Such an isomorphism uniquely determines a universal cone to F. Topological limits. Limits of functions are a special case of limits of filters, which are related to categorical limits as follows. Given a topological space X, denote by F the set of filters on X, x ∈ X a point, V(x) ∈ F the neighborhood filter of x, A ∈ F a particular filter and the set of filters finer than A and