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Limit (Category Theory)

A limit is the universal cone over a diagram. They are closely related to the concept of a colimit.

The Category of Cones

Let \(F: J \to C\) be a diagram of shape \(J\) over \(C\). There exists a functor \(\text{Cone}(-, F) \equiv \text{Hom}(\Delta(-), F): C^{\text{op}} \to \text{Set}\). This is to say, a functor which maps the opposite category of \(C\) to the category of sets by mapping each element \(c \in C\) to the set of natural transformations between the constant functor at \(c\) to \(F\). That is to say, this functor maps each element \(c \in C\) to the set of cones with summit \(c\) over the diagram \(F\). Likewise, morphisms in \(C\) are mapped to morphisms which take cones with one summit to those with another summit, over the same diagram of interest \(F\).

There are two ways to think about a universal cone: either as a representation for such a functor, or as a terminal object in its category of elements.

Representations of the Cone Functor

By the the Yoneda Lemma, a representation for \(\text{Cone}(-, F): C^{\text{op}} \to \text{Set}\) can be defined by an object \(\lim F \in C\) together with the universal cone \(\lambda: \Delta \lim F \Rightarrow F\), which is called the limit cone. These define a natural isomorphism \(C(-, \lim F) \cong \text{Cone}(-, F)\).

The Category of Elements of the Cone Functor

As with any functor, we can define the category of elements of this functor, \(\int \text{Cone}(-, F)\), which we call the category of cones. Because this is a hom-set functor, this category of elements is a slice category. The elements of this category are pairs \((c, x)\) where \(c \in C\) and \(x\) is a cone over \(F\) with summit \(c\).

Special Examples of Limits and Colimits

There are many special cases for which limits and colimits have particular names due to their use in less abstract branches of mathematics. Here a few examples are provided.

  • The limit of a diagram indexed by a discrete category with only identity morphisms is called a product.
    • In the special case in which the \(J\) is the empty category, the product is a terminal object. This is because a cone over an empty diagram is simply an object in the codomain category \(C\), and so the category of cones is isomorphic to \(C\). Limits are terminal objects in the category of cones, so these products are terminal objects in \(C\).
  • The limit of a diagram indexed by the parallel pair category is called an equalizer.
  • Consider the poset category \(J\) such that \(\text{Ob}(J) = \{ \alpha, \beta, \gamma \}\) and \(\text{Mor}(J) = \{ f: \alpha \to \gamma, g: \beta \to \gamma \} \cup \{\text{Identity morphisms}\}\). A diagram indexed by such a category is called a cospan, and the (co)limit of a cospan is called a pullback (pushout).

Limits in the Category of Sets

Let \(F: J \to \text{Set}\). A limit of such a diagram is a representation \(\text{Set}(X, \lim F) \cong \text{Cone}(X, F)\) of the functor which sends a set \(X\) to the set of cones over \(F\) with summit \(X\). The singleton set \(\{1\}\) represents the identity functor on the category of sets, so \(\text{Set}(X, \lim F) \cong \lim F\), which implies that \(\lim F \cong \text{Cone}(\{1\}, F)\).

A product of sets \(A_j\) indexed by some set \(j \in J\) is the set of cones over this collection of sets with summit \(\{1\}\), which is simply a \(J\)-tuple of elements in the sets. This is the cartesian product, as expected for the product of sets.

The terminal object in \(\text{Set}\) is the set of cones over the empty diagram with summit \(\{1\}\). Only one such cone exists, so the terminal object is the singleton set itself.

For two functions \(f, g: X \to Y\), their equalizer is the set of maps \(\{1\} \to X\) such that \(fx = gx\), which is to say, it is the set \(\left\{x \in X : f(x) = g(x) \right\}\).