Let $G$ be a functor from categories $\mathcal{I}$ to $\mathcal{C}$ . A limit of $G$ is a pair $(L,\tau)$ where

1. $L:\mathcal{I}\to \mathcal{C}$ is a constant functor,
2. $\tau: L\to G$ is a natural transformation,

Since we may identify a constant functor with its value, say an object $A$ , in $\mathcal{C}$ , a limit of $G$ may be viewed as an object $A$ in $\mathcal{C}$ , together with a collection of morphisms $A \to G(I)$ , or $G_I$ , for each object $I$ in $\mathcal{I}$ such that

if is a morphism, then

Remarks.

• A limit of a functor may or may not exist. If it exists, then it is unique up to natural equivalence. In other words, we may call it the limit of $G$ .
• In the literature, the limit of $G$ is variously known as the inverse limit, left limit, projective limit, root, or left root of $G$ . and is generally written $\liminv G$ .
• The category $\mathcal{I}$ above is usually called the index category, and the functor $G$ a diagram in $\mathcal{C}$ .
• The most common index categories are finite categories, discrete categories, partially ordered categories, and more specifically, directed categories.
• On the other hand, it turns out that if, in a category, if equalizers of any pairs of morphisms, and arbitrary products of any collections of objects exist, then it can be shown that every functor into this category has a limit.

By reversing all the arrows in $\mathcal{C}$ , we arrive at the dual concept of a limit, that of a colimit. Precisely, the colimit of $G:\mathcal{I}\to \mathcal{C}$ is a pair $(R,\tau)$ where $R:\mathcal{I}\to \mathcal{C}$ is a constant functor and $\tau: G\to R$ is a natural transformation, such that for any pair $(M,\varphi)$ of constant functor $M:\mathcal{I}\to \mathcal{C}$ and natural transformation $\varphi: G\to M$ , there is a unique natural transformation $\phi: R\to M$ such that the following diagram

Examples. Some common examples of limits (inverse limits) are products, terminal objects, pullbacks, equalizers, kernels, and kernel pairs. These examples can be readily verified. Let us verify that a terminal object is a limit:

Let $G$ be the empty functor (from the empty category) into an arbitrary category $\mathcal{C}$ . So the limit of $G$ is just an object $C$ in $\mathcal{C}$ , and that's it, as there are no objects in the empty category, there are no morphisms from $C$ in the limit of $G$ . If $A$ is any object in $\mathcal{C}$ , then there is a unique morphism $A\to C$ , and that's it. But this means that $C$ is just a terminal object of $\mathcal{C}$ .

Please see the verification of some of these examples in the attachments below.

Some examples of colimits (direct limits) are coproducts, initial objects, pushouts, coequalizers, cokernels, and cokernel pairs.