The adjunction is about functors($\mathcal{A}↔\mathcal{B}$), and the representable is about presheaves($\mathcal{A}^\text{op}→\textbf{Set}$). The limit, which is the main concept in this section, is about the structure inside the category $\mathcal{A}$ itself. This not only covers the numerous constructions of objects from other objects in categories, such as products in most mathematics, kernels in algebra, etc. but also unifies the common mathematic concepts among several mathematic fields.

Limits, Colimits

Products

It’s better to understand limits by observing some examples first, so we begin with the definition of the product, the simplest limit.

For given objects $X$ and $Y$, the product of two is the object $X \times Y$ with morphisms $p_X:X\times Y\rightarrow X$ and $p_Y:X\times Y\rightarrow Y$ (=projections) such that

$$ \forall f_X:A\rightarrow X,f_Y:A\rightarrow Y,\\ \exists!\overline{f}:A\rightarrow X\times Y \,s.t.\\ f_X=p_X\circ\overline{f},\,f_Y=p_Y\circ\overline{f}. $$

tikzedit11.svg

In other words, the product makes a one-to-one correspondence between $\mathcal{A}(A,\,X)\times\mathcal{A}(A,\,Y)$ and $\mathcal{A}(A,X\times Y)$, so it’s equivalent to think of the pair of morphisms into each object as a single morphism into the product.

Likewise, we can define a product of more than two objects by $\displaystyle\prod_{i\in I} X_i$ with morphisms $\displaystyle p_j:\prod_{i\in I} X_i \rightarrow X_j$.

Equalizers

The next type of limit is an equalizer, just like the equalizer in a $\textbf{Set}$. For given objects $X,\,Y$ with two morphisms $s,t:X\rightarrow Y$, a fork of $s$ and $t$ is an object $A$ with a morphism $f:A\rightarrow X$ such that $s\circ f= t\circ f$, and similar to the property of the product, the equalizer of $s$ and $t$ is a fork $(E,\iota)$ such that every fork $(A,f)$ amounts to the unique morphism $\overline{f}:A\rightarrow E$ satisfying $f=\iota\circ\overline{f}.$

tikzedit12.svg

Unlike the products, equalizers have certain morphisms as a target, and additional commuteness condition is required. On the other hand, we can make a multi-conditional equalizer using the product: The equalizer satisfying for all $s_i,t_i:X\rightarrow Y_i,\, i\in I$ is just an equalizer of $\displaystyle(s_i){i\in I},(t_i){i\in I}:X\rightarrow \prod_{i\in I} Y_i.$

Limits

The two examples above have some properties in common:

  1. its morphisms make some commuteness for each morphism in the target,
  2. Consist of an object $L$ and morphisms $p_{X_i}:L\rightarrow X_i$ for each $X_i$ in the target,
  3. is some kind of terminal object among those satisfying above.

With the definition of the limit, these properties get a precise language. We first set up a ‘target’: Let $\textbf{I}$ be a small category, then a functor $D:\textbf{I}\rightarrow \mathcal{A}$ is called a diagram in $\mathcal{A}$ of shape $\textbf{I}$, and this is our target for the limit. Since $D$ makes an $\textbf{I}$-looking diagram inside $\mathcal{A}$, it is well-termed.

Next, we define a concept in which the limit is the terminal; a cone. A cone on a diagram $D$ consists of a vertex $A$ and morphisms $f_I:A\rightarrow D(I)$ for all $I\in \textbf{I}$, satisfying the second condition above, described by:

$$ \forall u:I\rightarrow J,\, D(u)\circ f_I=f_J. $$

Placing the diagram on the bottom, it certainly looks like a cone, with edges commuting with all morphisms in the diagram.

Finally, a limit of $D$ is a universal cone $\left(p_I:L\rightarrow D(I)\right){I\in \textbf{I}}$(these morphisms are called projections). That is, for every cone $\left( f_I:A\rightarrow D(I)\right){I\in\textbf{I}}$, there exists an unique morphism $\overline{f}:A\rightarrow L$ such that:

$$ f_I=p_I\circ\overline{f},\:\forall I\in \textbf{I}. $$