Limits

Definition.

Fix a category $\mathsf{C}$ and an $\mathsf{J}$-shaped diagram $D : \mathsf{J} \to \mathsf{C}$ in $\mathsf{C}$. The limit of $D$ is the terminal cone over $D$ (should it exist), and dually the colimit of $D$ is the initial cocone under $D$ (should it exist). Explicitly, the limit consists of:

• An object $\ell \in \mathsf{C}$, often instead denoted $\lim_{\mathsf{J}} D$;
• A collection $(\lambda_i)_{i \in \mathsf{J}}$ of arrows $\lambda_i : \ell \to D i$, one for each object $i$ of $\mathsf{J}$^[2] [2] Unlike $\lim_{\mathsf{J}} D$, there seems to be no standard notation for this cone.;

Subject to the following constraints:

• $\lambda$ forms a cone; that is, for each $i \xrightarrow{f} j$ in $\mathsf{J}$, we require $D f \circ \lambda_i = \lambda_j$:

• $\lambda$ is the terminal such cone; that is, for any other cone $\alpha : a \Rightarrow D$, there exists a unique map $u : a \to \ell$ such that for all $i \in \mathsf{J}$, $\alpha_i = \lambda_i \circ u$:

The colimit consists of the dual data, obtained by replacing $\mathsf{C}$ by $\mathsf{C}^{\mathrm{op}}$; explicitly:

• An object $\ell \in \mathsf{C}$, often instead denoted $\mathrm{colim}_{\mathsf{J}} D$;
• A collection $(\iota_i)_{i \in \mathsf{J}}$ of arrows $\iota_i : D i \to \ell$;

Subject to the following constrants:

• $\iota$ forms a cocone; that is, for each $i \xrightarrow{f} j$ in $\mathsf{J}$, we require $\iota_j \circ D f = \iota_i$;
• $\iota$ is the initial such cocone; that is, for any other cocone $\alpha : D \Rightarrow a$, there exists a unique map $u : \ell \to a$ such that for all $i \in \mathsf{J}$, $\alpha_i = u \circ \lambda_i$:

Example. (Product of sets)

FIXME: MOVE THIS DISCUSSION TO A DEDICATED PAGE ON CO/PRODUCTS, AND HAVE THIS PAGE BE MORE FOCUSED ON LIMITS THEMSELVES! Fix $\mathsf{C} := \mathbf{Set}$ and let $\mathsf{J} := (\bullet\quad \bullet)$ be the category with two objects and no non-identity arrows. A diagram (i.e. a functor) $D : (\bullet\quad \bullet) \to \mathbf{Set}$ consists of two sets $A$ and $B$ — the images of the two objects in the indexing category $\mathsf{J}$. A cone over $D$ consists of the blue data in $\mathbf{Set}$ displayed below. This cone corresponds to a unique map $u$ (displayed in red) into the limit $L \xRightarrow{\lambda} D$ such that $\lambda_A \circ \textcolor{red}{\ell} = \textcolor{blue}{a}$ and $\lambda_B \circ \textcolor{red}{\ell} = \textcolor{blue}{b}$. Notice that these equations can also be used to determine the pair $\textcolor{blue}{(f, g)}$ knowing just $\textcolor{red}{\ell}$; thus, the maps $\textcolor{red}{\ell}$ correspond exactly to the pairs $\textcolor{blue}{(f, g)}$.

Recall that elements of a set $X$ are in bijection with functions $1 \to X$^[3] [3] Here, $1 := \{0\}$ is the singleton set, and the bijection sends an element $x \in X$ to the function $f : 1 \to X$ with $f(0) := x$.. We can use this to determine the elements of the limit $L$. Setting $\textcolor{magenta}{Z} := 1$, our elements $u : 1 \to L$ of the limit correspond exactly with pairs of elements $1 \xrightarrow{\textcolor{blue}{a}} A$ and $1 \xrightarrow{\textcolor{blue}{b}} B$. So, elements of $L$ could be defined to be pairs of elements of $A$ and $B$; that is, we could set $L := A \times B$, with $\textcolor{red}{u} := (\textcolor{blue}{a}, \textcolor{blue}{b})$. Now, the relationship $\textcolor{blue}{a} = \lambda_A \circ \textcolor{red}{u}$ — read in terms of elements rather than functions as $\textcolor{blue}{a} = \lambda_A (\textcolor{red}{u}) = \lambda_A (\textcolor{blue}{a}, \textcolor{blue}{b})$ — demands that $\lambda_A : A \times B \to A$ be the left projection map, and similarly $\lambda_B : A \times B \to B$ should be right projection map $\lambda_B : (\textcolor{blue}{a}, \textcolor{blue}{b}) \mapsto \textcolor{blue}{b}$.

Exercise.

Take instead $\mathsf{C}$ as one of the following categories of algebras: $\mathbf{Monoid}$, $\mathbf{Group}$, $\mathbf{Vect}_K$. Verify that the $(\bullet \quad \bullet)$-indexed limits in $\mathsf{C}$ are the relevant products of algebras (e.g. direct product of groups, or direct sum of vector spaces).