Category theory Notes • Limits & Colimits

Diagram

Let $\mathscr{A}$ be a category and $\mathbf{I}$ a small category. A functor $\mathbf{I} → \mathscr{A}$ is called a diagram in $\mathscr{A}$ of shape $\mathbf{I}$.

Cone

A cone on diagram $D : \mathbf{I} → \mathscr{A}$ is an object $A ∈ \mathscr{A}$ (the vertex of the cone) together with a family:

$$ (A \stackrel{f_I}{\rightarrow} D(I))_{I\in \mathbf{I}} $$

of maps in $\mathscr{A}$ such that for all maps $I \stackrel{u}{→} J$ in $\mathbf{I}$, the triangle

commutes.

Limit

A limit of D is a cone $(L \stackrel{p_I}{→} D(I))_{I∈\mathbf{I}}$ with the property that for any cone on D, there exists a unique map $\overline{f} : A → L$ such that $p_I ◦ \overline{f} = f_I$ for all $I ∈ \mathbf{I} $.

The maps $p_I$ are called the projections of the limit.

We denote $(L \stackrel{p_I}{→} D(I))_{I ∈ \mathbf{I}}$ is a limit as $L = \text{lim}_{\leftarrow \mathbf{I}} D$.

Example

Product

Take $\mathbf{I}$ be a category with only two objects and no non-trivial mophisms, and $\mathscr{A}$ be an arbitary category. It's obvious that functor $D: \mathbf{I} → \mathscr{A}$ is a diagram:

And $A$ together with map family $A \stackrel{f_1}{\rightarrow} X$ and $A \stackrel{f_2}{\rightarrow} X$ forms a cone.

The most universal cone, which is the limit in this shape. Is $X \times Y$:

Take

• $\mathscr{A} = \textbf{Set}$
• $X = \{1,2\}$
• $Y = \{a,b\}$

An example of a non-limit cone:

• $A' = \{1, 2\}$
• $f_1(x) = x$
• $f_2(x) = \text{if}\ x = 1\ \text{then}\ a\ \text{else}\ b$

We can see there exists a unique $\overline{f}(x) = (x, \text{if}\ x = 1\ \text{then}\ a\ \text{else}\ b)$ which makes this graph correct.

Equalizer

Take $\mathbf{I}$ be a category with only two objects and two parallel mophisms, and $\mathscr{A}$ be an arbitary category. It's obvious that functor $D: \mathbf{I} → \mathscr{A}$ is a diagram:

And $A$ together with map $A \stackrel{f}{\rightarrow} X$ forms a cone.

The most universal cone, which is the limit in this shape. Is the equalizer:

Take

• $\mathscr{A} = \textbf{Set}$
• $X = \{1,2,3\}$
• $Y = \{a,b\}$
• $$ s(x) = \begin{equation} \begin{cases} a & x = 1,3 \\ b & x = 2 \end{cases} \end{equation} $$
• $$ t(x) = a $$

An example of a non-limit cone:

• $A' = \{1, 2, 3\}$
• $g(x) = (f ◦ \overline{f})(x) = x, x \in \{1,2,3\}$

Note: $E = \{1,3\}$, $f(x) = x, x \in \{1,3\}$

We can see there exists a unique $\overline{f}(x) = x, x \in \{1,2,3\}$ which makes this graph correct.

Pullback

Take $\mathbf{I}$ be a category with three objects and two mophisms, point from different sources into one same target, and $\mathscr{A}$ be an arbitary category. It's obvious that functor $D: \mathbf{I} → \mathscr{A}$ is a diagram:

And $A$ together with map family

• $A \stackrel{f_1}{\rightarrow} X$ and
• $A \stackrel{f_2}{\rightarrow} Y$

forms a cone.

The most universal cone, which is the limit in this shape. Is the pullback:

Take

• $\mathscr{A} = \textbf{Set}$
• $X = \{1,2\}$
• $Y = \{a,b\}$
• $Z = \{c, d\}$
• $s(1) = c$, $s(2) = d$
• $t(a) = c$, $t(b) = d$

An example of a non-limit cone:

• $A' = \{1, 2\}$
• $f_1(1) = 1$, $f_1(2) = 2$
• $f_2(1) = a$, $f_2(2) = b$

Note: $P = \{(x,y) ∈ X × Y |s(x) = t(y)\} = \{(1, a), (2, b)\}$, $p_1((x, y)) = x$, $p_2((x, y)) = y$

We can see there exists a unique

$$ \overline{f}(x) = \begin{equation} \begin{cases} (1,a) & x = 1 \\ (2,b) & x = 2 \end{cases} \end{equation} $$

which makes this graph correct.

Colimit

Colimits are dual of limits, or, limits in the opposite category.

Colimits are denoted as $\text{lim}_{\rightarrow \mathbf{I}} D$.

Examples

Monos & limit / Epics & colimits

Recall we define monomorphism as:

In any category $C$, an arrow $f: A → B$ is called a monomorphism if for any $g, h: C → A$, $f\cdot g = f\cdot h$ implies $g = h$.

We can find out that saying $f$ is a monomorphism is equivalent with saying:

is a pullback.

Whenever we prove a result about limits, a result about monics will follow.

The dual concept of monos is epics, which is corresponding with a pushout colimit.