Limits and co-Limits: Some Cool Things

I’m not going to reiterate the definitions of limits and co-limits from the last post, so just look them up if you’re new here. They’re not too hard.

This post is mainly just about some random cool things I’ve noticed/ “remembered” / come across whilst playing with the notions of limit and co-limit in various categories.

Thing 1

Say we’re working in R-Mod for some ring $R$ , and let $A,B$ be $R$ -modules. Taking the limit of the “discrete” diagram $A,B$ gives us the “product diagram: ” $A \overset{p_A}{\leftarrow} A \times B \overset{p_B}{\to} B$ . We can then take the co-limit of this diagram, which is the quotient $A \oplus B / K$ , where $K$ is the module generated by elements of the form $(p_A((a,b)), -p_B((a,b))$ for $(a,b) \in A \times B$ . It then follows that the pushout is trivial. Strange.

EDIT: for some clarification, I want to show that for any diagram of the form $A \overset{f}{\leftarrow} C \overset{g}{\to} B$ , then the colimit is the object $A \oplus B/ K$ , where $K$ is the submodule generated by the elements $\{(f(c),-g(c)\}_{c \in C}$ . Clearly, it does fit into the appropriate diagram. Now let $D$ be any other module with maps $j_A: A \to D$ , $j_B: B \to D$ such that $j_A \circ f = j_B \circ g$ . Then, via the universal property of the direct sum, there is a unique map $F: A \oplus B \to D$ such that $j_A = F \circ i_A$ and $j_B = F \circ i_B$ (where $i_A: A \to A \oplus B$ and $i_B : B \to A \oplus B$ are the canonical inclusions). Then we have that $F \circ i_A \circ f = F \circ j_B \circ i_B$ , so $F((f(c),0) = F(0,g(c))$ for any $c \in C$ . Hence $F((f(c),-g(c)) = 0$ , i.e. $K \subseteq \text{ Ker} F$ . Via the universal property of the quotient, there is a unique homomorphism $\overline{F} : A \oplus B/ K \to D$ that makes the whole diagram commute.

In the case where $C = A \times B$ and $f = p_A$ and $g = p_B$ , $K = \langle (p_A(a,b),-p_B(a,b)) \rangle_{a \in A, b \in B} = \langle i_A(A) + i_B(B) \rangle \cong A \oplus B$ , so that the quotient $A \oplus B/ (A \oplus B) = \{0\}$ .

The dual construction is (obviously) similar, where we first take the co-limit of the discrete diagram, then the limit of the “co-product diagram.” It is also the zero module.

Thing 2

Adjunction spaces in Top, the category of topological spaces. Let $X,Y$ be topological spaces, $A \subseteq Y$ (represented as a monic $i: A \to Y$ ) be a subspace. Let $f: A \to X$ be a continuous function. Then the adjunction space obtained by gluing $X$ to $Y$ along $f$ is just the co-limit of the diagram $Y \overset{i}{\leftarrow} A \overset{f}{\to} X$ .

Thing 3

Let $X$ be a topological space, which has a natural poset structure on its collection of open sets. Formally, we turn $X$ into a category X whose objects are the open sets of $X$ and the morphisms are determined via $\text{Hom}(V,U) \neq \{ \emptyset \}$ iff $V \subseteq U$ . Let $V,U,W$ be elements of X such that $V \subseteq W$ and $U \subseteq W$ . Then the limit of the diagram $U \to W \leftarrow V$ is just the intersection $U \cap V$ .

Thing 4

Limits as functors. Turns out you can replace the notion of a “diagram in C” (where C is the category we’re looking at) with a functor $\mathcal{A} : \textbf{I} \to \textbf{C}$ , where I is a small category. Think about it! The limit of such a diagram is denoted by $\varprojlim \mathcal{A}$ .

Quick note: Adjoint Pairs of Functors

Say we have two functors $F: \textbf{C} \to \textbf{D}$ and $G: \textbf{D} \to \textbf{C}$ . We say $F,G$ form an adjoint pair if, for all $X \in \textbf{C}$ , $Y \in \textbf{D}$ , we have a bijection $\text{Hom}_{\textbf{D}}(F(X),Y) \cong \text{Hom}_{\textbf{C}}(X,G(Y))$ that is natural in $X$ and $Y$ . Furthermore, we say $F$ is left adjoint to $G$ , and similarly $G$ is right adjoint to $F$ . Also, $G$ is a right adjoint functor if it has a left adjoint, and likewise for left adjoint functors.

Thing 5

Right adjoint functors commute with Limits. Let $F: \textbf{C} \to \textbf{D}$ and $G: \textbf{D} \to \textbf{C}$ be an adjoint pair of functors, and let $\mathcal{A} : \textbf{I} \to \textbf{D}$ be a diagram. The statement is then that

$G(\varprojlim \mathcal{A}) = \varprojlim G \circ \mathcal{A}$

Awesome. The proof is actually pretty straightforward abstract nonsense, just take the definition of $\varprojlim \mathcal{A}$ as a limiting cone, apply $G$ , get a map $G(\varprojlim \mathcal{A}) \to \varprojlim G \circ \mathcal{A}$ . Then, use adjunction to get a map $F (\varprojlim G \circ \mathcal{A} ) \to \mathcal{A}(I)$ for all objects $I$ in I. The universal property of $\varprojlim \mathcal{A}$ gives us a map $\varprojlim G \circ \mathcal{A} \to G(\varprojlim \mathcal{A})$ by applying adjunction again. These maps are quickly seen to be inverses of each other (keep looking through universal properties and such). A good outline of this proof can be found in Paolo Aluffi’s “Algebra: Chapter 0”

Similarly, we have that Left adjoints commute with co-limits.

Math is awesome.

3 thoughts on “Limits and co-Limits: Some Cool Things”

Gabe says:

October 3, 2012 at 7:16 pm

You should be careful with your Thing 1. I’m pretty sure the co-limit is A x B, (which is A sum B in this case) not empty. If you look at the definition of the co-limit, you will get A cross B cross (A sum B) mod out by a relation which leaves A cross B. If I’m wrong though, I apologize.

Gabe says:

October 5, 2012 at 7:01 pm

Ah okay. I misread your post. I though you were taking an inverse limit of the diagram
A B using the arrows as your inverse system.
But that makes sense now; the colimit of A, B is empty.

1. Gabe says:
  
  October 6, 2012 at 1:03 am
  
  “A B”

Limits and co-Limits: Some Cool Things

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