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 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

  1. 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.

  2. 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.

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