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.

Author: brianhepler

I'm a third-year math postdoc at the University of Wisconsin-Madison, where I work as a member of the geometry and topology research group. Generally speaking, I think math is pretty neat; and, if you give me the chance, I'll talk your ear off. Especially the more abstract stuff. It's really hard to communicate that love with the general population, but I'm going to do my best to show you a world of pure imagination.

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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