Commutative Algebra 49

Morphism of Diagrams

Throughout this article \mathcal C denotes a category and J is an index category.

Definition

Given diagrams D, D' : J\to \mathcal C, a morphism D \to D' is a natural transformation T : D\Rightarrow D'.

Thus we have the category of all diagrams in \mathcal C of type J, which we will denote by \mathcal C^J.

For example if we write D and D’ as tuples:

\left((A_i)_{i\in J}, (\beta_{e}: A_i \to A_j)_{(e : i\to j)}\right), \quad \left((A'_i)_{i\in J}, (\beta'_{e}: A'_i \to A'_j)_{(e : i\to j)}\right)

a morphism D\to D' is a collection of morphisms \gamma_i : A_i \to A_i' such that

(e:i\to j) \implies \beta'_e \circ \gamma_i = \gamma_j \circ \beta_e : A_i \to A_j'.

In diagram form, we have the following, where all “rectangles” commute.

diagram_morphism_example

Exercise A

Suppose any diagram of type J in \mathcal C has a colimit. Prove that we get a functor \mathrm{colim} : \mathcal C^J \longrightarrow \mathcal C which takes a diagram in \mathcal C to its colimit. In other words show that a morphism of two diagrams of same type in \mathcal C induces a morphism of their colimits.

blue-lin

Category of Modules

Proposition 1.

Colimits always exist in the category of A-modules.

Proof

Suppose \left((M_i)_{i\in J}, (\beta_{e}: M_i \to M_j)_{(e : i\to j)}\right) is a diagram of type J. Let P = \oplus_{i\in J} M_i with canonical embeddings \nu_i : M_i \to P. Let Q\subseteq P be the submodule generated by all elements of the form \nu_i(m_i) - \nu_j(\beta_e(m_i)) over all m_i \in M_i and e : i\to j in J. We claim that P/Q satisfies our desired universal properties. Define

\epsilon_i : M_i \longrightarrow P/Q, \qquad M_i \stackrel {\nu_i}\longrightarrow \oplus_{i\in J} M_i = P \stackrel \pi \longrightarrow P/Q.

By definition \epsilon_j \circ \beta_e = \pi\circ \nu_j \circ \beta_e = \pi\circ \nu_i = \epsilon_i for all e:i \to j.

Now suppose we have a module N with linear maps (\alpha_i : M_i \to N)_{i\in J} such that for any e:i\to j we have \alpha_j \circ \beta_e = \alpha_i. The collection of \alpha_i induce, by definition of direct sum, a unique map g : \oplus_{i\in J} M_i \to N such that g\circ\nu_i = \alpha_i for each i\in J. Hence

e:i\to j \implies g\circ (\nu_j \circ \beta_e - \nu_i) = g\circ \nu_j \circ \beta_e - g\circ \nu_i = \alpha_j \circ \beta_e - \alpha_i = 0

so g factors through f:P/Q \to N such that f\circ \pi = g. Thus for each i\in J we have f\circ \epsilon_i = f\circ \pi\circ \nu_i = g\circ \nu_i = \alpha_i. ♦

Exercise B

Prove that colimits exist in the categories \mathbf{Set}, \mathbf{Top} and \mathbf{Gp}.

blue-lin

Another Functoriality

Definition.

Suppose F: J_0 \to J is a morphism of index categories. Composition then gives:

(D : J \to \mathcal C) \mapsto (D\circ F : J_0 \to \mathcal C).

Thus a diagram of type J gives us a diagram of type J_0. If J_0 is a subcategory of J, this is just the restriction of D to J_0, denoted by D|_{J_0}.

In fact we get a functor F: \mathcal C^{J} \to \mathcal C^{J_0}. Indeed, a morphism between diagrams D_1, D_2 : J \to\mathcal C is a natural transformation T : D_1\Rightarrow D_2. We let F take this T to

F(T) =  1_F * T : (D_1 \circ F) \Rightarrow (D_2\circ F),

where * is a form of “horizontal composition” of natural transformation (see the optional exercise here).

Although the abstract definition looks harrowing, the underlying concept is quite easy when J_0 is a subcategory of J, so it helps to keep this special case in mind. We denote the diagrams D_1, D_2 by the following tuples

((A_i)_{i\in J}, (\beta_e : A_i \to A_j)_{e:i\to j}), \quad ((A'_i)_{i\in J}, (\beta'_e : A'_i \to A'_j)_{e:i\to j})

so that a morphism D_1 \to D_2 is of a collection of morphisms \gamma_i : A_i \to A_i' in \mathcal C such that for any e:i\to j we have \beta'_e\circ \gamma_i = \gamma_j \circ \beta_e. Now the new diagrams D_1|_{J_0} and D_2|_{J_0} are the same tuples but with i\in J_0 and e:i\to j running through morphisms in J_0. Hence, F(T) is given by the same collection of \gamma_i, except now i runs through i \in J_0.

Proposition 2.

Let F : J_0 \to J be a morphism of index categories. Then for any diagram in \mathcal C of type J, denoted by the pair (A_i) and (\beta_e)_{e:i\to j}, we have an induced

f: \mathrm{colim}_{i_0 \in J_0} A_{F(i_0)} \longrightarrow \mathrm{colim}_{i\in J} A_i

assuming both colimits exist.

Proof

By definition A := \mathrm{colim}_{i\in J} A_i comes with a collection of morphisms (\epsilon_i : A_i \to A)_{i\in J} such that \epsilon_j \circ \beta_e = \epsilon_i for all e:i\to j in J.

Similarly A_0 := \mathrm{colim}_{i_0 \in J_0} A_{F(i_0)} comes with a collection of morphisms (\epsilon_{i_0} : A_{F(i_0)} \to A_0)_{i_0 \in J_0} such that \epsilon_{j_0} \circ \beta_{F(e_0)} = \epsilon_{i_0} for all e_0 : i_0 \to j_0 in J_0.

From restricting the first colimit, we get a collection (\epsilon_{F(i_0)} : A_{F(i_0)} \to A)_{i_0 \in J_0} such that \epsilon_{F(j_0)} \circ \beta_{F(e_0)} = \epsilon_{F(i_0)} for all e_0 : i_0 \to j_0 in J_0.

By universal property of the colimit A_0, this induces a unique morphism f:A_0 \to A such that f\circ \epsilon_{i_0} = \epsilon_{F(i_0)} . ♦

Example

By restricting the following diagram

diagram_embedding_example

we obtain a morphism B \coprod B' \to B \coprod_A B', assuming both objects exist. More generally we have \coprod_{i \in J} A_i \to \mathrm{colim}_{i\in J} A_i.

In fact, the proof of proposition 1 gives us a clue on how to construct general colimits. First take the coproduct, which corresponds to colimit over a diagram of vertices. Next we “add the arrow relations” by taking the coequalizer for each arrow.

blue-lin

Limits

Limits are the dual of colimits.

Definition.

Take a diagram in \mathcal C of type J, written as

((A_i)_{i\in I}, (\beta_e : A_i \to A_j)_{(e:i\to j)}

The limit of the diagram comprises of the following data:

(A, (\pi_i : A \to A_i)_{i\in J})

where A = \lim_{i \in J} A_i \in \mathcal C is an object, \pi_i : A \to A_i is a morphism in \mathcal C for each i\in J, such that for any arrow e:i\to j, we have \beta_e\circ\pi_i = \pi_j.

We require the following universal property. For any tuple

(B, (\alpha_i : B \to A_i)_{i\in J})

where B\in \mathcal C is an object, \alpha_i is a morphism for each i\in J, such that for any arrow e:i\to j, we have \beta_e \circ \alpha_i = \alpha_j, there is a unique morphism f : B \to A such that

\pi_i\circ f = \alpha_i \text{ for each } i\in J.

As before, we have the following special cases.

Example 1: Products

If J is obtained from an index set I, the limit is the product \prod_{i\in I} A_i.

Example 2: Pullbacks (Fiber Products)

If J is the following, the resulting limit is the pullback.

pullback_diagram

Example 3: Equalizers

Definition.

The equalizer of \beta_1, \beta_2 : A\to B in a category \mathcal C is the limit of the following diagram.

coequalizer_diagram

This is a pair (C, \pi : C\to A) such that \beta_1\circ \pi = \beta_2\circ \pi and, for any pair (D, \alpha : D\to A) such that \beta_1 \circ \alpha = \beta_2 \circ \alpha, there is a unique f:D\to C such that \pi\circ f = \alpha.

Exercise C

Prove that limits always exist in the category of A-modules.

blue-lin

Initial and Terminal Objects

Definition.

An object A\in \mathcal C is said to be initial (resp. terminal) if for any object B\in \mathcal C, there is a unique morphism A\to B (resp. B\to A).

Note

  • If A is initial or terminal, there is a unique morphism A\to A, i.e. the identity.
  • A is initial if and only if it is a (colimit / limit) of the empty diagram. [ Exercise: pick the right option and write the dual statement. ]

Exercise D

Prove that if A and A’ are initial, there is a unique isomorphism A\to A'. Dually, the same holds for terminal objects. In summary, initial (resp. terminal) objects are unique up to unique isomorphism.

The phrase “unique up to unique isomorphism” has been used multiple times while looking at universal properties. This is not a coincidence, for initial and terminal objects can be used to describe universal properties of various constructions. Here is an example.

Lemma 1.

Let M, N be A-modules. Consider the category \mathcal C(M, N), whose objects are pairs

(P, B : M\times N \to P)

where P is an A-module, B is an A-bilinear map. The morphisms

(B : M\times N \to P) \longrightarrow (B' : M\times N \to P')

are A-linear maps f:P\to P' such that f\circ B = B'. Then (M\otimes_A N, (m, n)\mapsto m\otimes n) is an (initial / terminal) object in \mathcal C. [ Exercise: pick the right option. ]

Proof

Follows directly from the definition. ♦

blue-lin

This entry was posted in Advanced Algebra and tagged , , , , , , , . Bookmark the permalink.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s