1.6.7 Isomorphisms

Now we apply the general categorical approach to some general problems that always arise when we work with the classes of objects. The first one is: which of them should be considered the "same"?

Definition 1.6.31   Given an arbitrary category $\mathcal{C}$ , we call two objects $X,Y\in\in\mathrm{Ob}\mathcal{C}$ isomorphic if there exist such morphisms $f:X\longrightarrow Y$ and $g:Y\longrightarrow X$ that $f\circ g =1_Y$ and $g\circ f = 1_X$ ; both $f$ and $g$ are called isomorphisms.

Control question 1.32   Prove that a connected graph cannot be isomorphic to a disconnected one.

Control question 1.33   Prove that the homology groups of isomorphic graphs are isomorphic. (The first word isomorphic is used in the categorical sense just introduced and the second is a traditional group-theoretic one). Does this assertion imply the assertion of the previous Control Question?

There is a concept between the concepts of small and large categories.

Definition 1.6.32   We call the category moderate if the isomorphism classes of its objects constitute a set (some authors call such categories the categories with a skeleton).

Definition 1.6.33   This set of isomorphism classes can be called the moduli space of the category. When makes sense, it will be denoted $\mathcal{M}[\mathcal{C}]$ .

Example 1.6.34   The category of finite sets is moderate and the set $\mathbf{N}$ of natural numbers is its moduli space. Note that the inclusion $0\in\mathbf{N}$ is necessary.

Exercise 1.6.35   Understand the last assertion.

Remark 1.6.36   We strongly advise our readers to understand why the category $\mathcal{GRPH}$ is moderate. ^1.2 Its moduli space $\mathcal{M}[\mathcal{GRPH}]$ is a countable set with lots of fascinating structures.

Project A. 1   List the isomorphism classes of graphs with $\le$ 3 edges.

(Recall that we have agreed to consider graphs without isolated vertices, otherwise the exercise would make no sense).

Exercise 1.6.37   Why?!