It is well known that morphisms between the objects of the category are structure-preserving, but I found that in a textbook it said that morphisms are often structure-preserving. Does this mean that there can be a morphism that is not structure-preserving?

  • $\begingroup$ It may be a naive question,because I was reviewing Homological Algebra recently and I found that I still have a few problems. $\endgroup$ – ToneyLiang Feb 15 at 16:05

I think the issue begins here:

It is well known that morphisms between the objects of the category are structure-preserving

This is not the case. The notion of a category generalises the notion of 'sets-with-structure and structure-preserving functions', such as groups and homomorphisms, or topological spaces and continuous maps.

But the extent of the generality is extreme: the objects of a category need not even be sets, and the morphisms of a category need not be functions.

For example, every monoid can be considered as a one-object category, where the elements of the monoid are the morphisms from the single object to itself. In this case, the 'object' is just a placeholder—it has no notion of 'structure'—and the morphisms are certainly not functions (in general).

Categories that 'look like' sets-with-structure and structure-preserving morphisms are called concrete categories. What this means is that the category $\mathcal{C}$ comes equipped with a faithful functor $U : \mathcal{C} \to \mathbf{Set}$. An object $A$ of $\mathcal{C}$ can be thought of has having 'underlying set' $U(A)$, and a morphism $f : A \to B$ can be thought of as having 'underlying function' $U(f) : U(A) \to U(B)$. However, concrete categories are still more general than sets-with-structure and structure-preserving morphisms. There may not actually be any structure to speak of.

  • $\begingroup$ Thanks a lot.I find my issue. $\endgroup$ – ToneyLiang Feb 15 at 16:10
  • $\begingroup$ So why the textbooks just say that the morphisms in the category of groups are homomorphism,and don't say anything else?Does it imply that the morphisms in the category of groups must be homomorphism?I think it is not right. $\endgroup$ – ToneyLiang Feb 19 at 13:20
  • $\begingroup$ @ToneyLiang: I don't know what textbooks you're using. The 'category of groups' is defined to be the category whose objects are groups and whose morphisms are group homomorphisms. That is one example of a category. $\endgroup$ – Clive Newstead Feb 19 at 13:34
  • $\begingroup$ I use a series of books called graduate texts in mathematics,maybe it is the requirement of the book to define the morphisms of the category of groups are homomorphism. $\endgroup$ – ToneyLiang Feb 19 at 13:46

A category doesn't have to consist of sets with some additional structure and maps between those preserving the structure.

Examples for categories that are not of this kind are:

  • Given any group $G$, we can form a category with one object $*$ and for each $g\in G$ a morphism $\varphi_g\colon *\to *$, where composition of morphisms is defined via the group operation and $\operatorname{id}_* = \varphi_{e}$ for $e\in G$ the identity element.
  • Given a poset $(P,\le)$ we can form a category with set of objects $P$ and exactly one morphism $x\to y$ for each $x,y\in P$ with $x\le y$.
  • The homotopy category of topological spaces, where the objects are topological spaces and a morphism $X\to Y$ is a homotopy class $[f]$ of a continuous map $f\colon X\to Y$.
  • $\begingroup$ Thanks a lot.I find my issue $\endgroup$ – ToneyLiang Feb 15 at 16:11
  • $\begingroup$ Note that my first two examples are still concrete categories (see Clives answer), and I can write out the faithful functors to $\mathbf{Set}$ if anybody is interested. $\endgroup$ – Christoph Feb 15 at 16:11

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