# Is there a meaningful distinction between “inclusion” and “monomorphism”?

The title pretty much says it all. As far as I can tell, the terms "inclusion" and "monomorphism" are equivalent. (Ditto for $\hookrightarrow$ and $\rightarrowtail$.) Is this the case?

Edit: Judging by the answer and comments I've received so far, I realize I'm working with definitions that may not be universal. My source is Mitchell's Theory of Categories (1965). On p. 6, he writes (emphasis in the original):

If $\alpha:A' \to A$ is a monomorphism, we shall call $A'$ a subobject of $A$, and we shall refer to $\alpha$ as the inclusion of $A'$ in $A$.

As far as this quote goes, I'd say that "inclusion" and "monomorphism" are essentially synonymous, differing only in the fine grammatical structures of the English phrases that can be formed with each; e.g. "$f$ is the inclusion of $X$ in $Y$" vs. "$f$ is a monomorphism from X to Y". The fact that Mitchell uses the definite article for "the inclusion", even though it was defined by the existence of "a monomorphism" (indefinite article), is a bit disconcerting, but Mitchell addresses this concern a few lines later:

...it is important to remember that in general there is more than one monomorphism from $A'$ to $A$, and that whenever we speak of $A'$ as a subobject of $A$ we shall be referring to a specific monomorphism $\alpha$.

On the basis of this additional clarification, I'd "fill out" Mitchell's definition of "inclusion" slightly: if $A'$ is a subobject of $A$ (as defined above), then the inclusion of $A'$ in $A$ is

the monomorphism $\alpha$ that our designation of $A'$ as a subobject of $A$ is based on. (✽)

In fact, I imagine that, given this context, one could use a phrase like "the monomorphism of $A'$ to $A$" as shorthand for the formula (✽) I just proposed. If so, "inclusion" and "monomorphism" would be quite synonymous. (At the very least, there is a bijection between the class of monomorphisms and the class of inclusions.)

Due to these considerations, I was puzzled by random comments I'd found online that implied some meaningful distinction between these two terms (e.g. statement of preference as to which type of arrows to assign in diagrams to inclusions and which to monomorphisms), and this prompted my original question.

I am now particularly interested in learning of examples of monomorphisms that are not inclusions, as alluded to by Zhen Lin in one of the comments below. This will give me a clue about the various non-synonymous meanings given to the term "inclusion".

I regret that this clarification of my original post is many times longer than the original, but after having erred perhaps on the side of brevity, now I'm making up for it by going verbose.

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Well, "monomorphism" is a term from category theory. In certain categories the idea of "inclusion" also makes sense (because of the inner nature of the objects there). In some cases these two notions coincide, but in other cases they do not coincide. –  GEdgar Sep 13 '11 at 14:35
Not every monomorphism is an inclusion (e.g. in the category of topological spaces), but I certainly hope every inclusion is a monomorphism... –  Zhen Lin Sep 13 '11 at 14:40
@kjo: Regarding your elaborated post, you may some of the discussion here about subobjects and monomorphisms useful. (In particular, Mitchell's definition of subobject is not standard.) –  Zhen Lin Sep 14 '11 at 3:01
It would be very much easier to answer your query if you gave a few examples of categories you're comfortable with. From your previous questions I'm not entirely sure if you have a good stock of examples to draw from. So: could you give a short list of categories you like to think about? E.g. topological spaces, modules, groups, rings, fields, etc? –  t.b. Sep 14 '11 at 3:02
If using “inclusion” in Mitchell's sense, these terms are different from the formal point of view. Monomorphism is a property of morphisms. Inclusion is a component of a subobject, i.e. the term “inclusion” extracts a morphism from any subobject. These terms have very different nature. –  beroal Sep 17 '11 at 11:19

Monomorphism is a categorical notion. Sere this wikipedia article. Its characterising property is left-cancellation.

The word inclusion doesn't have a strictly defined meaning. It often refers to the natural injection which is associated with a subobject, like a subgroup of a group or a subspace of a topological space or a vector space. This is in accordance with this wikipedia article.

As for the notation: I think both $\hookrightarrow$ and $\rightarrowtail$ are used to denote both inclusions and monomorphisms. Personally, I'd prefer $\hookrightarrow$ for inclusions and $\rightarrowtail$ for monomorphisms.

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As Rasmus points out, monomorphism is a purely categorial notion.

On the other hand, inclusion is "multi-paradigmatic" - its designed to bridge the classical and categorial approaches. In particular, if $A$ and $B$ are sets and $A \subseteq B$, then according to category theory, it would not be technically correct to say that $A$ is a subobject of $B$, because we don't know how $A$ "sits inside" $B$ until we specify an injection $\phi : A \rightarrow B$.

The idea behind the inclusion map is that there's an obvious choice of $\phi$. Namely, we can choose the unique $\phi : A \rightarrow B$ such that for all $a \in A$ it holds that $\phi(a)=a$, and we call this the inclusion of $A$ in $B$. In this way, we see that subsets in the classical sense of the word correspond naturally to subsets in the categorial sense of the word.

The same goes for subgroups, subrings etc.

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