# Do we really need separate tags for morphisms between certain objects when there's a tag for the objects?

Currently we have tags for , and and separate tags for , and . (The latter tag is used a total of 11 times, 6 times by the same user.)

In my opinion, the tags about the morphisms add little value to a question and should be made synonyms of the bigger tags.
While it is true that for example, not every group theory question focuses on the properties of group homomorphisms, it's difficult to find much group theory questions that don't involve group homomorphisms in some way. The set of users who are interested in reading or answering questions about group homomorphisms is almost surely identical with those that are interested in group theory.
If someone has a specific question about properties of group homomorphisms, then this can be made clear in the title and question text. I also don't think that the tag on group homomorphisms makes questions that are specifically about group homomorphisms easier to find, since it is difficult to ensure that it is consequently used on such questions (Currently, there are many questions about group homomorphisms, which only have the tag)

I could of course post this in the tag managment thread, but I feel like this is a general principle that should be discussed. If there's a consensus that seperate tags for morphisms are worthwhile, then I think it would be only consequential to create tags such as monoid homomorphisms and module homomorphisms etc.

Note that is already a synonym for and the proposal to make a synonym of is pending with 3 upvotes. I have also proposed as a synonym for . (I don't have the score to propose as a synonym to )

• I wold consider vector spaces to be a sufficiently important class of mathematical structures to leave the tag (linear-transformations) as a separate tag. (They are morphisms for this case.) – Martin Sleziak Feb 15 '18 at 16:06
• Since tags for morphisms were brought up, maybe we could also discuss whether separate tags for isomorphisms are needed? Or is it better to leave this for a separate discussion? There are tags (vector-space-isomorphism). (group-isomorphism), (ring-isomorphism) and (module-isomorphism). The tag (homeomorphism) is already a synynom of (general-topology). The tag (graph-isomorphism) is perhaps slightly different from the previously mentioned tags. – Martin Sleziak Feb 15 '18 at 16:11
• I don't see the point of the synonym. If the tag is not useful why keep it? (A synonym is useful in case somebody might not know the correct tag or if there is conflicting terminology. This is basically impossible in the given case.) – quid Feb 15 '18 at 23:27
• Tags are for searching. It is much easier to search for a single tag than multiple tags. Then the question is, how often would you want to search for morphisms in general, rather than specific "ring homomorphism." A "morphism" tag seems like it is a category theory question. – Thomas Andrews Feb 21 '18 at 17:57
• I suppose a second purpose of a tag is for badges. What achievement does it mean to get a "morphism" badge? It also seem unified, when you consider all the different kinds of morphism. Such a badge seems vague. – Thomas Andrews Feb 21 '18 at 18:05
• Given the voting below, I cancelled the (pending) synonyms. – quid Mar 31 '18 at 15:45

I am against this proposal. The tags and are pretty large and it is rather desirable to have some more tags to structure them.

The "homomorphism" tags to me seem reasonable, in particular for groups and rings (the semigroup version is maybe small).

At the beginning level there is a decent number of considerations that are somewhat specifically about this notion and derived ones (such as kernel).

The incoherence that ring-morphisms is already a synonym should better be resolved by canceling it and linking it together with homomorphism and maybe isomorphism tags.

The arguments put forward against the tags could be mounted against many a tag.

Tangentially, I think subgroups should not be a synonym either.

Finally, if the tags should be found to be useless they should better be merged and removed rather than being preserved as synonyms. Making them synonyms in my mind is worse then either keeping or removing them.

• I think you've constructed a straw man argument by insisting that the issue is whether "the tags should be found to be useless". A synonym is a word that has essentially the same meaning as another word. Here the motivation is not to deny "some more tags to structure" larger topics, but rather to ask whether "morphism" tags do that when applied in a purely categorical sense. I think existing tags (normal-subgroups) and (ideals) do more to provide structure. – hardmath Feb 19 '18 at 1:47
• But for most users (also among those that might use the tag) ring-homomophism does not at all have the same meaning as ring. Your entire argument for the syns is based on a false premise. Most users asking the relevant question would not even know what a category is. (The ring-morphism syn is basically useless; it worked exactly once, over an extended period of time.) [And please answer my question on you answer.] – quid Feb 19 '18 at 18:17

I agree with Mathein's suggestions about creating tag synonyms. Also I agree with the cases where Martin suggests not creating synonyms and want to express my thoughts about those.

Two considerations occur to me about the separate importance of tag linear-transformations. The first has to do with that most commonplace of problems, their representations, e.g. as matrices. There's a good bit of mathematical and algorithmic lore about the connections between these particular morphisms and representations, so much that it makes sense to identify that as a searchable topic.

The second point has to do with topological aspects of linear transformations. Even in the finite dimensional setting the notion of isometry connects us to some important approximation problems. The topic in topological vectors spaces is also a notable extension of the concept in vector spaces per se.

Martin also points out the tag graph-isomorphism as worth keeping separate. I agree, not only because it connects to the important open problem about algorithms for detecting isomorphic graphs, it helps to frame the counting issue regarding labelled vs. unlabelled graphs. So I feel this tag has significance beyond its base tag graph-theory.

• I agree with the views on graph isomorphisms and linear transformations. While I wrote "general principle" in the OP, I had mostly the examples in mind that I gave. (And possible other examples which don't exist yet, such as [monoid-homomorphism], [field-homomorphism] etc.) I didn't want to suggest that we should establish a strict rule that is applied mindlessly to everything, but I do think that we should treat ring homomorphisms and group homomorphisms (and possible tags on a similar level) in a consistent way. – MatheinBoulomenos Feb 15 '18 at 19:07
• Apologies for confusing you with @Martin, but I could hardly make a more flattering misidentification. – hardmath Feb 15 '18 at 19:23
• What's the point of the synonym? – quid Feb 15 '18 at 23:28
• @quid: Assuming this is for me, the point of synonyms here and generally is to cut down on the tag space's redundancy, which should improve search results. E.g. it would provide the same results whether one searches on [group-theory] or [group-homomorphism]. – hardmath Feb 16 '18 at 7:24
• Sorry I was a bit too brief here. (My comment on OP is a bit more detailed.) My point is, either we do not need the tags (then the just can be merged and removed), or they do serve some purpose (which they still serve once synonymised). Thus, let me start by asking what is in your opinion the purpose of the tags? After all you propose to keep the tags. – quid Feb 16 '18 at 9:41
• @quid: Ah, I get the point. I'm thinking that these are "natural" synonyms which are likely to occur again and again in various categories. As Eilenberg and MacLane noted originally, the morphisms can be used to do everything. So a user might well be thinking in terms of [group-homomorphism], and we'd silently convert that to "object oriented" term. – hardmath Feb 16 '18 at 15:56
• What do you think what would a hypothetical user you describe do in the absence of the tag? (a) be puzzled. (b) tag group-theory. (c) create the tag (if they can) (d) something else. In case of (b) there is no need for a syn, in case of (c) maybe there needs to be a reason provided that we override their choice. (I did not see any, and it is not as if the tag is all that small.) – quid Feb 18 '18 at 14:50

## Replace them by their relevant theory tag and the morphism tag

This all could be considered subsets of the tag, since just means any structure preserving map. Therefore, someone interested in group homomorphisms would just search for and .

This is a slight problem if you have a question related to ring theory and some kind of morphism other than ring homomorphisms (for example, if you are asking about ways to turn rings into group homomorphisms), but I don't think this will be a huge problem.

• It seems more likely that they'd search for (group-theory) and (homomorphism) (and now it seems strange that the latter isn't a synonym for (morphism) and indeed doesn't exist at all). – Toby Bartels Feb 24 '18 at 7:31