6:47 AM
Currently there are 3 questions tagged both adjoint and cateogry-theory.
In this case, this is clearly the intended tagging:
4

I'm an economist, not a mathematician. I've been trying to make sense of some concepts in functional analysis: dual, bidual, adjoint, natural mapping. The definitions of these notions come out of nowhere and I see no intuition. I thought that I should read category theory to see the big picture. ...

In these two cases I do not understand the post enough to judge. (But AFAICT the word adjoint is not used in the sense of linear algebra.)
3

Category theory is famous for its founder's motto that "adjoints are everywhere". While the original context of his quote was meant to encompass mathematics, we have since seen the concept of adjoints, a.k.a. duality, a.k.a. isomorphisms, extend far beyond the realm of pure mathematics into areas...

0

Consider the abelian cyclic group of twelve elements, C12, with generators $$C12^* = (e^1, e^5, e^7, e^{11}).$$ The cayley diagram generated by $e^1$ is a circle of twelve directed links for the twelve applications of the $e^1$ generator. Taking this as a small category, there must be an opp...

Phew! All questions that were in category-theory and adjoint have been retagged (except one which was about the relationship between adjoint functors and adjoint operators). Now all is left is renaming the tag. — Najib Idrissi Jan 17 at 11:42
Renaming the tag (in order to make clear that it is about linear algebra, adjoint operators, adjungate matrices, ...) definitely needs moderator intervention. And probably so does the synonym for [tag;chaos-theory]. I count only 5 users who have score at least 5 and could vote on the tag synonym.
To get moderators attention, I have mention this in Math Mods' Office (since at least one - or both - of the above issues cannot be done without mods):
in Math Mods' Office, 1 min ago, by Martin Sleziak
In tagging chatroom wythagoras reminded two older issues who gained some support in tag management thread, but they cannot be done without moderator's intervention (just by regular users). So if some of the mods has a time to look at them, perhaps you could comment there and - if appropriate - take some action.
And since I bother the mods, I will mention two more tag-related things which might need help from moderators.
Arjafi suggested to remove tag. And his suggestion is at score +7. (I am not sure whether he meant blacklisting, or just deleting the tag.)
7

There's another possibility that hasn't been mentioned in the OP, and the one I would prefer: kill it! I don't see subspaces as a tag that will ever be consistently used, and not something that has a concrete enough meaning to allow for an easy delineation of when it should be used. Should you a...

Whether we want to create separate tags for (linear-subspaces), (topological-subspaces), (normed-subspaces) is a different question. But I think that +7 is good enough to consider it consensus. (It is definitely more votes that tag-related posts usually get.)
And there are also ideals-related tags. There is a proposal to make synonyms $\to$ and $\to$ .
22

If you think that the tags maximal-ideals and prime-ideals should be synonyms of ideals, upvote this answer.

In this case, there are many users who have sufficient score to suggest the synonym.
I am one of those users, but I do not like the proposal. (Or at best I am indifferent to it.) So it would feel strange for me to suggest the synonym, when there are many others who can do it.
So basically I am asking whether there is some volunteer who wants to suggest the synonyms. (There is also possibility to earn a badge if it gets approved.)
And even if moderators decide to do this, I think it is perfectly ok. (The proposal has score +22, the alternative to leave them separate is at +6. If we only count upvotes, it is 25 to 11.)
I wrote a lot of text, but since there are several issues which need help from moderators (or at least some of them do), I thought this is better than to deal one of them at the time. (And each time ask moderators to come here.)
TL;DR: So to summarize, it would be nice to get some input from mods on: 1) adjoints, 2) chaos-theory, 3) subspaces, 4) (prime and maximal) ideals.

1 hour later…
8:29 AM
We already have the tags , , and on meta. Should we also add ?

2 hours later…
10:50 AM
I will point out that somebody suggested $\to$ synonym: math.stackexchange.com/tags/sequences-and-series/synonyms
Since the tag summation was explicitly created for the purpose to differentiate questions about finite sums from the questions about infinite sums, it would be really bad thing to mix up all questions together.
Which is why I wanted to request here some downvotes for the proposed synonym. But now I see that my votes was enough. (Score -2 is sufficient to remove the suggestion. Which I would have notice if I read carefully enough the link to the synonym I just posted above.)
@JonMarkPerry It seems that you have suggested the above synonym. See here why this would not be a very good idea:
4

I often want to improve the tags on questions like "Is it possible to make this expression shorter?". This question asks how to simplify or evaluate the finite sum $$\sum_{i=1}^n\left( 2^{i-1} a_i \sum_{k=0}^{n-i}10^k \binom{n-i}{k} \right)$$ Questions of this type seem to be fairly common,...

I have sent you an invitation to this room. I do not have much experience with "invite" feature in chat. But I hope this message will reach you.
Usually it is good to discuss synonyms in the tag-management thread first; at least if you are not absolutely sure that the two tags are really for the same purpose.

2 hours later…
1:01 PM
I am not sure belongs here. Remove?
15

The exponential function can be written as $$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots.$$ The subspace of all polynomials is $$\text{span}\{1, x,x^2, \dots \}$$ Sure $e^x$ is in this set?