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Great! I removed the redirects from complete atomic Boolean algebra, since there’s already extensive discussion of those at complete Boolean algebra.
Those redirects were my fault, sorry. I didn’t realize we already had discussion of them elsewhere. Thanks!
I wrote atom.
We should eventually add some remark, given that there is also the notion of atomic object.
@Urs: I added such a remark (see the section on categorification in atom).
Thanks!
I have added in two parenthetical remarks trying to briefly explain (to the reader who does not know this yet) in which sense this is a categorification. Please edit if you feel it breaks your text.
Just for completeness, I have split off an entry atomic category, and copied that therorem also over to presheaf topos.
atom gives a definition of “atomic” which elsewhere, e.g Wikipedia, is called “atomistic”:
A partially ordered set with a least element is called “atomic” if every non-zero element b > 0 has an atom a below it, i.e. b ≥ a :> 0.
A partially ordered set with least element 0 is called “atomistic” if every element is the least upper bound of a set of atoms.
For a Boolean lattice these two notions coincide, however I’ve worked with lattices which have atoms (atomic) but are not atomistic - they have a notion that might be called semi-atoms that exist in chains down to atoms or possibly infinite chains of semi-atoms that never terminate in an atom that covers the bottom.
I somewhat dislike the term “atomistic” and would prefer that that notion was referred to by “atomic” while the notion of “having atoms” took some marked form such as “atomed” or “atomful” or even “subatomic” since “atomistic” is a stronger notion.
Does current conventions really support the atomic/atomistic distinction? This is apparently the case in Lattice Theory at least as far back as Birkoff and von Neumann. I don’t know the use of “atomic” in other branches of mathematics other than they often use “prime” for a similar notion - I would suspect they have adopted “atomic” from Lattice theory.
If so many uses of “atomic” in the nLab need to be changed to “atomistic”.
If so many uses of “atomic” in the nLab need to be changed to “atomistic”.
Says who and whose army? ;-)
I added a bit more: in a (not necessarily complete) Boolean algebra, the notions of atom and atomic object coincide. They need not coincide for a general frame (an atom is an atomic object, but not conversely).
While I don’t have the backers or an army to forcefully decide the atomic/atomistic issue and maybe change lots of “atomic”s to “atomistic”s, I over compensated by adding the following small Properties section to atom.
If $a$ is an atom in an atomic (in the standard sense, not in the screwy definition currently on this page) lattice or more generally a meet semilattice and $b$ any other element then
$a \wedge b \in \{a, \bot\}$
This property is important to when I eventually get around to defining what I call a semi-atom (which includes atoms) with the property
if $a$ and $b$ are semi-atoms then
$a \wedge b \in \{a, b, \bot\}$
I don’t understand your property, Rod. What does it matter in what sense the semilattice is atomic? That’s irrelevant; if $a$ is an atom in any semilattice (even if it happens to be the only atom and there are lots of elements with no atoms beneath them) and $b$ is any other element, then $a \wedge b \leq a$ and so (classically) is equal to $a$ xor $\bot$. Indeed, $a \wedge b$ is the generic example of a $p$ such that $p \leq a$, so this is really just the definition of atom all over again.
Also: let’s get rid of the “screwy”. Remarks on names and where conventions differ are of course appreciated; pejoratives, not.
Toby, I don’t fully understand the notion of “property” as used in nLab. Maybe what I meant by calling it a “property” was just a simple formula that directly states what is implicit in the definition so maybe it should have been put under some other section name, but I don’t know what.
In my current perspective on mathematics I am inclined to move away from the axiomatic (definition and consequences) and to be more interested in a thick description where facts about a mathematical structure hold no matter how it is minimally axiomatized.
You are right that it doesn’t matter that the semi-lattice is atomic, just that $a$ is an atom in it. But without mentioning “atomic semi-lattice” how could I get a dig in about the wrong definition of “atomic”? :)
Under what heading would you record
$a \wedge b \in \{ a, \bot \}$
It is a useful and distinguished formula that can be separated from textual explanations and should exist somewhere. And I find it useful in my generalization to a semi-atom.
@Todd #15
Also: let’s get rid of the “screwy”. Remarks on names and where conventions differ are of course appreciated; pejoratives, not.
Umm, I put in “screwy” so that someone would be forced to edit and change things and maybe fix the atomic/atomistic problem.
Could I really not be understanding things such as category theory not having a atomic/atomistic distinction?
If so I’m sorry for trying to provoke a terminological decision and resolution.
I put in “screwy” so that someone would be forced to edit
Please don’t put entries intentionally in bad shape in order to force “someone” to do something. If there really is a termionology clash, it is easily resolved by having the entry discuss both meanings,and having its Properties-section explicitly refer (by number, if necessary) to whetever definition it needs.
I don’t fully understand the notion of “property” as used in nLab.
I don’t think what Toby said has anything to do with the entry property. Instead, he doubted the formulation of the statement that you made.
Added a subsection on terminology, written in a hurry, which at least points to variances between usages on the page and those elsewhere, and offers a justification for the usage on the page. I removed “screwy” from the Properties section but otherwise left it alone, although I agree with Urs and Toby that I don’t know what it’s doing there – it doesn’t seem to add anything.
I add my voice to Urs #17, and repeat what I said in #15 (also in #11: I don’t like being told what we “must do”). If one spots variance to other established terminology, the sensible thing to do is for that one to make a sober note of it on the page, and point to the note here. Then we can discuss.
I’m trying to understand why our meaning of ‘atomic’ (Wikipedia’s meaning of ‘atomistic’) categorifies to the notion of atomic category. I understand how atoms are categorified to atomic objects (even if it really only categorifies atoms in a boolean algebra), but where does this generating set come from, and where did the colimits (suprema) go?
Maybe this would make you happier. For any preorder $P$ with a small set of tiny objects (automatic if $P$ is small), the full inclusion $i: Tiny(P) \to P$ induces a functor $R: P \to 2^{Tiny(P)^{op}}$, by restricting a Yoneda embedding. If $P$ is small-cocomplete, this has a left adjoint $L$, taking a down-closed subset of $Tiny(P)$ to its supremum in $P$. If $P$ is atomic, i.e., if each element $p \in P$ is the supremum of the tiny objects below it, i.e., if $L$ is a reflector, i.e., if $L \circ R = 1$, then the claim is that the adjoint pair is an adjoint equivalence. (Or in other words, that an atomic sup-lattice is equivalent to a free sup-lattice, where “free” here means left adjoint to an underlying functor $sLat \to Preord$.)
The result on complete atomic Boolean algebras is a special case, since we have already established that tiny objects are the same as atoms in the context of Boolean algebras.
Let me write the proof of the claim in such a way to suggest how it is supposed to categorify. The reflector $L$ takes a $\mathbf{2}$-enriched functor $X: Tiny(P)^{op} \to \mathbf{2}$ to
$L(X) = \int^{a \in Tiny(P)} X(a) \cdot i(a)$where the coend is an appropriate supremum. We have
$\array{ R L(X) & = & P(i-, \int^a X(a) \cdot i(a)) \\ & = & \int^a X(a) \cdot P(i-, i(a)) \\ & = & \int^a X(a) \cdot Tiny(P)(-, a) \\ & = & X }$where the second equation comes from the fact that for tiny objects $t$, $P(t, -)$ preserves sups. The last line is a co-Yoneda lemma or whatever it is we call it. Thus $R L = 1$ as well.
The most obvious categorification (in the sense of moving from $(-1)$-$Cat$ enriched categories to $0$-$Cat$ enriched categories) would be to suppose we have a small-cocomplete category $C$ with a small dense subcategory of tiny objects such that every object $c$ of $C$ is the colimit over the obvious comma category of maps from tiny objects to $c$ (i.e., so that the left adjoint $L$ is a reflector of the canonical functor $C \to [Tiny(C)^{op}, Set]$). The condition appearing after the words “such that” should be what we mean by “atomic” (cocomplete) category. Again, the conclusion is that $C$ is equivalent to a presheaf category, by essentially the same argument as above.
Whatever I actually wrote down on the page atom is essentially a verbatim quote of Bunge of a result from her thesis that I found on the categories list, which presumably is a more minimal-looking set of hypotheses which should lead to the hypotheses as reformulated here; I suppose her hypotheses were formulated to satisfy her thesis needs.
Does this clarify the intention?
To address Toby’s question more satisfactorily, and to put “atomic object/category” in the proper context (which is enriched category theory), I have rewritten some relevant portions of atom. It could probably be smoothed over some more.
Thanks, Todd! So we learn that atomic categories are more like atomistic posets than atomic posets (to use Wikipedia’s terminology that distinguishes these), but they’re not quite (a categorification of) either, since they use (categorifications of) tiny elements instead of atoms.
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