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A base of a topological space $X$ need not be closed under intersection. But it is laxly closed in a sense. Does this concept of laxness have a name, perhaps in some categorified context?
Specifically, if $U, V$ belong to the base, then $U \cap V$ need not belong, but some subset of $U \cap V$ must belong. That is, we must have $W$ with $W \subseteq U \cap V$, that is $W \leq U \wedge V$, that is $W \to U \times V$. (We take it for granted that the power set of $X$ has intersections/meets/products. We are looking at a full subcategory of this power set and considering in what sense it’s closed under products.)
Cofiltered?
Well, a base is cofiltered, but that term says too much. I want to say that the base is [???] under intersection. There may not be a word for that. I could invent ‘lax-closed’.
What does “cofiltered” say that is more than what you want?
It says ‘intersection’. I want a word to put before ‘under intersection’.
OK, maybe you all are saying that ‘cofiltered’ is that word: cofiltered under intersection. Of course, that must be what you’re saying!
All right, but can I use this word if intersection is replaced with something that is not a semilattice operation? I feel like ‘cofiltered’ should only be used in that context, and that it generalises to non-semilattice posets rather than to non-semilattice binary operations.
Here’s another example. Take the operation of addition on the poset of real-valued functions of one real variable (which I’m just trying to keep from being too simple). Consider a set $X$ of such functions. Suppose that, whenever $a, b \in X$, we have some $c \in X$ with $c \leq a + b$. So $X$ is not closed under addition. Should I say that $X$ is lax-closed under addition, or is there anything better to say?
Ah, I see what you’re getting at. “Cofiltered” (or “codirected”) refers only to when the binary operation is intersection (and it makes sense even when intersections don’t exist).
Offhand I can’t think of a better name than “lax-closed”, except that perhaps “colax-closed” would be better. Since if $X\subseteq Y$ has this property and you choose such a $c$ for each $a,b\in X$ (or more generally for all $a_1, \dots ,a_n\in X$ — and maybe you need a separate assumption to be able to do this when $n=0$) then you’ll get a function $T X \to X$, where $T$ is the free-monoid monad, and a transformation
$\array{ T X & \hookrightarrow & T Y \\ \downarrow & \neArrow & \downarrow \\ X & \hookrightarrow & Y }$going in the direction that would make the inclusion $X\hookrightarrow Y$ into a colax map. (Although there’s probably no reason for $T X \to X$ to be a map of posets, or to make $X$ into a $T$-algebra (even a lax or colax one).)
Thanks, I’ll say ‘colax’.
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