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    • CommentRowNumber1.
    • CommentAuthorTobyBartels
    • CommentTimeSep 6th 2012

    A base of a topological space XX 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,VU, V belong to the base, then UVU \cap V need not belong, but some subset of UVU \cap V must belong. That is, we must have WW with WUVW \subseteq U \cap V, that is WUVW \leq U \wedge V, that is WU×VW \to U \times V. (We take it for granted that the power set of XX 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.)

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeSep 6th 2012

    Cofiltered?

    • CommentRowNumber3.
    • CommentAuthorTobyBartels
    • CommentTimeSep 7th 2012

    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’.

    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeSep 7th 2012

    What does “cofiltered” say that is more than what you want?

    • CommentRowNumber5.
    • CommentAuthorTobyBartels
    • CommentTimeSep 7th 2012

    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.

    • CommentRowNumber6.
    • CommentAuthorTobyBartels
    • CommentTimeSep 7th 2012

    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 XX of such functions. Suppose that, whenever a,bXa, b \in X, we have some cXc \in X with ca+bc \leq a + b. So XX is not closed under addition. Should I say that XX is lax-closed under addition, or is there anything better to say?

    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeSep 7th 2012

    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 XYX\subseteq Y has this property and you choose such a cc for each a,bXa,b\in X (or more generally for all a 1,,a nXa_1, \dots ,a_n\in X — and maybe you need a separate assumption to be able to do this when n=0n=0) then you’ll get a function TXXT X \to X, where TT is the free-monoid monad, and a transformation

    TX TY X Y\array{ T X & \hookrightarrow & T Y \\ \downarrow & \neArrow & \downarrow \\ X & \hookrightarrow & Y }

    going in the direction that would make the inclusion XYX\hookrightarrow Y into a colax map. (Although there’s probably no reason for TXXT X \to X to be a map of posets, or to make XX into a TT-algebra (even a lax or colax one).)

    • CommentRowNumber8.
    • CommentAuthorTobyBartels
    • CommentTimeSep 8th 2012

    Thanks, I’ll say ‘colax’.