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1. • CommentRowNumber1.
   • CommentAuthorTobyBartels
   • CommentTimeSep 6th 2012
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   Author: TobyBartels
   Format: MarkdownItexA [[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.)

   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 $W\le U\wedge V$, 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.)

2. • CommentRowNumber2.
   • CommentAuthorTodd_Trimble
   • CommentTimeSep 6th 2012
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   Author: Todd_Trimble
   Format: MarkdownItexCofiltered?

   Cofiltered?

3. • CommentRowNumber3.
   • CommentAuthorTobyBartels
   • CommentTimeSep 7th 2012
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   Author: TobyBartels
   Format: MarkdownItexWell, 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’.

   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’.

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeSep 7th 2012
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   Author: Mike Shulman
   Format: MarkdownItexWhat does "cofiltered" say that is more than what you want?

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

5. • CommentRowNumber5.
   • CommentAuthorTobyBartels
   • CommentTimeSep 7th 2012
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   Author: TobyBartels
   Format: MarkdownItexIt 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.

   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.

6. • CommentRowNumber6.
   • CommentAuthorTobyBartels
   • CommentTimeSep 7th 2012
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   Author: TobyBartels
   Format: MarkdownItexHere'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?

   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 $c\le a+b$. So $X$ is not closed under addition. Should I say that $X$ is lax-closed under addition, or is there anything better to say?

7. • CommentRowNumber7.
   • CommentAuthorMike Shulman
   • CommentTimeSep 7th 2012
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   Author: Mike Shulman
   Format: MarkdownItexAh, 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).)

   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 $TX\to X$, where $T$ is the free-monoid monad, and a transformation

   $$\begin{array}{ccc}TX& \hookrightarrow & TY\\ \downarrow & ⇗& \downarrow \\ X& \hookrightarrow & Y\end{array}$$

   going in the direction that would make the inclusion $X\hookrightarrow Y$ into a colax map. (Although there’s probably no reason for $TX\to X$ to be a map of posets, or to make $X$ into a $T$-algebra (even a lax or colax one).)

8. • CommentRowNumber8.
   • CommentAuthorTobyBartels
   • CommentTimeSep 8th 2012
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   Author: TobyBartels
   Format: MarkdownItexThanks, I'll say ‘colax’.

   Thanks, I’ll say ‘colax’.