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1. • CommentRowNumber1.
   • CommentAuthorTobyBartels
   • CommentTimeAug 19th 2013
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItexAt [[partition]], I\'ve defined partitions of sets, numbers, intervals, measure spaces, and unity on topological spaes, giving these all as special cases of a general concept defined in a monoid whose nonzero elements form an ideal (and possibly equipped with some notion of infinite sum).

   At partition, I've defined partitions of sets, numbers, intervals, measure spaces, and unity on topological spaes, giving these all as special cases of a general concept defined in a monoid whose nonzero elements form an ideal (and possibly equipped with some notion of infinite sum).

2. • CommentRowNumber2.
   • CommentAuthorRodMcGuire
   • CommentTimeAug 19th 2013
   • (edited Aug 19th 2013)
   • PermaLink
   Author: RodMcGuire
   Format: MarkdownItexI've been going to write something up about this (have you noticed me poking around in the nLab :)?) The point I wanted to promote is that (at least for sets) a partition is the opposite of a function. For $f: A \to B$ a function, $f^{op}: B \to A$ is a partition, $image(f)$ indexes the equivalence classes, and $f^{op}\circ f: A \to P(A)$ maps an element of $A$ to its equivalence class induced by $f$. When partitions are involved, equivalence classes are often prominent though this doesn't seem to be the case for one of the most famous partition in categories and graphs where $hom_X$ is the opposite of the function $(source, dest): arrows(X) \to objects(X)^2$ More explicitly say a $0graph$ is a graph (quiver) enriched in $2$ (a most 1 arrow between objects) - I chose the name to make the underlying graph of a $0category$ a $0graph$. A relation between sets, $R: A \to B$ is a $0graph$ with objects partitioned into sets $A$ and $B$ such that $hom(A,B) = arrows(R)$, and a function is a special type of relation. An equivalence relation $E$ on $A$ is the underlying graph (a 0graph) of $E$ considered as a category and this correspondance holds for all relations that are reflexive and transitive. The terminology gets confusing - what is often called the graph of a function is actually the $hom$ of the function's $0graph$. I've been poking around trying to find the correct words to describe the above (which probably aren't right as given), and also trying to figure out how the notion of fiber is involved, what to call the opposite of a function (a "cofunction"?), and on what nlab pages this should be mentioned.

   I’ve been going to write something up about this (have you noticed me poking around in the nLab :)?)

   The point I wanted to promote is that (at least for sets) a partition is the opposite of a function. For $f: A \to B$ a function, $f^{op}: B \to A$ is a partition, $image(f)$ indexes the equivalence classes, and $f^{op}\circ f: A \to P(A)$ maps an element of $A$ to its equivalence class induced by $f$.

   When partitions are involved, equivalence classes are often prominent though this doesn’t seem to be the case for one of the most famous partition in categories and graphs where $hom_X$ is the opposite of the function $(source, dest): arrows(X) \to objects(X)^2$

   More explicitly say a $0graph$ is a graph (quiver) enriched in $2$ (a most 1 arrow between objects) - I chose the name to make the underlying graph of a $0category$ a $0graph$.

   A relation between sets, $R: A \to B$ is a $0graph$ with objects partitioned into sets $A$ and $B$ such that $hom(A,B) = arrows(R)$, and a function is a special type of relation.

   An equivalence relation $E$ on $A$ is the underlying graph (a 0graph) of $E$ considered as a category and this correspondance holds for all relations that are reflexive and transitive.

   The terminology gets confusing - what is often called the graph of a function is actually the $hom$ of the function’s $0graph$.

   I’ve been poking around trying to find the correct words to describe the above (which probably aren’t right as given), and also trying to figure out how the notion of fiber is involved, what to call the opposite of a function (a “cofunction”?), and on what nlab pages this should be mentioned.

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeAug 19th 2013
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexPerhaps you want the word [cograph](http://ncatlab.org/nlab/show/graph+of+a+function#cograph) or [collage](http://ncatlab.org/nlab/show/cograph+of+a+profunctor)?

   Perhaps you want the word cograph or collage?

4. • CommentRowNumber4.
   • CommentAuthorTodd_Trimble
   • CommentTimeAug 21st 2013
   • (edited Aug 21st 2013)
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexRe #1: this is a nice page, Toby -- thanks! Re #2: the opposite $f^{op}$ of a function $f: A \to B$ (in this sense) would be (edited) a partition of $A$ with extra "[stuff](http://ncatlab.org/nlab/show/stuff,+structure,+property)" (corresponding to elements in $B$ not in the image); a partition without extra stuff would be opposite to a surjective function. The corresponding equivalence relation would be the [[kernel pair]] of $f$.

   Re #1: this is a nice page, Toby – thanks!

   Re #2: the opposite $f^{op}$ of a function $f: A \to B$ (in this sense) would be (edited) a partition of $A$ with extra “stuff” (corresponding to elements in $B$ not in the image); a partition without extra stuff would be opposite to a surjective function. The corresponding equivalence relation would be the kernel pair of $f$.