nForum - Search Results Feed (Tag: action) 2023-12-06T00:25:20+00:00 https://nforum.ncatlab.org/ Lussumo Vanilla & Feed Publisher Stabilizer group https://nforum.ncatlab.org/discussion/10042/ 2019-06-15T13:19:32+00:00 2019-06-16T20:24:36+00:00 jesuslop https://nforum.ncatlab.org/account/1486/ Hi, at Stabilizer Group, the notation for action groupoid used (one slash) seems not to fit with the one at action groupoid (two slashes). Not changing myself for the sake of prudence.

Hi, at Stabilizer Group, the notation for action groupoid used (one slash) seems not to fit with the one at action groupoid (two slashes). Not changing myself for the sake of prudence.

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epi (co?)subobjects and graphs or actions https://nforum.ncatlab.org/discussion/6392/ 2014-12-22T17:46:07+00:00 2014-12-23T04:04:01+00:00 RodMcGuire https://nforum.ncatlab.org/account/92/ Is there a different notion of subobject in a category that uses epis rather than monos? (which I am here calling cosubobject for lack of any better term) Here is a crude definition of subobject ...

Is there a different notion of subobject in a category that uses epis rather than monos? (which I am here calling cosubobject for lack of any better term)

Here is a crude definition of subobject (mono):

When an object, $o$, consists of sets $a, b, c$, then a subobject of it, $o'$, consists of sets $a', b', c'$ along with injections of them into those of $o$, namely $ia: a' \to a$, $ib: b' \to b'$, and $ic: c' \to c$ .

The co-definition replaces the injections with the surjections: $sa: a \to a'$, $sb: b \to b'$, and $sc: c \to c'$ .

Conceptually the standard mono definition forms subobjects by throwing away elements of sets while the epi definition forms them by merging elements together.

Both definitions seem to give the same results on the category $Set$.

The second seems to work better when the objects are graphs. Say I define an type of edge labeled directed graph object that consists of sets $verts, arrows, names$ where each $vert$ has one outgoing $arrow$ for every element of $names$.

$arrows: verts \times names \to verts$

(this is just an action of $names$ on $verts$. However the nLab page action is so abstract it doesn’t mention actions of sets, much less discuss subobjects of actions.)

Under the “throw away” definition, it is hard to form subobjects because if you throw away a $vert$ then you have to also throw away each $arrow$ that points to it, the $vert$s those $arrow$s come from, and so on. However throwing away a $name$ is simple - you only also have to throw away the $arrow$s with that $name$ and the deletion doesn’t cascade.

Under “merge together”, graph/action subobjects may make more sense. If you merge $v_{1}$ and $v_{2}$ then you have to merge the $arrow$s coming from them by $name$, and to merge same $name$d $arrow$s $a_{1}$ and $a_{2}$ you have to merge the $vert$s they point to. Merging $name$s is also somewhat simple.

The epi definition seems to give a better notion of subgraph, at least for what I am working on, though I might prefer a definition where $vert$s and $arrow$s are merged while $name$s are thrown away.

Then again I’m no graph theorist and may be confused about subobjects.

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