Timeline for functors of string diagrams in a monoidal category
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
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| Mar 28, 2017 at 10:49 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Feb 26, 2017 at 10:40 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Jan 27, 2017 at 10:25 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Dec 28, 2016 at 10:05 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Nov 28, 2016 at 9:40 | answer | added | Xuexing Lu | timeline score: 2 | |
| Dec 18, 2015 at 0:02 | comment | added | Gerrit Begher | pps.univ-paris-diderot.fr/~mellies/papers/functorial-boxes.pdf is this what you're looking for ? | |
| Jul 26, 2011 at 16:16 | comment | added | Ben Sprott | Qiaochu, Yes, I was thinking of functor as we all know it in that it maps identity to identity and composition to composition. This is why I think any string diagram will be a string diagram under any functor. | |
| Jul 26, 2011 at 16:13 | comment | added | Ben Sprott | sorry about superfluous "\" slashes | |
| Jul 26, 2011 at 16:12 | comment | added | Ben Sprott | Todd, My particular example is a frobenius algebra, or comonoid category, $C$ in a symmetric monoidal category, $X$. The comonoid category is defined as $(A, \f,\g)$ where $A$ is an object in $X$ and $\f, \g $ are $f: A \rightarrow \A \otimes \A$ $g: A \rightarrow \A \otimes I$ So, a forgetful functor $F: C \rightarrow X$ just forgets the extra morphisms and gives $A$. Both categories $X$ and $C$ have string diagrams and I want to know what a string diagram becomes under $F$. So this is very similar to what you have suggested where $M = Free(S)$ where $S$ is a symmetric monoidal cat. | |
| Jul 26, 2011 at 16:03 | comment | added | Ben Sprott | Hey, sorry. I was really hasty here. "It's functor" is bad language and I will quickly explain. I am trying to think of functors as mapping diagrams to diagrams. This is true in normal presentations of diagrams where dots are objects and arrows are morphisms. Any diagram in $A$, under $F:A \rightarrow B$ will map to a similarly shaped diagram in $B$. So, one can think very locally about functors in terms of just what a diagram shape gets mapped to. I was thinking this would all be true when working with string diagram presentations of "diagrams" eek. | |
| Jul 26, 2011 at 3:14 | comment | added | Qiaochu Yuan | Don't you need the functor to preserve the appropriate structure? (Monoidal, etc.) | |
| Jul 26, 2011 at 3:14 | comment | added | Todd Trimble | I'm not following. If string diagrams are thought of as representing morphisms in a freely generated monoidal category or 2-category of some sort (generated by a tensor scheme, or by a computad), call it $Free(S)$, then a monoidal functor $F: Free(S) \to M$ into another monoidal category will interpret the string diagram as a morphism in $M$. But I don't know what "its functor" is. Are you considering the case where $M = Free(S)$, so that the values of $F$ are string diagrams again? | |
| Jul 26, 2011 at 3:04 | history | asked | Ben Sprott | CC BY-SA 3.0 |