Hi,
If I have a string diagram, can I take its functor easily by drawing a new string diagram and just say the wires go to wires and boxes go to boxes?
$\begingroup$
$\endgroup$
7
-
5$\begingroup$ 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? $\endgroup$– Todd TrimbleCommented Jul 26, 2011 at 3:14
-
$\begingroup$ Don't you need the functor to preserve the appropriate structure? (Monoidal, etc.) $\endgroup$– Qiaochu YuanCommented Jul 26, 2011 at 3:14
-
$\begingroup$ 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. $\endgroup$– Ben SprottCommented Jul 26, 2011 at 16:03
-
$\begingroup$ 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. $\endgroup$– Ben SprottCommented Jul 26, 2011 at 16:12
-
2$\begingroup$ pps.univ-paris-diderot.fr/~mellies/papers/functorial-boxes.pdf is this what you're looking for ? $\endgroup$– Gerrit BegherCommented Dec 18, 2015 at 0:02
|
Show 2 more comments
1 Answer
$\begingroup$
$\endgroup$
Yes, string diagram indeed has some functorial properties, see for example my draft paper: Combinatorics and algebra of tensor calculus, https://arxiv.org/abs/1501.01790
