animation by Marius BuligaI’m helping organize ACT 2019, an applied category theory conference and school at Oxford, July 15-26, 2019.
If you’re a grad student or postdoc interested in this subject, you should apply for the ‘school’ before January 30th. Later people can submit papers for the conference:
Dear all,
As part of a new growing community in Applied Category Theory, now with a dedicated journal Compositionality, a traveling workshop series SYCO, a forthcoming Cambridge U. Press book series Reasoning with Categories, and several one-off events including at NIST, we launch an annual conference+school series named Applied Category Theory, the coming one being at Oxford, July 15-19 for the conference, and July 22-26 for the school. The dates are chosen such that CT 2019 (Edinburgh) and the ACT 2019 conference (Oxford) will be back-to-back, for those wishing to participate in both.
There already was a successful invitation-only pilot, ACT 2018, last year at the Lorentz Centre in Leiden, also in the format of school+workshop.
For the conference, for those who are familiar with the successful QPL conference series, we will follow a very similar format for the ACT conference. This means that we will accept both new papers which then will be published in a proceedings volume (most likely a Compositionality special Proceedings issue), as well as shorter abstracts of papers published elsewhere. There will be a thorough selection process, as typical in computer science conferences. The idea is that all the best work in applied category theory will be presented at the conference, and that acceptance is something that means something, just like in CS conferences. This is particularly important for young people as it will help them with their careers.
Expect a call for submissions soon, and start preparing your papers now!
The school in ACT 2018 was unique in that small groups of students worked closely with an experienced researcher (these were John Baez, Aleks Kissinger, Martha Lewis and Pawel Sobociński), and each group ended up producing a paper. We will continue with this format or a closely related one, with Jules Hedges and Daniel Cicala as organisers this year. As there were 80 applications last year for 16 slots, we may want to try to find a way to involve more students.
We are fortunate to have a number of private sector companies closely associated in some way or another, who will also participate, with Cambridge Quantum Computing Inc. and StateBox having already made major financial/logistic contributions.
On behalf of the ACT Steering Committee,
John Baez, Bob Coecke, David Spivak, Christina Vasilakopoulou
Posted by John Baez 
and 
and get a morphism 
In a monoidal category you can also tensor morphisms 
and 
and get a morphism 
This of course relies on your ability to tensor objects. In a symmetric monoidal category you also have 
And of course, there is more to it than this. But this is enough to get started.
to denote the tensor product in our symmetric monoidal category, as usual in chemistry.
so their associator:
and a category of ‘commutative monoidal categories’, which we call 
is a Petri net: that is, a set 
of places, a set 
of transitions, and source and target functions![s, t \colon T \to \mathbb{N}[S]](https://s0.wp.com/latex.php?latex=s%2C+t+%5Ccolon+T+%5Cto+%5Cmathbb%7BN%7D%5BS%5D+&bg=ffffff&fg=333333&s=0 "s, t \colon T \to \mathbb{N}[S]")
![\mathbb{N}[S]](https://s0.wp.com/latex.php?latex=%5Cmathbb%7BN%7D%5BS%5D&bg=ffffff&fg=333333&s=0 "\mathbb{N}[S]")
is the underlying set of the free commutative monoid on 
![\mathbb{N}[f] \colon \mathbb{N}[S] \to \mathbb{N}[S']](https://s0.wp.com/latex.php?latex=%5Cmathbb%7BN%7D%5Bf%5D+%5Ccolon+%5Cmathbb%7BN%7D%5BS%5D+%5Cto+%5Cmathbb%7BN%7D%5BS%27%5D&bg=ffffff&fg=333333&s=0 "\mathbb{N}[f] \colon \mathbb{N}[S] \to \mathbb{N}[S']")
![s', t' \colon T' \to \mathbb{N}[S']](https://s0.wp.com/latex.php?latex=s%27%2C+t%27+%5Ccolon+T%27+%5Cto+%5Cmathbb%7BN%7D%5BS%27%5D+&bg=ffffff&fg=333333&s=0 "s', t' \colon T' \to \mathbb{N}[S']")
Explicitly, it’s a 
such that for all objects 
and 
we have
and 
be the category whose objects are commutative monoidal categories and whose morphisms are strict monoidal functors.
to its underlying Petri net. This Petri net has the set of objects 
as its set of places and the set of morphisms 
as its set of transitions, and![s, t \colon \mathrm{Mor}(C) \to \mathrm{Ob}(C) \hookrightarrow \mathbb{N}[\mathrm{Ob}(C)]](https://s0.wp.com/latex.php?latex=s%2C+t+%5Ccolon+%5Cmathrm%7BMor%7D%28C%29+%5Cto+%5Cmathrm%7BOb%7D%28C%29+%5Chookrightarrow+%5Cmathbb%7BN%7D%5B%5Cmathrm%7BOb%7D%28C%29%5D+&bg=ffffff&fg=333333&s=0 "s, t \colon \mathrm{Mor}(C) \to \mathrm{Ob}(C) \hookrightarrow \mathbb{N}[\mathrm{Ob}(C)]")
is a finite multisubset of 
We can think of this as a way of placing finitely ‘tokens’—little black dots—on the elements of 
![\mathbb{N}[X]](https://s0.wp.com/latex.php?latex=%5Cmathbb%7BN%7D%5BX%5D&bg=ffffff&fg=333333&s=0 "\mathbb{N}[X]")
denote the set of markings of the set 
there is a reachability relation![\blacksquare P \subseteq \mathbb{N}[X] \times \mathbb{N}[Y]](https://s0.wp.com/latex.php?latex=%5Cblacksquare+P+%5Csubseteq+%5Cmathbb%7BN%7D%5BX%5D+%5Ctimes+%5Cmathbb%7BN%7D%5BY%5D+&bg=ffffff&fg=333333&s=0 "\blacksquare P \subseteq \mathbb{N}[X] \times \mathbb{N}[Y]")
feed those tokens into the Petri net 
move them around using transitions in 
leaving no tokens behind.
and move them around using transitions in 
that is ‘reachable’ from the original marking of 
extends to a ‘lax double functor’
is a double category having open Petri nets as horizontal 1-cells and 
is a double category having relations as horizontal 1-cells.
as objects,
as vertical 1-morphisms,
This double category has
as horizontal 1-cells from 
![\mathbb{N}(f) \colon \mathbb{N}[X] \to \mathbb{N}[Y]](https://s0.wp.com/latex.php?latex=%5Cmathbb%7BN%7D%28f%29+%5Ccolon+%5Cmathbb%7BN%7D%5BX%5D+%5Cto+%5Cmathbb%7BN%7D%5BY%5D&bg=ffffff&fg=333333&s=0 "\mathbb{N}(f) \colon \mathbb{N}[X] \to \mathbb{N}[Y]")
is a really a functor.)![\blacksquare P \colon \mathbb{N}[X] \to \mathbb{N}[Y]](https://s0.wp.com/latex.php?latex=%5Cblacksquare+P+%5Ccolon+%5Cmathbb%7BN%7D%5BX%5D+%5Cto+%5Cmathbb%7BN%7D%5BY%5D&bg=ffffff&fg=333333&s=0 "\blacksquare P \colon \mathbb{N}[X] \to \mathbb{N}[Y]")
is a lax double functor. This means that it does not send a composite open Petri net 
to the composite of the reachability relations for 
So, we do not have
to be this one:
is this:
is a proper subset of 
In 
a token can move all the way from point 1 to point 5. But it does not do so by first moving through 
It has to take a more complicated zig-zag path where it first leaves 
then comes back into 
Various choices of semantics for open Petri nets can be described using symmetric monoidal double functors out of 
for example this:
is to put the pictures together:
we have a new Petri net whose set of places is the disjoint union of those for 
which looks like this:
or 
This is why to get a category, with composition strictly associative, we need to use isomorphism classes of open Petri nets as morphisms. But there are advantages to avoiding this and working with open Petri nets themselves. Basically, it’s better to work with things than mere isomorphism classes of things! If we do this, we obtain not a category but a 
and 
‘objects’, 
and 
‘horizontal 1-cells’, and 
a ‘2-morphism’. We can compose vertical 1-morphisms to get new vertical 1-morphisms and compose horizontal 1-cells to get new horizontal 1-cells. We can compose the 2-morphisms in two ways: horizontally and vertically. (This is just a quick sketch of the ideas, not the full definition.)
with:
