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Hi... sorry, I seem to be having severe writer's block against cranking out new lectures. It's not that I don't love you folks anymore - I do, and I'm having some interesting conversations here. It's just something about figuring out what to do lecture about next, and how to do it. Writing is always a mind game. I'm pretty good at writing a lot, but sometimes I really have to fight myself.

I *almost* overcame my resistance today, but I had to write a blog article about the work I'm doing with Metron on the Complex Adaptive System Composition and Design Environment project, to help convince the people at DARPA that we're doing good work, and that seems to have sucked out today's supply of energy.

I will try again tomorrow morning.

Anyway, here is that blog article:

It's about some new work John Foley and are doing... a continuation of the earlier work described here:

Part 1. CASCADE: the Complex Adaptive System Composition and Design Environment.

Part 2. Metron’s software for system design.

Part 3. Operads: the basic idea.

Part 4. Network operads: an easy example.

Part 5. Algebras of network operads: some easy examples.

Part 6. Network models.

Part 7. Step-by-step compositional design and tasking using commitment networks.

There, that should give you something to read about applied category theory until tomorrow!

## Comments

Great, I've been really enjoying this series of articles. Will you be publishing more CASCADE papers? I remember you saying that the Network Models paper was just setting up the basic machinery.

`Great, I've been really enjoying this series of articles. Will you be publishing more CASCADE papers? I remember you saying that the Network Models paper was just setting up the basic machinery.`

Yes, we'll be putting out more papers. After the Network models paper came this one, which I haven't blogged about yet:

But we're finally getting ready to do some serious applications, and this has forced new thoughts about the basic machinery! So, everything is going slower than expected, but in the end I want some papers that that lead up to actual software.

`Yes, we'll be putting out more papers. After the [Network models](https://arxiv.org/abs/1711.00037) paper came this one, which I haven't blogged about yet: * Joe Moeller, [Noncommutative network models](https://arxiv.org/abs/1804.07402). But we're finally getting ready to do some serious applications, and this has forced new thoughts about the basic machinery! So, everything is going slower than expected, but in the end I want some papers that that lead up to actual software.`

Just thinking out loud, this seems like it would have some pretty interesting applications in mathematical/computational evolution. I've been thinking a little bit about this topic recently, and hope to find some time to work more on it.

I had a question, is an operad a sort of generalization of the idea of a simplicial complex? I picked up the book

Elementary Applied Topologyby Robert Ghrist, and there is a chapter on simplicial complexes as a way of building up structures. The final chapter is on 'Categorization', so I wondered how/if these might be related.`Just thinking out loud, this seems like it would have some pretty interesting applications in mathematical/computational evolution. I've been thinking a little bit about this topic recently, and hope to find some time to work more on it. I had a question, is an operad a sort of generalization of the idea of a simplicial complex? I picked up the book *Elementary Applied Topology* by Robert Ghrist, and there is a chapter on simplicial complexes as a way of building up structures. The final chapter is on 'Categorization', so I wondered how/if these might be related.`

Grant -

Categorization and categorification are completely different things, at least if people are using words correctly. People get confused sometimes. But it's quite possible Ghrist was using "categorization" to mean what it really means: the job of taking things and classifying them into different kinds, e.g. writing a program that can tell the difference between pictures of cats and pictures of dogs. Applied topology is sometimes used for things like this.

An operad is not like a simplicial complex. A simplicial complex is a space made out of simplices:

An operad is a gadget with a set of "n-ary operations" \(O_n\) for each \(n = 0,1,2,\dots\). For example, if you take \(O_n\) to have \(n!\) elements, it can describe all the ways you can take \(n\) elements in a monoid or ring and multiply them (in all possible orders).

I explained the more general 'typed operads' here:

and you get back to the ordinary operads if you take the set of types to have 1 element. We use typed operads in our CASCADE project to describe ways of assembling networks of various kinds of agents. If you have two kinds of agents - say, cars and trucks - then your set of types has two elements.

`Grant - [Categorization](https://en.wikipedia.org/wiki/Categorization) and [categorification](https://en.wikipedia.org/wiki/Categorification) are completely different things, at least if people are using words correctly. People get confused sometimes. But it's quite possible Ghrist was using "categorization" to mean what it really means: the job of taking things and classifying them into different kinds, e.g. writing a program that can tell the difference between pictures of cats and pictures of dogs. Applied topology is sometimes used for things like this. An operad is not like a simplicial complex. A simplicial complex is a space made out of simplices: <center><img src = "https://upload.wikimedia.org/wikipedia/commons/thumb/5/50/Simplicial_complex_example.svg/400px-Simplicial_complex_example.svg.png"></center> An operad is a gadget with a set of "n-ary operations" \\(O_n\\) for each \\(n = 0,1,2,\dots\\). For example, if you take \\(O_n\\) to have \\(n!\\) elements, it can describe all the ways you can take \\(n\\) elements in a monoid or ring and multiply them (in all possible orders). I explained the more general 'typed operads' here: * [Complex adaptive system design (part 3)](https://johncarlosbaez.wordpress.com/2017/08/17/complex-adaptive-system-design-part-3/) and you get back to the ordinary operads if you take the set of types to have 1 element. We use typed operads in our CASCADE project to describe ways of assembling networks of various kinds of agents. If you have two kinds of agents - say, cars and trucks - then your set of types has two elements.`

I see, I had wondered if there were general structures for the idea of 'complexification'...or building up the whole from parts. Ghrist actually used 'categorification', that was my error. I didn't realize there was a sharp distinction in usage, but that makes perfect sense. This helps me understand more clearly. Thanks

`I see, I had wondered if there were general structures for the idea of 'complexification'...or building up the whole from parts. Ghrist actually used 'categorification', that was my error. I didn't realize there was a sharp distinction in usage, but that makes perfect sense. This helps me understand more clearly. Thanks`

Thinking out loud also.. it seems that this issue https://en.wikipedia.org/wiki/Hydrograph#Unit_hydrograph to study rain/stream on a drainnage basis perhaps could be also approached as a very special case (simple) on the same frame of the PERT..perhaps..

`Thinking out loud also.. it seems that this issue https://en.wikipedia.org/wiki/Hydrograph#Unit_hydrograph to study rain/stream on a drainnage basis perhaps could be also approached as a very special case (simple) on the same frame of the PERT..perhaps..`

While I am not an expert, I have worked with some people in hydrology. The traditional way to study hydrographs is to model responses as a linear time invariant system. Essentially, output signals like stream flow are related to input signals like precipitation. When precipitation increases, stream flow also increases, but there's a time delay. You can estimate this time delay deconvolving the signals. Essentially, we imagine precipitation as a signal \(x : \mathbb{R} \to \mathbb{R}\) and stream flow as a signal \(y : \mathbb{R} \to \mathbb{R}\) where stream flow is the convolution of \(x\) and another function (called the impulse response, I think):

$$y(t) = \int_{-\infty}^t f(\tau)x(t-\tau) d\tau $$ Essentially, we suppose \(y\) equals a linear combination of all past values of \(x\), if you estimate \(f\) you learn how the system responds to any input signal and you can estimate time delays and so on.

I think John was working on math for diagrams of linear time invariant systems somewhere, but I can't remember if it was in CASCADE or not.

`While I am not an expert, I have worked with some people in hydrology. The traditional way to study hydrographs is to model responses as a linear time invariant system. Essentially, output signals like stream flow are related to input signals like precipitation. When precipitation increases, stream flow also increases, but there's a time delay. You can estimate this time delay deconvolving the signals. Essentially, we imagine precipitation as a signal \\(x : \mathbb{R} \to \mathbb{R}\\) and stream flow as a signal \\(y : \mathbb{R} \to \mathbb{R}\\) where stream flow is the convolution of \\(x\\) and another function (called the impulse response, I think): \[y(t) = \int_{-\infty}^t f(\tau)x(t-\tau) d\tau \] Essentially, we suppose \\(y\\) equals a linear combination of all past values of \\(x\\), if you estimate \\(f\\) you learn how the system responds to any input signal and you can estimate time delays and so on. I think John was working on math for diagrams of linear time invariant systems somewhere, but I can't remember if it was in CASCADE or not.`

@Scott thanks..i think.linear time invariant system links to passive linear circuits modelling that ,just like PERT ,is approached on the big frame of monoidal categories... perhaps the initial idea holds....and its possible related to the topology of the drainage network also..

`@Scott thanks..i think.linear time invariant system links to passive linear circuits modelling that ,just like PERT ,is approached on the big frame of monoidal categories... perhaps the initial idea holds....and its possible related to the topology of the drainage network also..`

@John, concerning frameworks for complex decision making, do you recommend some paper based on applied category ideally ( but not mandatory) specifically related to bioprospection like this "traditional" one below? Im contributing on a project on this context right now ( if perhaps you get interested on this project please let me know it ..and i would send this msg to the research coordinator) ..thanks in advance, best regards https://onlinelibrary.wiley.com/doi/abs/10.1002/9781118567166.ch4

`@John, concerning frameworks for complex decision making, do you recommend some paper based on applied category ideally ( but not mandatory) specifically related to bioprospection like this "traditional" one below? Im contributing on a project on this context right now ( if perhaps you get interested on this project please let me know it ..and i would send this msg to the research coordinator) ..thanks in advance, best regards https://onlinelibrary.wiley.com/doi/abs/10.1002/9781118567166.ch4`