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Article on application of category theory to biology

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  • 1.
    edited January 2014

    That's very helpful. David Spivak's Category theory for scientists, is one of the tomes I've been working through in Azimuth time. I keep thinking that ologs have something to contribute to the formalisation of topic maps/rdf epistemology which nad has just brought up on some other thread. I'm hoping this paper will give me a bit better traction on various kinds of Petri net categorical specifications. (Then we can discuss some reconciliation of interactive javascript options for the new year :)) Cheers

    Comment Source:That's very helpful. David Spivak's [Category theory for scientists](http://math.mit.edu/~dspivak/CT4S.pdf), is one of the tomes I've been working through in Azimuth time. I keep thinking that [ologs](http://en.wikipedia.org/wiki/Olog) have something to contribute to the formalisation of topic maps/rdf epistemology which [[nad]] has just brought up on some other thread. I'm hoping this paper will give me a bit better traction on various kinds of Petri net categorical specifications. (Then we can discuss some reconciliation of interactive javascript options for the new year :)) Cheers
  • 2.

    Jim, great, I'm glad that it's helpful. Unfortunately I don't have time to dig into this right now. I'd be interested to hear any review comments on this article.

    Here's my underlying question: what are the plausible, potential applications of research into the category theory of networks. I'm considering putting a small blurb about this into my qualitative introduction to Azimuth. But I should have a somewhat clearer picture of the possibilities before I try to explain them to someone else!

    Here are some initial thoughts, which are based around observations that John has made:

    • There are a multitude of disparate formalisms for networks

    • But there appears to be substantial common structure

    • Goal: find appropriate definitions for general categories of networks

    • Goal: use these to reformulate the existing ad hoc network formalisms

    • Functors may express equivalences between different ad hoc network types

    • Overall, the hope is to create a general theory of networks

    • Theorems of the general theory will then take on specific forms in each of the ad hoc categories

    • Simulation and reasoning tools map be developed using the general network language

    Are there other things that one might hope for?

    My goal is not to make promises to the reader about the outcomes, but to convey some motivation for the research quest.

    Comment Source:Jim, great, I'm glad that it's helpful. Unfortunately I don't have time to dig into this right now. I'd be interested to hear any review comments on this article. Here's my underlying question: what are the plausible, potential applications of research into the category theory of networks. I'm considering putting a small blurb about this into my qualitative introduction to Azimuth. But I should have a somewhat clearer picture of the possibilities before I try to explain them to someone else! Here are some initial thoughts, which are based around observations that John has made: * There are a multitude of disparate formalisms for networks * But there appears to be substantial common structure * Goal: find appropriate definitions for general categories of networks * Goal: use these to reformulate the existing _ad hoc_ network formalisms * Functors may express equivalences between different _ad hoc_ network types * Overall, the hope is to create a _general_ theory of networks * Theorems of the general theory will then take on specific forms in each of the _ad hoc_ categories * Simulation and reasoning tools map be developed using the general network language Are there other things that one might hope for? My goal is not to make promises to the reader about the outcomes, but to convey some motivation for the research quest.
  • 3.

    David wrote:

    Here’s my underlying question: what are the plausible, potential applications of research into the category theory of networks.

    I plan to give 4 talks on network theory at the computer science department in Oxford this spring. The first will be an overview, so I've been thinking about how to address your underlying question. People are bound to ask "what can you do with this stuff?"

    It's a bit tricky, since I tend to approach research by trying to understand what's going on, rather than trying to accomplish a specific task. Generally, I've found that understanding what's going on helps people accomplish useful tasks. But this particular chunk of research is still in its early stages. So, the specific applications aren't lined up yet, and I don't even want to worry too much about finding them — although, in fact, I'm always worrying about just that.

    Right now I'm very excited because I'm starting to see how

    • electrical circuit theory
    • control theory
    • chemical reaction network theory
    • evolutionary game theory
    • Bayesian network (= "belief network") theory

    all fit together in a single picture. I'm writing papers with 6 people on different aspects of this stuff, and I hope to finish the current batch of papers during my sabbatical. I'm way behind on explaining this stuff in the network theory series! I would like to catch up on network theory blog posts during my sabbatical, too.

    However, it'll probably take me another year to fully flesh out the connections here, in the form of theorems. I feel at that point it'll be easier to say what we might do with a general theory of networks. Of course I hope we'll be able to better understand biological and ecological systems as networks through which information and other things flow. But then the question is: how will that help?

    I have some ideas, of course, but I would love to hear suggestions and/or questions.

    By the way, after my introductory talk, I plan to talk about

    • electrical circuits and control theory
    • information and entropy, and
    • chemical reactions

    all using category theory.

    Comment Source:David wrote: > Here’s my underlying question: what are the plausible, potential applications of research into the category theory of networks. I plan to give 4 talks on network theory at the computer science department in Oxford this spring. The first will be an overview, so I've been thinking about how to address your underlying question. People are bound to ask "what can you _do_ with this stuff?" It's a bit tricky, since I tend to approach research by trying to _understand what's going on_, rather than trying to _accomplish a specific task_. Generally, I've found that understanding what's going on helps people accomplish useful tasks. But this particular chunk of research is still in its early stages. So, the specific applications aren't lined up yet, and I don't even want to worry too much about finding them — although, in fact, I'm always worrying about just that. Right now I'm very excited because I'm starting to see how * electrical circuit theory * control theory * chemical reaction network theory * evolutionary game theory * Bayesian network (= "belief network") theory all fit together in a single picture. I'm writing papers with 6 people on different aspects of this stuff, and I hope to finish the current batch of papers during my sabbatical. I'm way behind on explaining this stuff in the network theory series! I would like to catch up on network theory blog posts during my sabbatical, too. However, it'll probably take me another year to fully flesh out the connections here, in the form of theorems. I feel at that point it'll be easier to say what we might _do_ with a general theory of networks. Of course I hope we'll be able to better understand biological and ecological systems as networks through which information and other things flow. But then the question is: _how will that help?_ I have some ideas, of course, but I would love to hear suggestions and/or questions. By the way, after my introductory talk, I plan to talk about * electrical circuits and control theory * information and entropy, and * chemical reactions all using category theory.
  • 4.
    edited January 2014

    David wrote:

    • Goal: find appropriate definitions for general categories of networks

    • Functors may express equivalences between different ad hoc network types

    It's worth noting a subtlety here. A lot of mathematicians would be tempted to treat a network of some type as an object in some category.

    However, the really nice approach (in my opinion) is to think of a network not as an object, but as a morphism. That is, it's a way to go from some bunch of things ("inputs") to some other bunch of things ("outputs"). Whenever we draw diagrams like electric circuits we are implicitly making use of standard mathematical technology for drawing morphisms.

    You can see a partially worked-out example of this philosophy here:

    This is a kind of proposal for the Ph.D. thesis project he's doing at Oxford. He's doing his thesis with me, but being formally supervised by Jamie Vicary and Bob Coecke. The introduction is a nice overview of what he's trying to accomplish! I think it's quite readable, a lot more than most of what I've been writing so far!

    (Here's something less readable: if you think networks are objects in a category, and I think they're morphisms, how can we both be right? The answer is to use a 2-category, where we allow 2-morphisms between morphisms. In a structure with more layers, like this, one man's morphism can be another man's object.)

    Comment Source:David wrote: > * Goal: find appropriate definitions for general categories of networks > * Functors may express equivalences between different ad hoc network types It's worth noting a subtlety here. A lot of mathematicians would be tempted to treat a network of some type as an object in some category. However, the really nice approach (in my opinion) is to think of a network not as an object, but as a _morphism_. That is, it's a way to go from some bunch of things ("inputs") to some other bunch of things ("outputs"). Whenever we draw diagrams like electric circuits we are implicitly making use of standard mathematical technology for drawing morphisms. You can see a partially worked-out example of this philosophy here: * Brendan Fong, [A compositional approach to control theory](http://math.ucr.edu/home/baez/Brendan_Fong_Transfer_Report.pdf). This is a kind of proposal for the Ph.D. thesis project he's doing at Oxford. He's doing his thesis with me, but being formally supervised by Jamie Vicary and Bob Coecke. The introduction is a nice overview of what he's trying to accomplish! I think it's quite readable, a lot more than most of what I've been writing so far! (Here's something less readable: if you think networks are _objects_ in a category, and I think they're _morphisms_, how can we both be right? The answer is to use a 2-category, where we allow 2-morphisms between morphisms. In a structure with more layers, like this, one man's morphism can be another man's object.)
  • 5.

    By the way, David Tanzer: here's one way you might write a blog article: interview me, or conduct a dialogue of some sort! You can make me explain big-picture stuff in simple broad strokes, and ask more questions when I'm not making sense.

    Comment Source:By the way, David Tanzer: here's one way you might write a blog article: interview me, or conduct a dialogue of some sort! You can make me explain big-picture stuff in simple broad strokes, and ask more questions when I'm not making sense.
  • 6.

    That's a good idea, thanks for the suggestion. I'll mediate on to go about this. (I've never conducted an interview before.)

    Let me practice here, by asking you and inteview-ish question:

    In the meantime, let me get started with this interview question:

    Can you tell us about your experiences conducting interviews. How do you approach them? How much do you prepare beforehand, and how much do you prep the interviewee in advance. What recording and communication media have you found most conducive to a good interview?

    Comment Source:That's a good idea, thanks for the suggestion. I'll mediate on to go about this. (I've never conducted an interview before.) Let me practice here, by asking you and inteview-ish question: In the meantime, let me get started with this interview question: Can you tell us about your experiences conducting interviews. How do you approach them? How much do you prepare beforehand, and how much do you prep the interviewee in advance. What recording and communication media have you found most conducive to a good interview?
  • 7.
    edited January 2014

    Hi Dave, I'm sure John will give a more detailed and up-to-date reply, but he wrote a little bit about interviewing in this thread from a few years ago.

    Comment Source:Hi Dave, I'm sure John will give a more detailed and up-to-date reply, but he wrote a little bit about interviewing [in this thread from a few years ago](http://forum.azimuthproject.org/discussion/490/the-berkeley-earth-group-allies/).
  • 8.
    edited January 2014

    David wrote:

    Can you tell us about your experiences conducting interviews. How do you approach them? How much do you prepare beforehand, and how much do you prep the interviewee in advance. What recording and communication media have you found most conducive to a good interview?

    I've interviewed a bunch of people for This Week's Finds, and I do it by email. That

    1) avoids the problem of transcribing spoken interviews, which are always full of annoying hmm's and stupid-looking sentence fragments,

    2) lets the interviewer and interviewee think as long as they want, whenever they want (though if someone waits too long, a nudge is good),

    3) easily lets the interviewee go back and edit previous answers, within reason.

    We could do it either by email or on the Azimuth Wiki. The former method makes it really easy to alert the other guy when you've said something — but you need to reply using "Forward" instead of "Reply" to avoid a huge pile of nested quote bars. The latter method would let people here watch. I'm not sure they're so bored that this would be entertaining for them, but it could allow some interesting (or confusing) multi-person interactions.

    I don't prepare a whole lot before the interview, except that I know what themes I want to talk about. I work hard to make sure the conversation stays on track, covering one or more topics in discrete "blocks" rather than roaming wildly all over the place.

    When I interview people for This Week's Finds I try to avoid putting forth my own views: I see my role as eliciting theirs. However, there's another kind of thing we could do, which is more of a "dialogue".

    It's up to you, but it's probably good to be clear ahead of time what's going on!

    Comment Source:David wrote: > Can you tell us about your experiences conducting interviews. How do you approach them? How much do you prepare beforehand, and how much do you prep the interviewee in advance. What recording and communication media have you found most conducive to a good interview? I've interviewed a bunch of people for _This Week's Finds_, and I do it _**by email**_. That 1) avoids the problem of transcribing spoken interviews, which are always full of annoying hmm's and stupid-looking sentence fragments, 2) lets the interviewer and interviewee think as long as they want, whenever they want (though if someone waits too long, a nudge is good), 3) easily lets the interviewee go back and edit previous answers, within reason. We could do it either by email or on the Azimuth Wiki. The former method makes it really easy to alert the other guy when you've said something — but you need to reply using "Forward" instead of "Reply" to avoid a huge pile of nested quote bars. The latter method would let people here watch. I'm not sure they're so bored that this would be entertaining for them, but it could allow some interesting (or confusing) multi-person interactions. I don't prepare a whole lot before the interview, except that I know what themes I want to talk about. I work hard to make sure the conversation stays on track, covering one or more topics in discrete "blocks" rather than roaming wildly all over the place. When I interview people for _This Week's Finds_ I try to avoid putting forth my own views: I see my role as eliciting theirs. However, there's another kind of thing we could do, which is more of a "dialogue". It's up to you, but it's probably good to be clear ahead of time what's going on!
  • 9.
    edited January 2014

    David wrote:

    Here’s my underlying question: what are the plausible, potential applications of research into the category theory of networks?

    My previous reply was not very encouraging; I am dissatisfied with how poorly I've come to working out the potential applications, but it looks stupid to say I don't know what they are. Let me list three of my main answers:

    • Improving the "smart grid", where flows of electric power, water, natural gas and other commodities are tied together with flows of information that can be used to help regulate the other flows and "optimize" them.

    • Improving models of food webs, the carbon/nitrogen/oxygen/phosphorus/etc. cycles, and other "ecological" networks, so we can better understand how they'll respond to changes we impose on them.

    • Inventing "ecotechnology", which acts more like biological systems do than current technology.

    Very broadly, I want us to understand Nature and stop fighting it.

    Comment Source:David wrote: > Here’s my underlying question: what are the plausible, potential applications of research into the category theory of networks? My previous reply was not very encouraging; I am dissatisfied with how poorly I've come to working out the potential applications, but it looks stupid to say I don't know what they are. Let me list three of my main answers: * Improving the "smart grid", where flows of electric power, water, natural gas and other commodities are tied together with flows of information that can be used to help regulate the other flows and "optimize" them. * Improving models of food webs, the carbon/nitrogen/oxygen/phosphorus/etc. cycles, and other "ecological" networks, so we can better understand how they'll respond to changes we impose on them. * Inventing "ecotechnology", which acts more like biological systems do than current technology. Very broadly, I want us to understand Nature and stop fighting it.
  • 10.
    edited January 2014

    I don’t prepare a whole lot before the interview, except that I know what themes I want to talk about.

    By the way, it sucks when interviewers have a list of questions already prepared and keep marching through that list instead of responding like a human being to the last thing the interviewee said. It's stiff and dull. I hate it when some very interesting person is being interviewed by someone who misses all the chances to let that person be interesting.

    It's better to react to the situation at hand: know where you're heading and keep on track, but still taking some opportunities to probe more deeply into interesting things the interviewee said!

    Comment Source:> I don’t prepare a whole lot before the interview, except that I know what themes I want to talk about. By the way, it sucks when interviewers have a list of questions already prepared and keep marching through that list instead of responding like a human being to the last thing the interviewee said. It's stiff and dull. I hate it when some very interesting person is being interviewed by someone who misses all the chances to let that person be interesting. It's better to react to the situation at hand: know where you're heading and keep on track, but still taking some opportunities to probe more deeply into interesting things the interviewee said!
  • 11.
    edited January 2014

    John wrote:

    It's worth noting a subtlety here. A lot of mathematicians would be tempted to treat a network of some type as an object in some category.

    However, the really nice approach (in my opinion) is to think of a network not as an object, but as a morphism. That is, it's a way to go from some bunch of things ("inputs") to some other bunch of things ("outputs"). Whenever we draw diagrams like electric circuits we are implicitly making use of standard mathematical technology for drawing morphisms.

    Great, that makes a lot of sense. I have assimilated it into my thinking.

    Here is the first portion of a popular-style paragraph that introduces the category theory of networks:

    Besides applications such as the Leaf, there is ongoing research into the mathematics of networks. This includes the study of categories of networks. Here, a category, in the technical sense, means a collection of entities of the same type, along with a system of relationships between them. This subject, which is known as category theory, is fundamentally about how relationships are composed together. Now, networks can be concatenated to form composites, and hence each network is a kind of "composable relationship." Category theory is therefore a natural setting in which to look for the general logic of network composition.

    Comment Source:John wrote: > It's worth noting a subtlety here. A lot of mathematicians would be tempted to treat a network of some type as an object in some category. > > However, the really nice approach (in my opinion) is to think of a network not as an object, but as a _morphism_. That is, it's a way to go from some bunch of things ("inputs") to some other bunch of things ("outputs"). Whenever we draw diagrams like electric circuits we are implicitly making use of standard mathematical technology for drawing morphisms. Great, that makes a lot of sense. I have assimilated it into my thinking. Here is the first portion of a popular-style paragraph that introduces the category theory of networks: > Besides applications such as the Leaf, there is ongoing research into the mathematics of networks. This includes the study of <i>categories</i> of networks. Here, a category, in the technical sense, means a collection of entities of the same type, along with a system of relationships between them. This subject, which is known as category theory, is fundamentally about how relationships are composed together. Now, networks can be concatenated to form composites, and hence each network is a kind of "composable relationship." Category theory is therefore a natural setting in which to look for the general logic of network composition.
  • 12.
    edited January 2014

    This text shows some techniques that I find useful for making difficult subjects sound less scary.

    First, try to build up the reader's confidence before plunging into the deeper points that you want to make. Here, I am building on the fact that everyone knows what a category is, in the informal sense. The first usage of the word "categories" is an unannounced segue into the technical definition. In the next sentence, when I begin defining a category, I expect the reader to be thinking, I know what a category is, this is a piece of cake. This confidence should be reinforced by the next clause, which is very evident -- that a category consists of things -- but then the kicker comes in the last clause, which says that this kind of category is not just a sack of objects, but it includes the relationships as well. It's a ramp leading to this last point.

    Some further principles:

    • When introducing a term, search for the synonym which will best resonate with the reader, while still being accurate. Here I figured that a morphism, or an arrow, or a way to go between things, is such a generic idea, that it might as well be called a relationship -- and this term is already implanted in the reader's lexicon.

    • Don't go deeper than is appropriate for the given dialog with the reader. I could have gone on to talk about domain and codomain, etc., but that would lose many many readers, without any substantial gain.

    • Consider the rhythm of the text. There are strong beats and weak beats. There are positions within the skeleton of the sentences that are emphasized, and positions which are de-emphasized. For example, the last words of the sentences are in an emphatic position, and text in a subordinate clause is de-emphasized.

    • Put the scary words on the off-beats, so that they can be flowed over without the reader having a chance to freak out. That is why I put the phrase "which is known as category theory" in subordinate clause.

    • Pad the scary words with some easy to digest phrases. Here I added the buffer words "which is known as" before the scary phrase "category theory." Like wrapping a pill in some bread.

    • When introducing a potentially scary topic, try to use a non-chalant tone, so they won't be tipped off that something big is coming up. I aimed for a mellow tone in the phrase "This subject, which is known as."

    By way of contrast, consider the sentence: Category theory is the study of objects and the morphisms that connect them. Fine for a different audience, but not as good for the lay-person.

    I once read, somewhere, some principles of writing, which I will elaborate upon here. Always maintain a picture of your imagined reader's state of mind, and even emotional state, at each point in the flow of their reading. Of course every real reader is different, but picture one that you are talking to, as you are writing. At each point, there will be some thought, or question, which is most prominent in the reader's mind. Your next sentence should be on the path to addressing this question, at the same time that you are weaving in new material in the exposition. To maintain the flow of the reader's attention, make the first words of the next sentence resonate with that most prominent point that is in their mind.

    Comment Source:This text shows some techniques that I find useful for making difficult subjects sound less scary. First, try to build up the reader's confidence before plunging into the deeper points that you want to make. Here, I am building on the fact that everyone _knows_ what a category is, in the informal sense. The first usage of the word "categories" is an unannounced segue into the technical definition. In the next sentence, when I begin defining a category, I expect the reader to be thinking, _I_ know what a category is, this is a piece of cake. This confidence should be reinforced by the next clause, which is very evident -- that a category consists of things -- but then the kicker comes in the last clause, which says that this kind of category is not just a sack of objects, but it includes the relationships as well. It's a ramp leading to this last point. Some further principles: * When introducing a term, search for the synonym which will best resonate with the reader, while still being accurate. Here I figured that a morphism, or an arrow, or a way to go between things, is such a generic idea, that it might as well be called a relationship -- and this term is already implanted in the reader's lexicon. * Don't go deeper than is appropriate for the given dialog with the reader. I could have gone on to talk about domain and codomain, etc., but that would lose many many readers, without any substantial gain. * Consider the rhythm of the text. There are strong beats and weak beats. There are positions within the skeleton of the sentences that are emphasized, and positions which are de-emphasized. For example, the last words of the sentences are in an emphatic position, and text in a subordinate clause is de-emphasized. * Put the scary words on the off-beats, so that they can be flowed over without the reader having a chance to freak out. That is why I put the phrase "which is known as category theory" in subordinate clause. * Pad the scary words with some easy to digest phrases. Here I added the buffer words "which is known as" before the scary phrase "category theory." Like wrapping a pill in some bread. * When introducing a potentially scary topic, try to use a non-chalant tone, so they won't be tipped off that something big is coming up. I aimed for a mellow tone in the phrase "This subject, which is known as." By way of contrast, consider the sentence: _Category theory_ is the study of _objects_ and the _morphisms_ that connect them. Fine for a different audience, but not as good for the lay-person. I once read, somewhere, some principles of writing, which I will elaborate upon here. Always maintain a picture of your imagined reader's state of mind, and even emotional state, at each point in the flow of their reading. Of course every real reader is different, but picture one that you are talking to, as you are writing. At each point, there will be some thought, or question, which is most prominent in the reader's mind. Your next sentence should be on the path to addressing this question, at the same time that you are weaving in new material in the exposition. To maintain the flow of the reader's attention, make the first words of the next sentence resonate with that most prominent point that is in their mind.
  • 13.
    edited January 2014

    John wrote:

    However, the really nice approach (in my opinion) is to think of a network not as an object, but as a morphism. That is, it’s a way to go from some bunch of things (“inputs”) to some other bunch of things (“outputs”).

    I see that Brendan Fong, in the proposal you cited above, talks about the limits of the input-output perspective for networks. Since the morphism perspective introduces a bias towards and input-output view -- or more generally towards a binary relationship between the domain side and the co-domain side -- how would this bias most naturally be overcome in the category theory of networks? Do n-categories help out here?

    Comment Source:John wrote: > However, the really nice approach (in my opinion) is to think of a network not as an object, but as a morphism. That is, it’s a way to go from some bunch of things (“inputs”) to some other bunch of things (“outputs”). I see that Brendan Fong, in the proposal you cited above, talks about the limits of the input-output perspective for networks. Since the morphism perspective introduces a bias towards and input-output view -- or more generally towards a binary relationship between the domain side and the co-domain side -- how would this bias most naturally be overcome in the category theory of networks? Do n-categories help out here?
  • 14.

    I have a lot of excitement and interest in the applications of category theory to complex systems and networks. I think it's the way to go.

    Comment Source:I have a lot of excitement and interest in the applications of category theory to complex systems and networks. I think it's the way to go.
  • 15.
    edited January 2014

    On another thread, I asked about to what extent the various climate models can be considered as network models. Thanks for the replies there.

    On another level, every system can be viewed as a network of relationships among the entities in the system.

    Even though this is a very general idea, that doesn't necessarily mean that it is an empty abstraction, like a philosopher who says that everything exists, and he has a theory of existence, and hence everything is an application of his theory. That would be vacuous, of course.

    But there really is a specific theory, math, and applications -- including a software industry of relational databases --that have to do with relations. I've wondered to what extent category theory really has things to say about the world of relations.

    There are intimations that it does have something to say, besides things that we knew already beforehand. For instance, consider a network of binary relations between some sets. This is, of course, very naturally represented as a diagram in the category of binary relations under ordinary composition. So in this view, networks are diagrams. Can this be unified with the view of networks as morphisms?

    More interestingly, the limit of this diagram gives the natural join of the relations. This is the composite relation, the composite network, the conjunction. What I find noteworthy here is that a fundamental construct of category theory is producing a fundamental operation on the network.

    Comment Source:On another thread, I asked about to what extent the various climate models can be considered as network models. Thanks for the replies there. On another level, every system can be viewed as a network of relationships among the entities in the system. Even though this is a very general idea, that doesn't necessarily mean that it is an empty abstraction, like a philosopher who says that everything exists, and he has a theory of existence, and hence everything is an application of his theory. _That_ would be vacuous, of course. But there really is a specific theory, math, and applications -- including a software industry of relational databases --that have to do with relations. I've wondered to what extent category theory really has things to _say_ about the world of relations. There are intimations that it does have something to say, besides things that we knew already beforehand. For instance, consider a network of binary relations between some sets. This is, of course, very naturally represented as a diagram in the category of binary relations under ordinary composition. So in this view, networks are diagrams. Can this be unified with the view of networks as morphisms? More interestingly, the limit of this diagram gives the natural join of the relations. This is the composite relation, the composite network, the conjunction. What I find noteworthy here is that a fundamental construct of category theory is producing a fundamental operation on the network.
  • 16.
    edited January 2014

    I never crossed the bridge into topos theory, but I gather that by taking other categories besides Set as the "ground category," we could cross into "non-Cartesian" realms where we would have, for example, theories of relations that were not built upon the construction of relations as sets of tuples. And, since processes are spatial-temporal relations, there would be non-Cartesian concepts of processes. Category theory could then have something to say about networks of generalized processes.

    Comment Source:I never crossed the bridge into topos theory, but I gather that by taking other categories besides Set as the "ground category," we could cross into "non-Cartesian" realms where we would have, for example, theories of relations that were not built upon the construction of relations as sets of tuples. And, since processes are spatial-temporal relations, there would be non-Cartesian concepts of processes. Category theory could then have something to say about networks of generalized processes.
  • 17.

    David wrote:

    I see that Brendan Fong, in the proposal you cited above, talks about the limits of the input-output perspective for networks. Since the morphism perspective introduces a bias towards and input-output view – or more generally towards a binary relationship between the domain side and the co-domain side – how would this bias most naturally be overcome in the category theory of networks?

    In a dagger-category every morphism $f : x \to y$ has a "dagger" $f^\dagger : y \to x$, which you think of as $f$ with the roles of input and output reversed. So, dagger-categories allow us to use categories without getting hung up on what the input and what's the output. (Think of an electrical circuit with two wires coming out: we can choose to treat either one as the input.)

    In a compact category we can have morphisms with many inputs and many outputs, e.g.

    $$ f: A_1 \otimes A_2 \to B_1 \otimes B_2 \otimes B_3 $$ and then "flip" some inputs to being outputs, or vice versa, e.g. $f$ above gives

    $$ \tilde{f} : A_1 \otimes A_2 \otimes B_3^* \to B_1 \otimes B_2 $$ (Think of a box with a bunch of wires coming in at top and a bunch coming out at bottom, where we can grab a wire at bottom and bend it around to the top, or vice versa.)

    All this is best explained with lots of pictures, as in my Rosetta Stone paper with Mike Stay. Most of the categories whose morphisms are "networks" are compact dagger-categories. Certainly this is true of the categories Brendan Fong and I are studying.

    All this stuff is considerably less stressful than $n$-category theory, since we've still just got objects and morphisms: things and processes. With $n$-categories we go on to 2-morphisms, 3-morphisms, etc. and the pictures become higher-dimensional. These could be used to formalize some things about how computers simulate a 3-dimensional atmosphere, for example. (I'm not claiming this can be helpful yet... but someday it may be.)

    Comment Source:David wrote: > I see that Brendan Fong, in the proposal you cited above, talks about the limits of the input-output perspective for networks. Since the morphism perspective introduces a bias towards and input-output view – or more generally towards a binary relationship between the domain side and the co-domain side – how would this bias most naturally be overcome in the category theory of networks? In a [dagger-category](http://en.wikipedia.org/wiki/Dagger_category) every morphism $f : x \to y$ has a "dagger" $f^\dagger : y \to x$, which you think of as $f$ with the roles of input and output reversed. So, dagger-categories allow us to use categories without getting hung up on what the input and what's the output. (Think of an electrical circuit with two wires coming out: we can choose to treat either one as the input.) In a [compact category](http://en.wikipedia.org/wiki/Compact_closed_category) we can have morphisms with many inputs and many outputs, e.g. $$ f: A_1 \otimes A_2 \to B_1 \otimes B_2 \otimes B_3 $$ and then "flip" some inputs to being outputs, or vice versa, e.g. $f$ above gives $$ \tilde{f} : A_1 \otimes A_2 \otimes B_3^* \to B_1 \otimes B_2 $$ (Think of a box with a bunch of wires coming in at top and a bunch coming out at bottom, where we can grab a wire at bottom and bend it around to the top, or vice versa.) All this is best explained with lots of pictures, as in my [Rosetta Stone](http://math.ucr.edu/home/baez/rosetta.pdf) paper with Mike Stay. **Most of the categories whose morphisms are "networks" are compact dagger-categories**. Certainly this is true of the categories Brendan Fong and I are studying. All this stuff is considerably less stressful than $n$-category theory, since we've still just got objects and morphisms: things and processes. With $n$-categories we go on to 2-morphisms, 3-morphisms, etc. and the pictures become higher-dimensional. These could be used to formalize some things about how computers simulate a 3-dimensional atmosphere, for example. (I'm not claiming this can be helpful yet... but someday it may be.)
  • 18.

    I like your points regarding how to write, David! Especially this one:

    Consider the rhythm of the text. There are strong beats and weak beats. There are positions within the skeleton of the sentences that are emphasized, and positions which are de-emphasized. For example, the last words of the sentences are in an emphatic position, and text in a subordinate clause is de-emphasized.

    A lot of scientific prose is painful to read, because the authors don't think about the "music" of sentences: the way they sound and flow. People understand the meaning of a sentence better when it has the right music.

    Comment Source:I like your points regarding how to write, David! Especially this one: > Consider the rhythm of the text. There are strong beats and weak beats. There are positions within the skeleton of the sentences that are emphasized, and positions which are de-emphasized. For example, the last words of the sentences are in an emphatic position, and text in a subordinate clause is de-emphasized. A lot of scientific prose is painful to read, because the authors don't think about the "music" of sentences: the way they sound and flow. People understand the meaning of a sentence better when it has the right music.
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