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November 15, 2007

Category Theory and Biology

Posted by David Corfield

Some of us at the Centre for Reasoning here in Kent are thinking about joining forces with a bioinformatics group. Over the years I’ve caught glimpses of people trying out category theoretic ideas in biology, so naturally I’ve wanted to take a closer look. An initial foray has revealed some intriguing work: André Ehresmann and Jean-Paul Vanbremeersch on Memory Evolutive Systems and Gerhard Mack (somewhere near Urs in Hamburg) on Universal Dynamics, a Unified Theory of Complex Systems: Emergence, Life and Death. Climbing the n-category ladder, Nils Baas who has ideas on abstract matter, has worked with Ehresmann and Vanbremeersch on ‘Hyperstructures and memory evolutive systems’, and with Torbjorn Helvik on higher-order cellular automata.

The forefather of biological category theory is Robert Rosen. I haven’t had a chance to look at his work yet, but for an easy (for Café regulars) introduction to some of his ideas, try Juan-Carlos Letelier et al. Organizational invariance and metabolic closure. In particular, it discusses the representation of enzyme metabolism by arrows in categories, taking into account the thesis:

Organisms are closed to efficient causes.

In this case, the thesis translates to the idea that there must be an internal process to repair or replace the enzymes which are metabolising the input molecules. So not only do we have processes mapping inputs to outputs. There must also be processes mapping outputs to the original process effectors. The paper explains conditions on evaluation maps.

I would imagine that a higher proportion of biologists are bewildered by this kind of work than their physicist colleagues by similar work in their field.

Posted at November 15, 2007 12:11 PM UTC

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Re: Category Theory and Biology

Nils Baas has been talking to me about the idea of “hyperstructures” a lot lately – and I have tried to think hard about it.

To me it seems like the main basic idea is this:

We might want to have something like an nn-graph and equip it with a notion of composition (“fusion”) which does not distinguish between source and targets.

Tom Leinster once told me that this is pretty close to saying “cyclic operad”, as far as I rememeber. But it seems to me that there might still be a good point in looking for more:

whatever the right notion of “nn-graphs with fusion” is, the “morphisms” between them should not simply be morphisms, but should be “bonds” (nn-graph elements), too.

With Konrad Waldorf I was talking about this a bit. We came up with the following idea which might be a good guiding example:

Let CC be any category with all pullback. Then we know that spans in CC form a bicategory.

But now, what if I considered multispans in CC?

Here a multispan is, clearly, an object on CC equipped with an arbitrary number of morphisms out of it.

Given two multispans, I can check if they have any “leg” in common, pull them back along this common leg and obtain a new multispan.

Clearly, the structure of multispans together with this fusion operation does not form a category – unless one artificially introduces labels that mark certain legs as incoming and other legs as outgoing.

Moreover, it is pretty clear that we can consider multispans of multispans in the obvious way, ad infionitum.

So I am guessing that multispans in a category CC might be a good guiding example for a definition of hyperstructure.

Even more so, since the other main motivating example that Nils Baas is emphasizing is hyperstructures of cobordisms with corners and singularities. Given that a cobordism can be regarded as a cospan, and naturally actually as a multi-cospan if we stop distinguishing between in- and out-parts of its boundary, this might actually be a special case of the more general multi(co)-span situation I just described.

Does anyone have any ideas on this??

I remeber John Baez pointing out how mutlispans of groupoids naturally appear as higher-rank tensors in the groupoidification program. Possibly there is some room for convergence here.

Where do multispans live?

Posted by: Urs Schreiber on November 15, 2007 1:09 PM | Permalink | Reply to this

Re: Category Theory and Biology

Where do multispans live?

You’d kind of want them in multicategories with morphisms represented as lines connecting labelled points on spheres.

Is the periodic table here pointing in that direction?

Posted by: David Corfield on November 15, 2007 5:48 PM | Permalink | Reply to this

Re: Category Theory and Biology

Where do multispans live?

Wouldn’t they form a planar algebra, or maybe a categorified version thereof?

Posted by: John Armstrong on November 16, 2007 12:32 AM | Permalink | Reply to this

Re: Category Theory and Biology

The best general theory about multispans I know is developed in Mark Weber’s paper “Yoneda Structures from 2-toposes” (Appl. Cat. Str. v.15, n.3, 2007). He has a definition of an elementary 2-topos (definition 4.10) and one of the
attribute of a 2-topos is what he calls “a classifying discrete opfibration” :
which plays the role of a subobject classifier in a topos.

Some paricular examples of 2-topoi (example 4.7) are the 2-category of categories ( CAT , with Set as \Omega ) and the 2-categories of CAT-presheafs on a small category C ( CAT(\hat(C) ). There is a 2-functor
E: CAT(\hat(C)) —> CAT
where E is given by some sort of Grotheindieck construction. This functor has a left 2-adjoint Sp_C.

What can be called an “object of multispans” is Sp_C(\Omega) which itself is \Omega for CAT(\hat(C)).

In a particular case of C being freely generated by a directed graph 1 ==> 2 (sorry, I do not know how to write it nicely without Tex, but I think everybody undertand that this is a site for the category of directed graphs) Sp_C is just
the category of usual spans. In the case G= Gl (site for globular sets) Sp_C is my category of higher spans. If you take C the site for cubical categories you will produce Marco Grandis’s cubical spans.

Mark proves that \Omega has a lot of nice properties. It is always a cartesian pseudomonoid and with some minor assumption it is even a cartesian closed object of your 2-topos. A particular nice situation is when there is a 2-monad T acting on your 2-topos which preserves discrete opfibrations with small fibers. In this case Mark proves that \Omega is a pseudomonoid in the category of psedoalgebras of T. This is the case of all examples I mentioned before. It gives a structure of composition of usual spans and a structure of augmented monoidal globular category of higher spans and cubical categories of Grandis’s spans together with coherence result for such structures. In particular this alows to develop a theory of operads inside \Omega, which is the basis of my definition of weak \omega-categories.

I hope this helps.


Posted by: michael on November 16, 2007 7:28 AM | Permalink | Reply to this

Re: Category Theory and Biology

Quite a while ago Michael Batanin gave me crucial information on multispans in the above entry. It took me until today to come back to this.

I started reading Yoneda structure for 2-toposes today but have to interrupt now.

I got the point about multispans on pages pages 37-38 and would like to better understand now how the composition of (multi)spans is induced from a pseudomonoid structure.

I am wondering if/how this recovers the notion that for two multispans there are in general many different ways to compose them, both because at height nn it may have more (or less) than two (n1)(n-1)-legs, but also because one can compose at different heights.

I’ll keep looking through the literature on this. I am just mentioning this in case anyone wants to join my little quest here (as advisor or as comrade-in-arms).

I am thinking that a good notion of multispans will be crucial for higher groupoidification.

Posted by: Urs Schreiber on January 21, 2009 10:44 PM | Permalink | Reply to this

Re: Category Theory and Biology

What’s higher groupoidification? When you turn a 2-linear map into a span of 2-groupoids?

Posted by: David Corfield on January 22, 2009 4:45 PM | Permalink | Reply to this

Re: Category Theory and Biology

What’s higher groupoidification? When you turn a 2-linear map into a span of 2-groupoids?

I am just thinking of multispans. Spans of spans, etc. Essentially arbitrary small diagrams in a category with pullbacks, really. At every point in the diagram which looks like aFca \leftarrow F \rightarrow c we can regard aa and bb as representing spaces of sections given by collections of spans from some fixed object into aa and into bb, respectively.

If this sounds too cheap to be of any use, let me describe the following example, which is one reason why I am considering this. It will start with a sentence or two on QFT, for motivational purposes, but the example itself at the end is pure combinatorics / finite group theory.

So, the idea is that QFT in full abstract generality is a continuous functor from multi-spans in some category S opS^{op} with finite limits to multi-spans in a category VV with finite limits, where

QFT:MultSpan(S op)MultSpan(V), QFT : MultSpan(S^{op}) \to MultSpan(V) \,,

where a multi-span in S opS^{op}, being a multi-co-span in SS, is interpreted as an extended cobordism, along the lines described here, whereas a multispan in VV is regarded as a “geometric”/”groupoidified” linear map.

So let me present an example which shows that we want to pull-push non only along simple spans, but also along spans-of spans.

Before starting, one more idea from QFT:

suppose S opS^{op} is VV-enriched so that [S op,V][S^op,V] is VV-enriched. Then every P[S op,V]P \in [S^op, V] induces a functor

QFT P:MultSpan(S op)MultSpan(V) QFT_P : MultSpan(S^{op}) \to MultSpan(V)

given by

QFT P:=[,P]:S opV QFT_P := [-, P] : S^op \to V

regarded as a map from MultSpan(S^{op}) to MultSpan(V) by applying levelwise. Notice that QFT PQFT_P is indeed a continuous functor in that it sends small limits in S opS^{op} (pushout composition of multi-co-spans in SS) to small limits in VV (pullback composition of multi-spans in VV) and hence respects the “extended” composition law in our bare-bone “extended QFT”.

Now, it turns out that not only the “parameter space” of cobordisms is naturally described in terms of spans, but also “target space” PP in general is: a span P XP QP YP_X \leftarrow P_Q \to P_Y of “target spaces” (in [S op,V][S^op,V], but never mind) is a correspondence space or “bi-brane” which can for instance encode the transition from a space P XP_X to its T-dual space P YP_Y (namely in the case that PP itself is a Poincaré line bundle etc. pp. – but never mind here, I just mention it to indicate the context of how to think of a target space span).

So a general σ\sigma-model QFT pairs parameter space cospans with target space spans

[ Σ in out Σ in Σ out, P Q P X P Y]=[Σ in,P X] [Σ in,P Q] [Σ in,P Y] [Σ,P X] [Σ,P Q] [Σ,P Y] [Σ out,P X] [Σ out,P Q] [Σ out,P Y] \left[ \array{ && \Sigma \\ & {}^{in}\nearrow && \nwarrow^{out} \\ \Sigma_{in} &&&& \Sigma_{out} }, \array{ && P_Q \\ & \swarrow && \searrow \\ P_X &&&& P_Y } \right] = \array{ [\Sigma_{in}, P_X] &\leftarrow & [\Sigma_{in}, P_Q] &\rightarrow & [\Sigma_{in}, P_Y] \\ \uparrow && \uparrow && \uparrow \\ [\Sigma, P_X] &\leftarrow & [\Sigma, P_Q] &\rightarrow & [\Sigma, P_Y] \\ \downarrow && \downarrow && \downarrow \\ [\Sigma_{out}, P_X] &\leftarrow & [\Sigma_{out}, P_Q] &\rightarrow & [\Sigma_{out}, P_Y] }

to produce multi-spans in VV which we are to br read groupoifiedly as generalized linear maps.

To amplify, let’s tell the story encoded in the diagram in the right. It says:

suppose you have a field configuration on Σ in\Sigma_{in} for a σ\sigma-model QFT with target space P XP_X. Then you can, in general,

- either propagate the field along Σ\Sigma withing the QFT coming from P XP_X to a field configuration on Σ out\Sigma_{out}, and then apply a duality transformation to end up with a field configuration of Σ out\Sigma_{out} with respect to a QFT coming from target space P YP_Y

- or you can first dualize the field configuration from one wrt P XP_X to one wrt P YP_Y and then start propagating the field configuration from P XP_X to P YP_Y in that theory.

- Or, in fact, something weird in between, where we pull back to Σ\Sigma and dualitze there and only then “path integrate” down to Σ out\Sigma_{out}.

These three operations need not coincide! The first one corresponds to having a defect line at Σ X\Sigma_X, the second having one at Σ Y\Sigma_Y. If both operations coincide we’ll presumeably say that we have a “topological defect” which may be moved around on the worldvolume at will.

Okay, now finally the concrete example:

Consider bi-branes in Dijjkgraaf-Witten theory. Meaning, consider S=TopS = Top, VV = something like nCatn Cat with its Crans-Gray closed monoidal structure and let target space be

P X=X=BG, P_X = X = \mathbf{B}G \,,

the total space of the trivial nn-bundle over the groupoid BG\mathbf{B}G, which is the one-object groupoid coming from some finite group GG.

Let I={ab}VI = \{a \to b\} \in V be the standard interval and consider the bi-brane

[I,BG] d 0 d 1 BG BG. \array{ && [I, \mathbf{B}G] \\ & {}^{d_0}\swarrow && \searrow^{d_1} \\ \mathbf{B}G &&&& \mathbf{B}G } \,.

We could regard this as the “time evolution bi-brane”

=[ I in out pt pt,BG] \cdots = \left[ \array{ && I \\ & {}^{in}\nearrow && \nwarrow^{out} \\ pt &&&& pt } \,, \mathbf{B}G \right]

obtained by mapping the interval regarded as an extended cobordism into target space. From this perspective we would regard this span as usual in groupoidification as a linear map itself, which acts on “states over the point”, given by spans

Ψ pt [pt,BG]BG \array{ && \Psi \\ & \swarrow && \searrow \\ pt &&&& [pt, \mathbf{B}G] \simeq \mathbf{B}G }

by compposition of spans. This indeed describes then time evolution in the DW-model (albeit along the interval it is a bit boring).

But, and now I finally come to the point, instead of regarding [I,BG][I, \mathbf{B}G] here as a groupoidified linear map, we can regard itself as a placeholder for generalized vector spaces and consider spans of that.

These very naturally arise here: using the natural co-category structure on the interval II, of course [I,BG][I, \mathbf{B}G] naturally carries the structure of a category internal to VV. The same is still true if we close the interval at its ends, so that it becomes B\mathbf{B}\mathbb{Z}.

That means there is a composition operation which is a morphism from this span [B,BG]× BG[B,BG] [B,BG] [B,BG] BG BG BG \array{ &&&& [\mathbf{B}\mathbb{Z}, \mathbf{B}G] \times_{\mathbf{B}G} [\mathbf{B}\mathbb{Z}, \mathbf{B}G] \\ &&& \swarrow && \searrow \\ && [\mathbf{B}\mathbb{Z}, \mathbf{B}G] &&&& [\mathbf{B}\mathbb{Z}, \mathbf{B}G] \\ & \swarrow && \searrow && \swarrow && \searrow \\ \mathbf{B}G &&&& \mathbf{B}G &&&& \mathbf{B}G }

back to [B,BG][\mathbf{B}\mathbb{Z}, \mathbf{B}G]. I won’t try to draw this here in its 2-dimensionality, but you see that this gives a span of spans which on the top hierarchical level is

[B,BG]× BG[B,BG] [B,BG]×[B,BG] [B,BG]. \array{ && [\mathbf{B}\mathbb{Z}, \mathbf{B}G] \times_{\mathbf{B}G} [\mathbf{B}\mathbb{Z}, \mathbf{B}G] \\ & \swarrow && \searrow \\ [\mathbf{B}\mathbb{Z}, \mathbf{B}G] \times [\mathbf{B}\mathbb{Z}, \mathbf{B}G] &&&& [\mathbf{B}\mathbb{Z}, \mathbf{B}G] } \,.

So we can pull-push through this guy any pair of “2nd order sections” given by two spans

Ψ 2 pt [B,BG]. \array{ && \Psi_2 \\ & \swarrow && \searrow \\ pt &&&& [\mathbf{B}\mathbb{Z}, \mathbf{B}G] } \,.

If you look at this, you’ll see that this pull-push operation indeed describes a crucial operation on DW bi-branes, namely the fusion product of bi-branes.

In the simple setup I described where we have an entirely trivial bundle P X=X=BGP_X = X = \mathbf{B}G whose fibers are points such a bibrane is just a categorified class function on GG. In proper Dijkgraaf-Witten theory we have a 3-vector bundle

P X=(EBG) P_X = (E \to \mathbf{B}G)

coming from a group 3-cocycle represented by a weak 3-functor

BGBBimod. \mathbf{B}G \to \mathbf{B} Bimod \,.

If you then go through this gymnastics you’ll see that the sections

Ψ 2 Ω ptBBimod=Bimod [B,P X] \array{ && \Psi_2 \\ & \swarrow && \searrow \\ \Omega_{pt} \mathbf{B} Bimod = Bimod &&&& [\mathbf{B}\mathbb{Z}, P_X] }

that we are talking about come from representations of [B,BG][\mathbf{B}\mathbb{Z}, \mathbf{B}G] on vector spaces – which is representations of the (untwisted) Drinfeld double, and the multi-span pull-push operation yields the fusion product on this representation category.

Sorry again for the long-winded reply. I had thought of sprinkling in nnLab links for further details, but then I thought I’ll maybe instead collect those here at the end, in case anyone is actually interested.

Maybe for toy examples-purposes it may be useful to think of the interpretation of plain catgory algebras as pull-psuh of bibranes as described at the end of nnLab: category algebra. There at the end it just says “push-forward natural transformations”. It is clear in these simple examples what this push-forward operation is. The above is one way to formalize it, I think.

Then, a little bit of info on the bi-brane imagery is at nnLab: bi-brane with further links.

Posted by: Urs Schreiber on January 22, 2009 6:27 PM | Permalink | Reply to this

Re: Category Theory and Biology

The fact that a “category” of multispans would need to have “legs” labelled as incoming or outgoing shows up already in the (bi)category of spans, which are also symmetric. In that setting, once you introduce a convention distinguishing between “source” legs and “target” legs of a span, it lets you see the same span in two ways, so that Span(C)Span(C) has duals for morphisms.

Presumably something similar will happen with multispans, except somewhat more complicated. As with multicategories generally, you might hope that they reduce to monoidal categories, so a morphism with nn inputs and mm outputs can be seen as a morphism from a monoidal product of nn things to one of mm things. In this case, I would imagine you’re just looking at a monoidal category with duals, or some such thing. If the multicategory doesn’t break down to something of that form, the kind of “dual” structure you have is just a little more complicated, since you can dualize any collection of inputs/outputs - duals of various valences.

Is there a name for these?

Posted by: Jeffrey Morton on November 16, 2007 7:24 PM | Permalink | Reply to this

Re: Category Theory and Biology

I asked

Where do multispans live?

and was fortunate enough to receive a couple of interesting replies.

In fact, the replies were so interesting that it turns out I’d need to sit down and study them further to really digest them.

Right now I don’t have the time to do that. Hopefully this will be an excuse to instead come back with another question/remark to the experts:

One important reason why categories are interesting as opposed to dull is because the morphisms spaces between two categories are themselves categories. It’s that recursive self-referential behaviour which makes categories powerful.

So when I am asking “where do multispans live”, I am looking for a similarly interesting answer.

What I don’t want is a category of multispans! If they are supposed to get at all close to Nils Baas’s idea of “hyperstructures”, then we don’t want morphisms from one multispan to another.

Instead, we’d want bonds of multispans: something like multispans of multispans, which connect a bunch of multispans in some way, without sayin gwhich one is incoming, which one is outgoing.

Right now I cannot tell if such a property of the “home” of multispans is already implcit if we’d follow John Armstrong’s proposal to think of them in terms of planar algebras. I simply don’t know enough about planar algebras. But maybe you can tell, John?

Similarly, I cannot tell at all at the moment if my above desideratum would be met by the formalism that Michael mentions above.

Is it clear what i am looking for? If so, does anyone have an idea how to find it?

Posted by: Urs Schreiber on November 21, 2007 3:05 PM | Permalink | Reply to this

Re: Category Theory and Biology

I simply don’t know enough about planar algebras. But maybe you can tell, John?

I must admit I don’t know enough about planar algebras either. There’s Vaughan Jones’ paper introducing them, and Scott Morrison over at the Secret Blogging Seminar has talked a bit about canopoleis (while anglicizing the plural to “canopolises”), which are supposed to be the next dimension up.

Basically here’s what I know: We think of a monoidal category with duals by putting morphisms in boxes with the inputs running in the top and out the bottom. Duals let us pull an input to an output or vice versa. Then we can compose morphisms by stacking boxes top-to-bottom or tensor them by stacking boxes side to side.

Planar algebras say that all that fiddling with the boxes just gets in the way. Instead, put your gadgets into circles, and connect them with “spaghetti and meatballs diagrams”. Such a diagram consists of an outer circle with a finite collection of marked points (the plate), a bunch of inner circles with their own collection of marked points (the meatballs), and a bunch of nonintersecting arcs connecting up all the marked points, possibly along with nonintersecting loops (the spaghetti).

Now there’s some collection of gadgets you can put in for any meatball. Here’s where our multispans would fit. Then any diagram with its meatballs filled with gadgets is itself a gadget, and can be used to fill meatballs in other diagrams.

The upshot is that there’s no preference for “in” and “out”. All the boundary points are equivalent – as they should be in multispans – and we can have as many of them as we want.

Posted by: John Armstrong on November 21, 2007 5:28 PM | Permalink | Reply to this

Re: Category Theory and Biology

Okay, thanks. That makes sense.

Now: can we form a planar algebra of planar algebras?

Posted by: Urs Schreiber on November 21, 2007 6:16 PM | Permalink | Reply to this

Re: Category Theory and Biology

Hi, Urs.

I am not 100% sure I understand what you want from multispans. But I believe Weber’s spans have some aspects of your desideratum. For example, the right structure on them is not a category structure but an internal cartesian object in a 2-topos + pseudo T-algebra structure. In a particular case of my higher spans this means an augmented monoidal globular category but the word category is the least important up to some extent. The most important think is that a n-span should be interpreted as a span between (n-1)-dimensional spans. Is this is not what you want from multispans?

Yes, I understand that this is not precisely what you want because we have prefered direction of higher spans (source and target). But we also have a natural reversing operation in higher span, and we can compose in any direction. So, the existence of prefered direction is not a problem.

More serious problem is that there are only two legs in an n-span (or 2^n in Grandis spans). We would like to have an arbitrary legged spans and spans of such spans and so on.

This can be overcome, I suspect, by using multicategory and opetopes. This also can be included in Mark’s theory. As a result we will have an n-span being diagram of sets of the shape of a barycentric subdivision of an n-opetope. It already looks like your multispan (or multispan of multispan) and the structure is pseodomulticategory (in generalised sense) structure. But in this case we do have a prefered direction. One of the legs should be a target, other are sources of your multispan.

By the way, the planar algebras, at least
how I understood from John’s explanation also have some prefered directions. First, we will have a circular order on the boundary of the outer circles and second, this outer circle is distinguished. This situation reminds me our description of n-opetopes with Kock, Joyal and Mascari.

We also have circles and a tree , which seats inside this nesting family of circles. A nesting family of circles is another name for a rooted tree. Yet, I believe, we can make the same construction by considering nonrooted trees (no distinguished vertex ) and circles not on a plane but on a sphere, so we will not have an outer circle either. Then we will have freedom to substitute at any circle. I do not know what sort of structure we produce in this way but I suspect that this is something like iterated cyclic multicategory. Then we turn on Mark’s mashine and produce the corresponding multispans. Of course, we should check some number of axioms to do it.


Posted by: Michael on November 22, 2007 5:25 AM | Permalink | Reply to this

Re: Category Theory and Biology

not on a plane but on a sphere

As I mentioned, to get the freedom to move the legs of multispans wherever you like, the constructions suggested by Noah Snyder seem promising. After spaghetti and meatballs comes lasagna and meatballs!

Posted by: David Corfield on November 22, 2007 9:49 AM | Permalink | Reply to this

Re: Category Theory and Biology

Closed circles won’t occur if one restricts to the annular tangles of Conne’s category. (Any planar algebra is naturally a cyclic module in the sense of Connes). See the last paragraph on page 2 of this paper by Vaughan Jones on general planar algebras which I referred to earlier in a different thread.

So do general planar algebras have any sense of preferred direction? Thanks.

Posted by: Charlie Stromeyer Jr on November 22, 2007 2:06 PM | Permalink | Reply to this

Re: Category Theory and Biology

I am not 100% sure I understand what you want from multispans.

I am not entirely sure, either. It was an attempt to obtain an instructive example for what Nils Baas thinks of as a “hyperstructure”.

To my mind the point seems to be:

while it is true that we can emulate most everything with interchangeable multi in- and out-puts using categories and constructions with them, like multicategories, operads, etc., this will typically involve first having inputs distinguished from outputs and then somehow forgetting that information again.

Hence one might suspect: maybe there is a more direct and fundamental description of higher structures of this sort.

Here is another attempt, which might indicate that there is room for some simplification:

one nice thing about not distinguishing between in- and outputs is that we can build everything from two rather fundamental operations: disjoint union and “fusion of legs”.

Let me try to formalize it:

an h-structure is

- an \mathbb{N}-graded set deg:B \mathrm{deg} : B \to \mathbb{N} i.e. B=B 0B 1B 2 B = B_0 \cup B_1 \cup B_2 \cdots (with B nB_n the set of nn-“bonds”)

- a degree -1 endomorphism :BB\partial : B \to B (the boundary map, sending each nn-bond to the (n1)(n-1)-bond it cobounds)

- an operation :B× BB \cup : B \times_\mathbb{N} B \to B (which takes two nn-bonds and regards them as a single nn-bond)

respecting the boundary map in that B× B B × B× B B \array{ B \times_\mathbb{N} B &\stackrel{\cup}{\to}& B \\ \;\;\; \downarrow^{\partial \times_\mathbb{N} \partial} && \downarrow^\partial \\ B \times_\mathbb{N} B &\stackrel{\cup}{\to}& B }

So far this would be an hh-structure without a notion of composition. My impression was that for applications in biology, a structure like this is assumed to be potentially useful.

Then on top of that we may want a “fusion” operation, i.e. an undirected notion of composition.

Whenever an nn-bond has two coinciding boundary parts, these may be fused. The space of nn-bonds with coinciding boundary parts is the pullback B FB_F in

B F B B× B IdxΔ B× B× B (Id×) B \array{ B_F &\to& \to &\to& B \\ \downarrow &&&& \downarrow^\partial \\ B \times_\mathbb{N} B &\stackrel{\mathrm{Id}x \Delta}{\to}& B \times_\mathbb{N} B \times_\mathbb{N} B &\stackrel{ \cup \circ(\mathrm{Id}\times \cup)}{\to}& B }

Fusion is a map

F:B FB F : B_F \to B

such that

B F F B B× B p 1 B \array{ B_F &\stackrel{F}{\to}& B \\ \downarrow && \downarrow^\partial \\ B \times_\mathbb{N} B &\stackrel{p_1}{\to}& B }

which says that after fusion the two (n1)(n-1)-bonds along which we have fused are no longer part of the boundary.

There would be some further consistency conditions on this.

And then I would have to see if the structure of all such h-structures is itself an h-structure.

But not right now…

Posted by: Urs Schreiber on November 22, 2007 7:59 PM | Permalink | Reply to this

Re: Category Theory and Biology


Could you give “us” some advice pertaining to where undergraduates would start if they were interested in someday working in this area of mathematics (namely Lie algebra and categorical theory). Courses available to the uninformed include linear algebra, algebraic structures, field theory… etc. What books would one read as a starting point?

Are there resources available online? I’ve never heard these topics mentioned in class and wonder when I would come across them. When do people who pursue such topics in graduate school learn about these topics?

Posted by: Curious Undergrad on November 15, 2007 3:50 PM | Permalink | Reply to this

Re: Category Theory and Biology

Dear Curious,

One of the wonderful things about category theory is that it connects to so many different parts of mathematics – and physics, and computer science, and even (as David points out) biology. So you can come at it from lots of different angles. In particular, you don’t need to learn about Lie algebras, or indeed any kind of algebra, in order to learn category theory.

Historically, category theory grew up in the context of algebraic topology. Partly for this reason, books on category theory tend to draw most of their examples from algebra and topology. Mac Lane’s classic book Categories for the Working Mathematician is in this tradition. Excellent though it is, I wouldn’t particularly recommend it for someone at your level.

An introduction to category theory that doesn’t demand knowledge of any advanced mathematics is Lawvere and Schanuel’s book Conceptual Mathematics. When you first flick through it you might think it looks trivial, but it’s definitely not!

If you’re feeling brave and want to plunge straight into some lecture notes – and be warned, these go fast – you might try those of Cheng or Cáccamo, Hyland and Winskel, or some old ones of mine.

Good luck!

Posted by: Tom Leinster on November 15, 2007 5:05 PM | Permalink | Reply to this

Re: Category Theory and Biology

C. U. wrote:

Could you give “us” some advice pertaining to where undergraduates would start if they were interested in someday working in this area of mathematics (namely Lie algebra and categorical theory). Courses available to the uninformed include linear algebra, algebraic structures, field theory… etc. What books would one read as a starting point?

Check out the list of my favorite books on math and physics. I include a lot on Lie groups and Lie algebras. A good university library should have most of these.

To get started on category theory, try Lawvere and Schanuel’s Conceptual Mathematics. Then try Goldblatt’s Topoi: a Categorial Analysis of Logic, which is available for free online, and darn cheap from Dover. It’s more elementary than the title makes it sound. Adamek, Herrlich and Strecker’s Abstract and Concrete Categories: the Joy of Cats is also elementary and free online.

When I was a student, I practically lived in the university library. There are dozens of books on even the most abstruse subjects, and you just have to keep looking around before you find the ones that are ideal for you at a given moment. (As you keep developing, what counts as ideal keeps changing.)

Posted by: John Baez on November 16, 2007 2:00 AM | Permalink | Reply to this

Re: Category Theory and Biology

Grant Malcolm and the late Ray Paton at the University of Liverpool explored the application of category theory to biology.

See, for example:

Michael J. Fisher, Grant Malcolm and Ray C. Paton: Spatio-logical processes in intracellular signalling. Biosystems, 55(1-3): 93–105, 2000.

available from:

Posted by: Peter on November 15, 2007 11:32 PM | Permalink | Reply to this

Re: Category Theory and Biology

This book might be worth looking at:

I know Plotkin’s work on category theory and computer science, but lately he’s been applying some of the same ideas to biology.

Lately Mike Stay has been trying to understand and explain the pi calculus, which is some mutant version of the lambda calculus. He might enjoy this article in the above volume:

  • Céline Kuttler, Simulating bacterial transcription and translation in a stochastic pi calculus, pp. 113-149.
Posted by: John Baez on November 16, 2007 2:23 AM | Permalink | Reply to this

Re: Category Theory and Biology

I think this is a very good idea.
I remain curious about math [BA] with an MD.

I have been reading both physics and engineering literature.
Biomathematics appears to be more oriented toward engineering than physics.
For example, pharmacology has some mechanisms of action that are poorly understood, yet somehow work.
This is more consistent with a Richard Bellman insight than with rigor or precision.

Posted by: Doug on November 16, 2007 3:10 AM | Permalink | Reply to this

Re: Category Theory and Biology

Something that intrigues me about the adoption of a mathematical theory, such as category theory, into a discipline like biology is that it illuminates understandings, and tensions between these understandings, of what constitutes success in that discipline.

As disciplines unfold, new understandings of success develop, old tensions may be resolved, and new tensions appear. I think the key task for the philosophy of science is to bring out what it is about this process, if anything, which is rightly described as rational.

Of course, this characterisation of the key task may put me at odds with other philosophers, so we have our own tensions to resolve. And we have to do our own thinking about the rationality of philosophy.

Posted by: David Corfield on November 16, 2007 10:35 AM | Permalink | Reply to this

ICCS, complexity, Re: Category Theory and Biology

Rashevsky and Rosen influenced my choice of PhD Thesis topic, indirectly, through reading their work. Particularly Dynamical Systems Theory in Biology [1970] by Rosen.

Since I was doing what today would be called metabolic network systems biology, and taking a couple of Graph Theory courses, and a Category Theory course in grad school (1973-1977) I was motivated to establish that Birth and Death were adjoint (a hunch that never panned out).

I went after explaining dynamical properties of metabolisms with the a priori belief that they were complex systems, in the sense that Rosen described.

I used classical means in my proof – Krohn Rhodes decomposition of the semigroup of differential operators of the nonlinear system of Michaelis-Menten equations of the metabolism, and finding eigenfunctions corresponding to waves in phase space (waves later rediscovered by Prigogine).

My first proof of my main result was 30 pages of matrix exponentials and the like, but that collapsed into a one-liner with the proper representation and semigroup decomposition, and restriction to “Physical Systems” (i.e. causality).

But, back then, only a handful of biologists had enough Math to follow this (some corresponded with me from Edinburgh and Russia). Only a handful of mathematicians knew enough biology (John Holland, in particular, on the Genetic Algorithm side).

But I think that Biology is ready for Category Theory today, due to the number of people in Complex Systems who know enough Biology and enough math, and care enough about system dynamics.

I urge that John Baez or someone else associated with this blog to make such a presentation at the 8th International Conference on Complex Systems, May 2009 (18 months from now), sponsored by the New England Complex Systems Institute.

The head of NCSI (Yaneer Bar-Yam) is a hard scientist, with several publications in Science. His father (also on the Board) has a named chiar of Physics at U.Massachusetts. Yaneer’s brother-in-law is also published in Science. The quality of the people and papers at ICCS is high.

Flimsy as my Physics credentials are, I chaired both Physics sessions at the 7th International Conference on Complex Systems (28 Oct-2 Nov 2007) because, in part, I have gotten to know the Executive Committee and staff rather well, and believe in extensive preparation for session chairs (i.e. reading several papers by each presenter, so as to introduce them well and in context of conference and session themes).

This interdisciplinary conference is the perfect venue for such a paper and dialogue. The default attendee is a recovering Physicist. Computer and Math (Graph Theory, Chaos) background is common. Appreciation of aesthetically strong theory is present, as is an appreciation for how complex Biological systems really are. Top biologists, from Harvard Medical School and many countries, are present.

Stuart Kauffman’s ideas are familar (he’s been a plenary speaker there).

I’m not in a position this time to make a formal invitation, as I have other responsibilities assigned for ICCS-2009, but do encourage this community to google “NECSI ICCS” and explore the notion.

Posted by: Jonathan Vos Post on November 16, 2007 5:19 PM | Permalink | Reply to this

Re: Category Theory and Biology

I have more a philosophical comment regarding David’s statement “I would imagine that a higher proportion of biologists are bewildered by this kind of work than their physicist colleagues by similar work in their field”.

I remember about 10 years ago, people started talking about something “biological physics” or “biophysics”. Someone wrote a letter to Physics Today claiming that biophysics was a subfield of physics, in the same way as condensed matter physics or particle physics, and that it should be a required class for all physics majors. I love biology, but clearly “biophysics” is a subfield of biology, not physics. If your goal is to increase our understanding of life on this planet, you’re doing biology. If your goal is to increase our understanding of the Universe, you’re doing physics.

So why did people invent the term “biophysics”? Throughout history, from Ancient Greece to the late 20th Century, physics required knowledge of advanced mathematics. In fact, the driving force behind the advancement of mathematics was physicists. However, biology required very little mathematical knowledge. Charles Darwin, even though he was a genius in his field, could not possibly have begun to understand the mathematics of Felix Klein or the physics of James Clerk Maxwell, even though Felix Klein and James Clerk Maxwell could easily have understood Darwin’s work. One of the reasons why the work of Gregory Mendel languished in obscurity is because it was a work in biology that required mathematical knowledge and understanding.

In fact all the way up until the late 20th Century, biology required very little mathematics. Physicists would be astonished how little mathematics biology required. However, in the late 20th Century, that changed. Finally, a measurable amount of biology required advanced mathematics. However, the biologists did not possess the mathematical knowledge to do it. Most biologists can’t even do highschool calculus. Therefore, they had to get physicists to do it for them. The physicists would not do it if you told them they were doing “biology” so you have to tell them they are doing “physics”. That’s why they invented the term “biophysics”.

I think the solution is not to get physicists to do biology. If they want to do it, that’s fine, but be honast that it’s a field of biology, not physics. The solution is to require biology students to take more math and physics classes. Right now you can get a BS in biology with no math classes at all. I think biology majors, actually any science major, including chemistry, geology, etc. should be required to take all the same math classes that physics majors are required to take.

Jeffery Winkler

Posted by: Jeffery Winkler on November 16, 2007 7:02 PM | Permalink | Reply to this

Re: Category Theory and Biology

Surely increasing our understanding of how life on this planet works DOES increase our understanding of the Universe. Namely, our understanding of the properties of matter which allow it to have bulk states that behave in ways typical of living things. Biophysics is presumably a subfield of *both* biology, *and* physics - demonstrating that the two fields are not disjoint. As, indeed, you would not expect them to be unless you believe in some sort of metaphysical vitalism, which as I understand it has long since become unfashionable in biology, let alone physics.

None of which is to deny your main point, which is that biologists would be well served by learning more math. Though I must add that I have taught math classes which were required courses for all life-science majors, so clearly not every BS program in biology lets you avoid them entirely.

Posted by: Jeffrey Morton on November 16, 2007 7:17 PM | Permalink | Reply to this

Re: Category Theory and Biology

@Tom Leinster & John Baez:

Thank you for your helpful suggestions! My background (and current area of study) is computer science and pure mathematics. I’m interested in exploring the possibility of applying new concepts to the field of neuroscience (not to mention broadening my horizons in algebra for other research).

I will (try to) read Conceptual Mathematics and come back booming with questions.

Thanks Again!

Posted by: Curious Undergrad on November 16, 2007 9:30 PM | Permalink | Reply to this

Re: Category Theory and Biology

Although I remain intrigued about the application of Category Theory to the broad field of Biology, there seem to be some areas where advances in biology exceed those of HEP Physics.

Biological Imaging techniques have been awarded Nobel Prizes [some multidimensional].
Nuclear Magnetic Resonance [NMR] for molecular structures: Physics 1952, Chemistry 1991 and 2002.
Magnetic Resonance Imaging [MRI] for body imaging: Physiology/Medicine 2003.
A Harvard group has applied MRI to Astronomy with “AstroMed”

Robotics through electrical and mechanical engineering application of mathematical pursuit evasion game theory has significantly advanced Medical Prostheses and the understanding of animal mechanics.

Research of animal swarm mechanics through the application of theoretical physics has been done by Iain Couzin

Carl Woese [BA math, physics; PhD biophysics] is probably best known for his work in Microbiology.

Posted by: Doug on November 23, 2007 2:35 AM | Permalink | Reply to this

Re: Category Theory and Biology

I’ve been thinking lately that category theory might be a viable way to make more rigorous the talk of “emergent properties” one hears so often in biophysics and other areas of the amorphous domain called “complex systems”. See John Armstrong’s random thoughts on Hofstadter.

Even if the biologists find themselves bewildered by category theory, they’re wanting to steal the word functor.

Posted by: Blake Stacey on November 28, 2007 11:08 PM | Permalink | Reply to this

Re: Category Theory and Biology

category theory might be a viable way to make more rigorous the talk of “emergent properties”

Nils Baas and Claus Emmeche once tried to do precisely that, see their On Emergence and Explanation

Posted by: Urs Schreiber on November 29, 2007 12:02 AM | Permalink | Reply to this

Re: Category Theory and Biology

Another physicist Seymour Benzer [BS Brooklyn, PhD Purdue] who made his mark in biology [E coli and fruit flies] recently died.

Carl Zimmer, science writer, has an ecellent “Farewell” tribute at The Loom

Posted by: Doug on December 1, 2007 11:06 PM | Permalink | Reply to this
Read the post Hierarchy and Emergence
Weblog: The n-Category Café
Excerpt: Emergence in hierarchies
Tracked: July 18, 2008 2:12 PM

Re: Category Theory and Biology

I started dreaming up some basic definitions at nnLab:multispan for the concept “multispan” and “multispan composition”. I want to eventually describe some motivating examples indicated above, but for the moment I ran out of time. But maybe you wanna have a look. All help is appreciated.

Posted by: Urs Schreiber on January 23, 2009 10:13 PM | Permalink | Reply to this

Re: Category Theory and Biology

I started an entry nnLab:hyperstructures about Nils Baas’ concept of that name discussed in the above entry.

Posted by: Urs Schreiber on January 24, 2009 2:39 PM | Permalink | Reply to this

Re: Category Theory and Biology

I am thinking about the following notion supposed to capture some \infty-cartegorical/hyperstructural flavor. Please give me sanity checks.

First of all, can I assume that every small poset DD has a cocompletion D¯\bar D with respect to pushout diagrams, and that D¯\bar D is again a poset.

I started searching for a bit of literature on this and got the impression that this ought to be true. But I am not sure.

Let me assume for the rest of this comment that it is true. If not, I need to go back and see how to fix this.

Here is a proposed definition:

Hyperstructure (an attempt)

For VV a suitable enrichment category, a VV-hyperstructure is a presheaf C:Posets opVC : Posets^{op} \to V such that for every diagram D 1D glueD 2 D_1 \leftarrow D_{glue} \to D_2 of posets the canonical morphism from C(cocompletion(D 1 glueD 2))C(cocompletion(D_1 \sqcup_{glue} D_2)) to C(D 1)× glueC(D 2)C(D_1) \times_{glue} C(D_2) is epi.

Posted by: Urs Schreiber on February 2, 2009 5:16 PM | Permalink | Reply to this

Re: Category Theory and Biology

For the free pushout completion of a poset, I think the following should work. First, the free cocompletion of a poset PP is given by the “Yoneda embedding”

P2 P opP \to 2^{P^{op}}

where 2 P op2^{P^{op}} can be identified with the set D(P)D(P) of downward-closed subsets of PP. Second, just take the closure of PP in D(P)D(P) with respect to pushouts (the intersection of all pushout-complete subposets of D(P)D(P) that contain PP).

To be slightly more explicit: this closure should consist of all downward-closed subsets that are finite unions of principal downward-closed subsets

hom(,a 1)hom(,a n)\hom(-, a_1) \cup \ldots \cup hom(-, a_n)

where you can get from any a ia_i to any other a ja_j by following a zig-zag path: a chain of diagrams which locally look like

a ma pa na_m \geq a_p \leq a_n

Posted by: Todd Trimble on February 2, 2009 7:23 PM | Permalink | Reply to this

Re: Category Theory and Biology

Hi Todd,

thanks for your comment!

I keep fiddling around with this idea. Now I think that I want/need full cocompletion, not just with respect to pushouts.

And I have another question. Here is what I am currently imagining as a definition:

let PosetsPosets be the category of small (maybe I want finite, even) posets and PosetsPosetsPosets' \subset Posets the category of small posets with all colimits.

Let ():PosetsPosets(\cdot)' : Posets \to Posets' be free cocompletion, i.e. Yoneda embedding.

Def. (second attempt):

A hyperstructure or bond structure KK is a (set-valued) presheaf on PosetsPosets' such that for all diamgrams D 1D glueD 2D_1 \leftarrow D_{glue} \to D_2 in PosetsPosets' the canonical morphism K((D 1 glueD 2))K(D 1)× K(D glue)K(D 2) K((D_1 \sqcup_{glue} D_2)') \to K(D_1) \times_{K(D_{glue}) }K(D_2) is epi.

Now, I want the following to be an example:

For CC a category with all colimits, the bond structure of multi-cospans in CC is the presheaf

Cat(,C)/ :(Posets) opSet Cat(-,C)/_\sim : (Posets')^{op} \to Set which sends each cocomplete poset DD to the isomorphism classes of functors DCD \to C.

I want to claim that this satisfies the gluing condition:

first of all, for K=Cat(,C)/ K = Cat(-,C)/_\sim we have K(D 1)× glueK(D 2)K(D 1 glueD 2). K(D_1) \times_{glue} K(D_2) \simeq K(D_1 \sqcup_{glue} D_2) \,.

Next, we always have an extension (and now please check that I am not hallucinating) of any fK(D 1 glueD 2)f \in K(D_1 \sqcup_{glue} D_2) to an f^K((D 1 glueD 2))\hat f \in K((D_1 \sqcup_{glue} D_2)'). Namely the left Kan extension of ff. It exsists because by assumption all colimits in CC exist.

More in detail, the Kan extension whould be given by

f^(B)= aD 1 glueD 2hom(a,B)f(a). \hat f(B) = \int^{a \in D_1 \sqcup_{glue} D_2} hom(a,B) \cdot f(a) \,.

Here the hom is in the cocompleted category and takes values in 2={}2 = \{\bottom \to \top\} and CC is taken to be canonically tensored over 22.

Have to run now…

Posted by: Urs Schreiber on February 3, 2009 3:01 PM | Permalink | Reply to this

Re: Category Theory and Biology

I have started typing the definition of bond structure or hyperstructure that I am imagining here into the nnLab. Currently this sits at the end of the former entry nnLab: hyperstructure in a Laboratory section.

I had thought about doing this in my “private web” area of the nnLab, but then this felt too reclusive. I am thinking that as long as it is clearly marked as “in laboratory stage” and not a final truth, it should sit in the nnLab proper to encourage more cooperation. After all, that’s the idea of the nnLab in the first place.

And I am prepared to discard all of this eventually should it turn out not to work out. But for the moment I am beginning to grow fond of what is there so far. So if anyone thinks this is heading for a dead-end I’d appreciate critical comments. I’ll also include further motivational examples.

Posted by: Urs Schreiber on February 4, 2009 10:52 AM | Permalink | Reply to this

Biohemical Flux Matroids and Hypergraphs; Re: Category Theory and Biology

Since undertaking my PhD dissertation research (1973-1977) I have been more interested in the foundational Math underlying metabolisms of reactions than the reactions or enzymes and metabolites themselves. This relates philosophically to the question of whether “relationships” are as “Real” as “Objects.” In my current research, assumptions are made in SBML (Systems Biology Markup Language). More precisely, we have fine papers such as the below.

New submissions for Fri, 6 Feb 09

[19] arXiv:0902.0847 [pdf, other]
Title: Hypergraphic Oriented Matroid Relational Dependency Flow Models of Chemical Reaction Networks
Authors: C. G. Bailey (1), D. W. Gull (2), J. S. Oliveira (2) ((1) Victoria University Wellington, Wellington, NZ, (2) Pacific Northwest National Laboratory, WA, USA)
Comments: 16 pages, 4 figures
Subjects: Combinatorics (math.CO); Quantitative Methods (q-bio.QM)

In this paper we derive and present an application of hypergraphic oriented matroids for the purpose of enumerating the variable interdependencies that define the chemical complexes associated with the kinetics of non-linear dynamical system representations of chemical kinetic reaction flow networks. The derivation of a hypergraphic oriented matroid is obtained by defining a closure operator on families of n-subsets of signed multi-sets from which a “Z-module” is obtained. It has been observed that every instantiation of the closure operator on the signed multiset families define a matroid structure. It is then demonstrated that these structures generate a pair of dual matroids corresponding respectively to hyperspanning trees and hypercycles obtained from the corresponding directed hypergraphs. These structures are next systematically evaluated to obtain solution sets that satisfy systems of non-linear chemical kinetic reaction flow networks in the MAP Kinase cascade cell-signaling pathway.

Posted by: Jonathan Vos Post on February 6, 2009 6:01 PM | Permalink | Reply to this

Re: Category Theory and Biology

the free cocompletion of a poset P is given by the “Yoneda embedding” P2 P op P \to 2^{P^{op}}

I was starting to write down some simple examples of free cocompletions of posets for myself, when I noticed that I might be mixed up about something.

If I onsider the pushout-poset

P={a b f g c} P = \left\{ \array{ a &&&& b \\ & {}_f\nwarrow && \nearrow_g \\ && c } \right\}

and try to write down the poset 2 P op2^{P^{op}} with

2={α} 2 = \{\bottom \stackrel{\alpha}{\to} \top\}

I seem to get, unless I am already too tired to think straight,

2 P op={ const (fα,gα) (fα,gId ) (fId ,gα) const } 2^{P^{op}} = \left\{ \array{ && const_{\top} \\ && \uparrow \\ &&(f\mapsto \alpha, g \mapsto \alpha) && \\ & \nearrow && \nwarrow \\ (f\mapsto \alpha, g \mapsto Id_\bottom) &&&& (f\mapsto Id_\bottom, g \mapsto \alpha) \\ & \nwarrow && \nearrow \\ && const_{\bottom} } \right\}

while I am expecting the free cocompletion to look like

{ } \left\{ \array{ &&\bullet && \\ & \nearrow && \nwarrow \\ \bullet &&&& \bullet \\ & \nwarrow && \nearrow \\ && \bullet } \right\}

I must be making some elementary mistake. Which one?

Posted by: Urs Schreiber on February 3, 2009 7:50 PM | Permalink | Reply to this

Re: Category Theory and Biology

To answer the last question first: would you expect the free cocompletion of the diamond (which is already cocomplete) to be the diamond again?

You shouldn’t, because the Yoneda embedding cannot be essentially surjective on objects. For example, the empty or initial object in 2 P op2^{P^{op}} is never isomorphic to something of the form hom(,x)\hom(-, x), because this always has the element 1 xhom(x,x)1_x \in hom(x, x).

So the free cocompletion in your example has five elements which form a diamond with an extra bottom attached. (You have an extra top attached.) The top element should be all of PP, regarded as a downward-closed subset of itself. The next rung down consists of hom(,a)\hom(-, a) and hom(,b)\hom(-, b) (the down-closed subsets {a,c}\{a, c\} and {b,c}\{b, c\}), then the next rung is hom(,c)\hom(-, c) (the down-closed {c}\{c\}), and at the bottom is the empty down-closed subset.

(In terms of arrow assignments, it looks like maybe you got turned around due to variance issues.)

Posted by: Todd Trimble on February 3, 2009 8:52 PM | Permalink | Reply to this

Re: Category Theory and Biology

it looks like maybe you got turned around due to variance issues

Oops, yes.

All right, so instead of erasing the above wrong picture of 2 P op2^{P^{op}}, which was actually the picture of 2 P2^{P}, I should at least post the correct one, just for the record:

2 P op={ const (f opα,g opId ) (f opId ,g opα) (f opα,g opα) const } 2^{P^{op}} = \left\{ \array{ && const_\top && \\ & \nearrow && \nwarrow \\ (f^{op}\mapsto \alpha, g^{op} \mapsto Id_\top) &&&& (f^{op}\mapsto Id_\top, g^{op} \mapsto \alpha) \\ & \nwarrow && \nearrow \\ && (f^{op} \mapsto \alpha, g^{op}\mapsto \alpha) \\ && \uparrow \\ && const_{\bottom} } \right\}

Posted by: Urs Schreiber on February 3, 2009 9:10 PM | Permalink | Reply to this

Re: Category Theory and Biology

Petri net modelling of biological networks provides a useful overview. And, it appears, Petri nets are monoids.

Posted by: David Corfield on January 29, 2009 9:26 AM | Permalink | Reply to this
Read the post Dendroidal Sets and Infinity-Operads
Weblog: The n-Category Café
Excerpt: Notes from a talk by Ieke Moerdijk on dendroidal sets, with a few remarks on presheaves on the category of posets.
Tracked: February 6, 2009 1:03 AM

Re: Category Theory and Biology

Now, as Alan Calvitti informed me, it’s the turn of symmetric monoidal (bi)categories to shed light on biology. Hmm, perhaps not too much light as yet.

If you want to see what systems biologists get up to, try Luca Cardelli’s talks.

Posted by: David Corfield on April 24, 2009 2:23 PM | Permalink | Reply to this

Re: Category Theory and Biology

There might be another appearance of category theory in biology, which I haven’t seen mentioned yet: population genetics! In a certain model to be described, sexual reproduction is given by the multiplication map of a population genetics monad – or at least something which closely resembles a monad…

Here’s how this works in detail. Let us consider a single genetic locus for simplicity. We start with the category Set and think of a set A as the set of possible alleles at the genetic locus. A genotype is then given by an unordered pair {a,b}\{a,b\} of elements of AA, where a=ba=b is allowed. A population in turn is determined by a formal conical combination of unordered pairs over AA. Let us denote the set of possible populations by M(A)M(A). We have M(A)={ iλ i{a i,b i},λ i0} M(A)=\left\{\sum_i\lambda_i\{a_i,b_i\},\:\lambda_i\geq 0\right\} Note that there is a natural map AM(A)A\rightarrow M(A) which assigns to each aAa\in A the population consisting of one individual with genotype {a,a}\{a,a\}.

Now let’s assume that we don’t distinguish the sexes and consider mating between (possibly) different populations. Then, a potential mating combination is nothing but an unordered pair of elements in M(A)M(A). We need to specify how many children one gets from each pair of mating populations in terms of a formal conical combination of unordered pairs of elements of M(A)M(A), where the number of children is encoded as the coefficient in the conical combination. But such a thing simply is an element of M(M(A))M(M(A))! Let us write it as iα i{ jλ ij{a ij,b ij}, kμ ik{c ik,d ik}} \sum_i\alpha_i\left\{\sum_j\lambda_{ij}\{a_{ij},b_{ij}\},\sum_k\mu_{ik}\{c_{ik},d_{ik}\}\right\} Each ‘mating data’ in the sense of an element of M(M(A))M(M(A)) yields a next-generation population in terms of an element of M(A)M(A). By the usual rules of genetics, this element is given by i,j,k14α iλ ijμ ik({a ij,c ik}+{a ij,d ik}+{b ij,c ik}+{b ij,d ik}) \sum_{i,j,k}\frac{1}{4}\alpha_i\lambda_{ij}\mu_{ik}\left(\{a_{ij},c_{ik}\}+\{a_{ij},d_{ik}\}+\{b_{ij},c_{ik}\}+\{b_{ij},d_{ik}\}\right) There should be a monad hiding here! Has anything like this been noticed before?

It is clear that MM actually is a functor on Set. One might indeed hope that the maps defined above yield a monad structure on MM. The right unit condition is obious. Associativity holds as long as we change the definition of MM to consist of convex combinations of unordered pairs, which coresponds to normalizing population size to be unity. However the left unit law requires that a population is invariant under breeding with itself – would it be sensible to restrict the allowed populations to only those satisfying the Hardy-Weinberg law?

PS: the formatting looks poor, how could I make the \sum’s bigger?

Posted by: Tobias Fritz on May 11, 2009 3:22 PM | Permalink | Reply to this

Re: Category Theory and Biology

Hmm, it looks like it might be included in Voevodsky’s work that David had mentioned here.

Posted by: Tobias Fritz on May 12, 2009 6:02 PM | Permalink | Reply to this

Re: Category Theory and Biology

Hmm, it looks like it might be included in Voevodsky’s work that David had mentioned here. Does anyone know of more detailed references?

Posted by: Tobias Fritz on May 12, 2009 6:05 PM | Permalink | Reply to this

More citations online; Re: Category Theory and Biology

Briefings in Bioinformatics 2001 2(3):258-270; doi:10.1093/bib/2.3.258
FREE Full Text (PDF) Freely available
© Henry Stewart Publications
Special issue papers
Systems biology: The reincarnation of systems theory applied in biology?
Olaf Wolkenhauer
Holds a joint appointment between the Department of Biomolecular Sciences and the Department of Electrical Engineering and Electronics (Control Systems Centre) at UMIST. His research interests include mathematical modelling and identification of dynamic systems with particular consideration of uncertainty in modelling, data and prediction.
[there’s Category Theory lurking here, although Rosen did not invoke it in 1960s]

Natural Transformations Models in Molecular Biology
IC Baianu - SIAM Natl. Meeting, Denver, CO, USA, 1983
This is available as a Word file from
which can be downloaded by plugging the title into Google Scholar

Olaf Wolkenhauer, Department of Biomolecular Sciences and Department of Electrical Engineering and Electronics, Control Systems Centre, UMIST, Manchester M60 1QD, UK Tel/Fax: +44 (0)161 200 4672 E-mail:

With the availability of quantitative data on the transcriptome and proteome level, there is an increasing interest in formal mathematical models of gene expression and regulation. International conferences, research institutes and research groups concerned with systems biology have appeared in recent years and systems theory, the study of organisation and behaviour per se, is indeed a natural conceptual framework for such a task. This is, however, not the first time that systems theory has been applied in modelling cellular processes. Notably in the 1960s systems theory and biology enjoyed considerable interest among eminent scientists, mathematicians and engineers. Why did these early attempts vanish from research agendas? Here we shall review the domain of systems theory, its application to biology and the lessons that can be learned from the work of Robert Rosen. Rosen emerged from the early developments in the 1960s as a main critic but also developed a new alternative perspective to living systems, a concept that deserves a fresh look in the post-genome era of bioinformatics.

Keywords: genomics, systems biology, causality, relational biology, (M,R)-systems, linear systems theory

Volume 64, Issues 1-3, January 2002, Pages 63-72

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Mobilising knowledge models using societies of graphs
Studies in Multidisciplinarity, Volume 2, 2005, Pages 135-146
R.C. Paton

This chapter discusses a way of mobilising knowledge by using models of knowledge based on graphs. These models are informal and geared to ease of use. A key feature of the approach is concerned with the idea that as knowledge about a domain unfolds, a society of graphs can be used to seed, generate and elaborate the emerging model. This society can help the exploration of a domain to unfold and the graphs satisfy a number of roles that we describe in terms of some key metaphors. A simple case study is followed to illustrate the approach concerned with the notion of a network.

Metaphors, models and bioinformation
Biosystems, Volume 38, Issues 2-3, 1996, Pages 155-162
Ray Paton

The notion of bioinformation is central to the biosciences. This short paper examines a number of metaphors which are intimately related to this idea. These include metaphors about ‘system’ as well as metaphors associated with biosystem as ‘text’. A framework is presented which allows ideas about biosystems and computer systems to be displaced and a number of specific topics are then discussed. Firstly, information processing in non-neural tissues is given as an example of parallel distributed processing. This is followed with a number of metaphors associated with ‘text’, including ‘glue’, ‘verbs’ and ‘interpretation’. The paper concludes with a proposal on how to integrate general ideas of bioinformation using the idea of the ‘ecology of domains’.

Process, structure and context in relation to integrative biology

Ray Paton
Corresponding Author
Department of Computer Science, The University of Liverpool, Liverpool L69 3BX, UK

Received 8 May 2001;
revised 18 July 2001;
accepted 19 July 2001.
Available online 18 December 2001.


This paper seeks to provide some integrative tools of thought regarding biological function related to ideas of process, structure, and context. The incorporation of linguistic and mathematical thinking is discussed within the context of managing thinking about natural systems as described by Robert Rosen. Examples from ecology, protein networks, and liver function are introduced to illustrate key ideas. It is hoped that these tools of thought, and the further work needed to mobilise such ideas, will continue to address a number of issues raised and pursued by Michael Conrad, such as the seed-germination model and vertical information processing.

Author Keywords: Ecology; Proteins; Category theory; Modelling; Function; Liver
Article Outline

1. Introduction
2. Graphs, processes, and objects
3. Context, ecology, and collection concepts
4. Proteins in context—graphs and ‘glues’
4.1. Proteins as verbs
4.2. Proteins, verbs and ‘glue’
5. ‘Glue’, categories, and functions
6. Concluding remarks

Posted by: Jonathan Vos Post on May 11, 2009 6:48 PM | Permalink | Reply to this

Re: Category Theory and Biology

Do you just mean that biologists are less likely to know category theory than physicists?

Posted by: human mathematics on December 29, 2011 4:23 AM | Permalink | Reply to this

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