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• CommentRowNumber1.
• CommentAuthorBen_Sprott
• CommentTimeSep 21st 2011
Hi,

If you have ever seen any of my other posts, you will see that I have been trying to capture a notion of continuously changing categories. This idea comes from two intutions. First is how we can have a discrete, ordered structure like a Domain, and have a topology and thus a continuous map on it. Those are the Scott topology and the Scott Continuous maps. This can be generalized up to categories by saying that the elements of the Domain are the objects and the ordering relations are the morphisms. Thus, we should expect a similar kind of topology on a category and notion of a continuous map. In fact, the definition of a continuous functor (which we can find on nlab) looks a bit like a generalization of the definition of a scott continuous map on a Domain. The second notion that I have been following is that of a domain of groups. In this case, we see how the compact elements of a domain of groups are the finitely generated groups. As far as I can tell, if we can apply this reasoning to categories, then we are going to see how the concept of finite element goes towards categories themselves.

I have a new idea in this vein which brings the notion of continuous functor closer to the group theoretic idea. We start by noting that a category can be given in terms of its morphisms only, and a long (or short or empty) list of equations over the morphisms. Thus, we see that a category can be given as a partial monoid and a large collection of data detailing all the compositions that are possible. This list would, of course, be simplified into axioms which are the universal properties of the category. However, this creates a distinction. There are those partial monoids with finitely many axioms and those that don't have this property. Next, we want to create a domain of partial monoids. Naturally, we could have a tiny Domain, like an empty set, but we could also have a relly big domain of partial monoids. This would also be a Domain of categories. On this domain we could define, in a straightforward way, a Scott topology and a continuos map.

I guess, I want to answer the following question: if we take the definition of compact element straight out of the definitions on a dcpo, then translate it this way into categories, what does the definition say about a category. I really want to say that the category will have finitely many axioms. The axioms could be given either in the standard object/morphism presentation or this arrows only presentation...I think.

I hope people find this post relevant and interesting. Let me know.
• CommentRowNumber2.
• CommentAuthorTodd_Trimble
• CommentTimeSep 21st 2011
• (edited Sep 21st 2011)

Ben, I’m having a hard time following all of this, but I would like to direct your attention to a generalization of some of these domain-theoretic ideas to categories, in the form of accessible categories and locally presentable categories. There is some material in the nLab on this, but some of the articles are in need of improvement; you might want to look at the text Locally Presentable and Accessible Categories by Adámek and Rosicky, and also Accessible Categories: the Foundations of Categorical Model Theory by Makkai and Paré.

In brief, the analogue of a dcpo is a filtered-cocomplete category $C$ (see filtered colimit), and the analogue of a compact element in a dcpo is an object $c$ of $C$ such that $\hom(c, -): C \to Set$ preserves filtered colimits. Such objects may be called finitely presentable; for $C = Grp$, these objects are indeed the finitely presentable groups. (There is a similar statement for the category of models of any algebraic theory.) If $C$ is cocomplete and every object is canonically a filtered colimit of fnitely presentable objects, we say $C$ is locally finitely presentable. Such categories generalize algebraic lattices.

$Cat$ (the category of small categories) is locally finitely presentable. This is a category of models of an essentially algebraic theory (a theory involving a list of partially defined algebraic operations where the domain of an operation is equationally definable in terms of operations occurring earlier in the list). It is a theorem that essentially algebraic categories are basically the same thing as locally presentable categories (Adámek and Rosicky, chapter 3).

I hope this provides food for thought – I strongly recommend these texts.

• CommentRowNumber3.
• CommentAuthorBen_Sprott
• CommentTimeSep 24th 2011
Sorry, but I am going to ask one more or two more pointed questions here because a quick answer might really help me.
If I have a dcpo, I can see it as a category, X, ( as I can for any ordered structure with the right axioms like transitivity etc.). The ordering relations are the morphisms in the category. Now, in a dcpo, we have suprema and if you draw a simple diagram of a supremum for a directed set, you basically get the diagram of a limit. The question is this: does the category X have all finite limits? Does it have equalizers? When looking at the definition of a compact element in terms of just the drawings with the ordering relation, what categorical property do the compact elements display? This last question is quite vague, but if you start at the revelation you get by looking at the diagrams for the suprema, and try to apply it to the compact elements, something must appear. I guess, in short, what are the axioms of the category X?

btw: Todd, thanks for the references. Once I get to the library I will get these book.
• CommentRowNumber4.
• CommentAuthorFinnLawler
• CommentTimeSep 24th 2011

Todd has already answered most of these questions, Ben. You might also try Adámek's paper 'A categorical generalization of Scott domains', Math. Struct. in Comp. Sci. 1997.

Finite limits are given by finite products (= meets in a poset) and equalizers. I don't think an arbitrary dcpo will have meets. Equalizers in a poset are trivial, because all parallel arrows are equal by definition.

• CommentRowNumber5.
• CommentAuthorTodd_Trimble
• CommentTimeSep 24th 2011

I believe I’ve already answered the most important of these questions.

if you draw a simple diagram of a supremum for a directed set, you basically get the diagram of a limit

No, you get the diagram for a colimit.

The question is this: does the category X have all finite limits?

Do you mean: does it have all finite colimits? No, consider a set as a discrete preorder. Every directed subset is a singleton (!), and the colimit of a singleton is its unique element. So sets are dcpo’s. They are clearly neither finitely cocomplete nor finitely complete.

Does it have equalizers?

Vacuously, yes. An equalizer is a limit of a diagram consisting of a pair of parallel arrows. The vacuous case is when the parallel arrows are identical; the equalizer is the identity arrow on the domain. (If you actually meant coequalizer, then it would be the identity on the codomain.) Notice that in a poset, parallel arrows are equal because in a poset, there is at most one arrow $x \to y$ between any two given objects.

what categorical property do the compact elements display?

An element $x$ in a dcpo is compact if $\hom(x, -): P \to Set$ (or equivalently for a poset, $\hom(x, -): P \to \{\bot, \top\}$) preserves filtered (aka directed) colimits. I had mentioned this in comment 2, but perhaps I should amplify.

An element $x$ is compact if, for any directed subset $D \subset P$, we have

$x \leq \sup D \Leftrightarrow \exists_{d \in D} x \leq d$

Notice the direction $\Leftarrow$ is automatic; only the direction $\Rightarrow$ has content. Here $\sup D$ is a directed (filtered) colimit, and the left side holds if $\hom(x, colim D) = \top$. The right side holds iff $\{\hom(x, d): d \in D\}$ contains $\top$, i.e., if the sup or colimit of $\{\hom(x, d): d \in D\} \subseteq \{\bot, \top\}$ is $\top$. So the compactness condition translates to saying

$\hom(x, colim_{d \in D} d) = \top \Leftrightarrow colim_{d \in D} \hom(x, d) = \top$

which is equivalent to $\hom(x, -)$ preserving colimits of directed subsets $D$.

what are the axioms of the category X?

That it is a filtered cocomplete poset. See comment 2.

thanks for the references

There’s an embarrassment-of-riches problem here, but another book in my library that I find useful is Paul Taylor’s Practical Foundations of Mathematics. He has some material on the stuff you’re asking about. Another good book is Johnstone’s Stone Spaces. There are many others, of course.

• CommentRowNumber6.
• CommentAuthorTobyBartels
• CommentTimeSep 25th 2011
• (edited Sep 25th 2011)

what are the axioms of the category X?

That it is a filtered cocomplete poset.

As a list of conditions on a category, you might prefer to say that it has all filtered colimits and is thin.

• CommentRowNumber7.
• CommentAuthorTodd_Trimble
• CommentTimeSep 25th 2011

Thanks. I’m not sure I’ve ever heard the word ’thin’ in this sense. Can you point me to a reference outside the nLab for this?

• CommentRowNumber8.
• CommentAuthorBen_Sprott
• CommentTimeSep 26th 2011
Todd, thanks again for the info.
• CommentRowNumber9.
• CommentAuthorTodd_Trimble
• CommentTimeSep 26th 2011

Sure, no problem Ben. Hopefully I’ve given you satisfactory answers to the questions you asked in #3.

• CommentRowNumber10.
• CommentAuthorBen_Sprott
• CommentTimeOct 1st 2011
Here is a fun question:
Is an internal category in Fdhilb necessarily small? Alternatively replace fdhilb with monoidal category.
• CommentRowNumber11.
• CommentAuthorBen_Sprott
• CommentTimeOct 1st 2011
• (edited Oct 2nd 2011)
I think I would prefer to know if there are internal categories in SET (which cat of sets is a good question. I'll take the chance here and say the cat of all small sets) that are are not $k$-presentable for any regular $k$. And further, the same question except for categories internal to fdhilb.

Ok so small and k-presentable are the same concept.
• CommentRowNumber12.
• CommentAuthorBen_Sprott
• CommentTimeOct 1st 2011
• (edited Oct 2nd 2011)
The reason I am asking all this is that I am interested in "continuously" reasoning over all structures. To do this we need to traverse, if only along the way, cats that are not k-presentable. If we try to do our reasoning in a cat of cats where every object is small or k-presentable, then we restrict our reasoning in ways that do not take advantage of the full power of our intuitions. I am coming at this from the perspective of a physicist. In mathematics, I would wager, we present in SET but we do not intuit in set. We can also reason in set, though this reasoning may be too restrictive for physics for the reasons I have mentioned.
• CommentRowNumber13.
• CommentAuthorBen_Sprott
• CommentTimeOct 2nd 2011
• (edited Oct 2nd 2011)
It is a trend in certain circles to speak of physics as structures on or in a monoidal category. Take Vicary's work and Coecke's and there are others. Of extreme interest right now are the internal categories of monoids and comonoids. My contribution has been to point out that an aparatus is a structured object ie an object of some known category. An object under study is imprecisely known and measurements are functors used to "see" structural properties of the system under study in terms of "local" diagrams in the known category. The arrows of these diagrams are transformations of your aparatus (turning knobs and pressing buttons and lights flashing and changing colour).
Science evolves by changing or expanding the available transformations and this includes completely changing the category which you "know of" as your aparatus. A big picture might point out that each of these newfangled aparatus categories is internal to the (fundamental, ultimate reality) category FDHilb.
• CommentRowNumber14.
• CommentAuthorBen_Sprott
• CommentTimeOct 2nd 2011
Thus, I am interested in showing that there are categories internal to FDHilb that are not k-presentable. Alternatively, I would say that the "ultimate reality category", which some take as FDHilb, would have to have most internal categories non-k-presentable.
• CommentRowNumber15.
• CommentAuthorBen_Sprott
• CommentTimeOct 2nd 2011
Here is a thoerem:

The category of all internal categories in FDHilb is not locally presentable.
• CommentRowNumber16.
• CommentAuthorTodd_Trimble
• CommentTimeOct 2nd 2011

@Ben: I think we should slow down a moment, because it’s not so clear to me that your intended meanings of words and phrases are the ones I or other people might be receiving. So allow me to ask a few questions.

By FDHilb, I take it you mean the category finite-dimensional (complex) Hilbert spaces and linear maps. But how are you understanding this? As an ordinary category, or as a symmetric monoidal category, or as a $\dagger$-symmetric monoidal category, or what?

This is actually quite important. Normally when you say ’internal category’ in a category $C$, people usually assume you mean $C$ has finite limits and we are using the standard notion of internal category in a finitely complete category. Note that $FDHilb$ is finitely complete! (As a mere category, it is equivalent to the category of finite-dimensional vector spaces over $\mathbb{C}$.) However, I have a feeling, based on previous discussions of yours, that this is not the notion of internal category you mean.

There is a much more sophisticated notion of internal category in a monoidal category, which might be the notion you meant, but before we proceed I have to make sure. Once again, I don’t see that the $\dagger$-structure is being used, so that for all intents and purposes for this notion, one could just as well use $FDVect$ with its standard tensor product, which is monoidally equivalent to $FDHilb$.

I think there are a bunch more questions I could ask, but I’ll stop here for now. Please give mathematically precise answers; I will take a very dim view of phrases like “ultimate reality category”, “transformations of the apparatus”, and vague mention of work of Coecke and Vicary.

• CommentRowNumber17.
• CommentAuthorTobyBartels
• CommentTimeOct 2nd 2011

If you don’t make it a $\dagger$-category, then you ought to use only the linear maps with norm at most $1$ as morphisms. (That way, you get the correct isomorphisms.) Then Hilb is distinguished from Vect.

However, the internal categories in this FDHilb are (if I’m not messing up) all discrete, so it’s still not what we want.

• CommentRowNumber18.
• CommentAuthorBen_Sprott
• CommentTimeOct 2nd 2011
• (edited Oct 2nd 2011)
Hi Todd,

You are right that I was not clear on how to define the internal category for my purposes. I have just looked again at the nlab post based on Marcelo Aguiar’s thesis. Thank you again for your patience. I am going to go ahead and say that I am indeed interested in the internal cats of Aguiar's. Furthermore, I am looking at the composition of the forgetful and free functors RQ in Vicary's harmonic oscillator paper as a candidate monad. This particular adjunction is deeply fundamental. In any case, I am now curious why this adjunction is so special or unique. Where are the rest of them? Afterall, I am talking about all monads in Comod(M) as a category of categories so there should be lots of monads.

I have a paper in the works where I am just at the point of defining an aparatus as a category, all using this particular adjunction RQ. I really apreciate the help. I need help with this paper but I don't want to post the draft in the open right now. I would be proud to send you a copy. This would elucidate the vague phrases.

As an aside:
It would be great,also, to get back to our discussion on how to start a presentation of the theory of categories in this context. I am at a loss there except that we have a definition of "category" as a monad.
• CommentRowNumber19.
• CommentAuthorTodd_Trimble
• CommentTimeOct 3rd 2011

Okay, great. Maybe you could start by telling me what is this $R$ and $Q$, or at least giving me a link to the paper of Vicary you are referring to. I haven’t thought deeply about internal categories in the monoidal category $FDVect$, but this doesn’t seem too scary, since $FDVect$ is self-dual and people have a lot of experience with f.d. algebras over $\mathbb{C}$, f.d. bimodules between them, and so on.

I have quite a bit going on right now, so progress could be slow at first.

• CommentRowNumber20.
• CommentAuthorBen_Sprott
• CommentTimeOct 3rd 2011
Hey Todd,

Here is the paper by Vicary that I am really focusing on:

http://arxiv.org/abs/0706.0711

In particular, the adjunction (RQ) that I am interested in is given in definition 4.1. It is a forgetful and free pair. I think Vicary focuses on the endofunctor on the base category $C$, but I am interested in the reverse. I am interested in an endofuctor on Comod(C).
• CommentRowNumber21.
• CommentAuthorBen_Sprott
• CommentTimeOct 3rd 2011
Here are the papers that outline what I am trying to do in physics. I will take them down in a few days as they are not very good and one is only partially finished.

Here is the one I have referred to in post 18:
http://dl.dropbox.com/u/6040582/CategoricalPhysics.pdf

This one is pretty bad but gets some ideas flowing:
http://dl.dropbox.com/u/6040582/EvolvingUniverseFeb24.pdf
• CommentRowNumber22.
• CommentAuthorBen_Sprott
• CommentTimeOct 6th 2011
Hi,

Could someone give an example of a simple (the simpler the better) category that is an internal category defined as a monad on Comod(M), the category of comonoids in a monoidal category. Todd or Bart, any cat would be fine as long as it is explicitly given as an explicit monad on Comod(M). It would be greatly appreciated.

Ben
• CommentRowNumber23.
• CommentAuthorTodd_Trimble
• CommentTimeOct 6th 2011

Ben, to understand this definition, one should consider first the case where the monoidal product is a cartesian product. In fact, one should start with $Set$ with its cartesian product. Do you know what happens in this case? In other words, how you would characterize the notion of “internal category in the monoidal category” $(Set, \times)$ in more familiar terms?

(By the way, TobyBartels is Toby – not Bart. (-: )

• CommentRowNumber24.
• CommentAuthorBen_Sprott
• CommentTimeOct 7th 2011
Hey Todd, Toby (sorry about the name mixup)

So, I am trying to do this example in Set. It is a good suggestion as there some starting data in the nLab page on these internal cats. Thanks again.
• CommentRowNumber25.
• CommentAuthorTodd_Trimble
• CommentTimeOct 7th 2011
• (edited Oct 7th 2011)

What you might want to do first is to understand what comonoids in the monoidal category $(Set, \times)$ are like. (Hint: they are forced to be something very simple.) This is not meant to be a difficult exercise at all, but it’s something pretty fundamental to a general understanding of (cocommutative) comonoids – the basic idea has far-reaching implications, which form a bridge between symmetric monoidal categories and cartesian categories.

• CommentRowNumber26.
• CommentAuthorBen_Sprott
• CommentTimeOct 10th 2011
• (edited Oct 10th 2011)

Hey, I am trying to do this example.

In the case of $(Set, \times)$, I think sets are the objects, with a cartesian product and singleton as the unit and morphisms of these comonoids are spans.

$\mu : a \rightarrow a \times a$

And

$\nu : a \rightarrow I$

The objects of the category of comonoids each come with structural maps mu and nu. nu morphisms come from the fact that, there is exactly one map from any set to a singleton. mu maps are confusing me, its some kind of injection into the cartesian product…. I see, $\mu (a)= (a,a)$.

Composition of spans is due to the presence of pullbacks in Set. The bicategory nature means that the “hom-collections” are actually categories of spans and….span morphisms (?) and i don’t know what a span morphism would look like…..except that it is probably a function from the spanning set to another spanning set. The span morphisms have to preserve the mu and nu morphisms and that means making a diagram commute. I have a little drawing here where a span from comonoids $A$ and $B$ ($A \leftarrow Q \rightarrow B$) is commuting with mu and nu. It involves the projection $\pi_{Q}:Q \times Q \rightarrow Q$

Why spans? Why are the comonoid morphisms spans in this example?

Meanwhile, a monad, according to nlab, in a bicategory is an object, an endomap and various 2-cells with two important diagrams as axioms for the endomap. I am trying to cook up some calculations with a 3 element set, but it is all a little unfamiliar. I guess, given any set, and an appropriately chosen endomap and 2-cell, one ends up with a monad. What is confusing is, first, how we are allowed to see these monads as categories. Second, doing a simple little example like a three element set is confusing because there are so many possible spans that one can use as the endomap. I guess every span will give a different monad…?

Struggling with it…

• CommentRowNumber27.
• CommentAuthorTodd_Trimble
• CommentTimeOct 10th 2011
• (edited Oct 10th 2011)

You seem to be getting the hang of some of it, anyway. Let me lend a hand.

First, suppose that $(X, \delta: X \to X \times X, \varepsilon: X \to 1)$ is a comonoid in $(Set, \times)$. (N.B.: we are not a priori assuming this $\delta$ is the diagonal map. Here it is merely notation for the comonoid comultiplication.) Then $\varepsilon = !: X \to 1$ has to be the unique map to the 1-point set ($1$ is terminal). Now $\delta: X \to X \times X$, being a map to a cartesian product, is uniquely determined by the pair of maps $\pi_1 \circ \delta: X \to X$, $\pi_2 \circ \delta: X \to X$, where $\pi_1, \pi_2$ are the two projections from $X \times X$ to $X$. But notice that

$(\pi_1: X \times X \to X) = (X \times X \stackrel{id_X \times !}{\to} X \times 1 \cong X); \qquad \qquad (\pi_2: X \times X \to X) = (X \times X \stackrel{! \times id_X}{\to} 1 \times X \cong X)$

and so, for example,

$(X \stackrel{\delta}{\to} X \times X \stackrel{\pi_1}{\to} X) = (X \stackrel{\delta}{\to} X \times X \stackrel{id_X \times \varepsilon}{\to} X \times 1 \cong X)$

but the right side is the identity on $X$, according to one of the axioms for a comonoid (one of the counit axioms). By similar reasoning, $\pi_2 \circ \delta = id_X$, using the other counit axiom. Therefore

$\delta = \langle id_X, id_X \rangle: X \to X \times X;$

in other words, we have proved that $\delta$ is indeed the diagonal map! Conclusion: there is exactly one comonoid structure on any object $X$, whose comultiplication is the standard diagonal and whose counit is the projection $!: X \to 1$. Notice the same argument applies to any category with cartesian products, in place of $Set$. It is a simple, but in some sense important and far-reaching argument. Notice also that these comonoids are cocommutative.

Now that we know the nature of comonoids in $(Set, \times)$, we can consider right comodules, left comodules, and bicomodules. But these can be dispatched with, using arguments very similar to the above. For example, suppose $(C, \beta: C \to X \times C)$ is a left comodule with respect to the unique comonoid structure on $X$. Again, $\beta$, being a map to a cartesian product, is uniquely determined by a pair of maps: $\beta = \langle \beta_1, \beta_2 \rangle: C \to X \times C$. Using one of the comodule axioms, it may be shown that $\beta_2 = id_C$. You should check that the other comodule axiom is automatically satisfied, for any choice of $\beta_1$. Therefore, a (left) comodule over $X$ is nothing but an object $C$ equipped with a map $f = \beta_1: C \to X$.

Similarly, it may be shown that the data of right comodule over $Y$ amounts to an object $C$ equipped with a map $g: C \to Y$. A left-$X$ right-$Y$ comodule (or bicomodule) amounts to the same thing as a pair of maps

$X \stackrel{f}{\leftarrow} C \stackrel{g}{\to} Y$

(you should check that the compatibility condition between the separate left comodule and right comodule structures, as required by the definition of bicomodule, are automatically satisfied in the cartesian case). In other words, a bicomodule between comonoids $X$ and $Y$ amounts to a span from $X$ to $Y$.

One should go on to check that bicomodule composition amounts to the usual span composition. I’ll save this for another time (too busy today). Thus, a monad in the bicategory of comonoids and bicomodules (w.r.t. a finitely complete category) amounts to a monad in the bicategory of spans. The next thing we should discuss after that is how a monad in the bicategory of spans may be seen as amounting to a category structure (but, again, I’m too busy to do more about that today).

• CommentRowNumber28.
• CommentAuthorBen_Sprott
• CommentTimeOct 10th 2011

I think there is a typo in the nlab section on bicomodules

http://ncatlab.org/nlab/show/bicomodule

$\rho_c : M \rightarrow C \otimes M$

There’s a D instead of a C.

• CommentRowNumber29.
• CommentAuthorTodd_Trimble
• CommentTimeOct 10th 2011

Fixed. If you see obvious errors, please feel free to fix them yourself. :-)

• CommentRowNumber30.
• CommentAuthorTobyBartels
• CommentTimeOct 10th 2011

Since Todd didn’t say it explicitly, Ben, I will: you were correct. (At least your version as edited 5 hours ago is correct.)

• CommentRowNumber31.
• CommentAuthorBen_Sprott
• CommentTimeOct 11th 2011

Todd, I would be honoured.

• CommentRowNumber32.
• CommentAuthorBen_Sprott
• CommentTimeOct 24th 2011

Hey Guys,

I want to thank you both for your help. I am busy at work and have found a few hours to follow along with Todd. He suggested I show that the second comonoid axiom is satisfied for any $\beta_1$ and this can be shown using function/element notation:

Let $\beta_1$ be any function $\beta_1 : C \rightarrow X$

The second axiom states that

($\epsilon_X \times id_C ) \dot \beta = id_c$

So, we would expect that, for $c \in C$

$\beta(c) = (\beta_1(c), c)$

and

$(\epsilon_X \times id_c)(\beta(c))= (\{ \star \}, c) = c$

and this is true for any function $\beta_1$

• CommentRowNumber33.
• CommentAuthorBen_Sprott
• CommentTimeOct 24th 2011

For the next part, we want to check the compatibility between the left and right comodules. Again, using function, element notation:

$(f \times Id_Y)(g(c)) = (f \times Id_Y)(c,g(c))=(f(c), c, g(c))$

and

$(Id_x \times g)(f(c)) = (Id_x \times g)(f(c),c) = (f(c),c,g(c))$

and this is true for any $f,g$.

• CommentRowNumber34.
• CommentAuthorTodd_Trimble
• CommentTimeOct 26th 2011
• (edited Oct 27th 2011)

Hi, Ben. I have just a couple of comments, before we press on.

By the “other axiom” for comodule structures $\beta: C \to X \times C$, I meant the one which asserts commutativity of the diagram

$\array{ C & \stackrel{\beta}{\to} & X \times C \\ \mathllap{\beta} \downarrow & & \downarrow \mathrlap{id_X \times \beta} \\ X \times C & \underset{\delta_X \times id_C}{\to} & X \times X \times C }$

The other comment is notational. I use the notation $f \times g$, where $f: A \to B$ and $C \to D$ are maps, to mean the map $f: A \times C \to B \times D$. Then, given $f: A \to B$ and $g: A \to C$ (same domain), I use $\langle f, g \rangle$ to denote the composite

$A \stackrel{\delta_A}{\to} A \times A \stackrel{f \times g}{\to} B \times C$

Many people, including you in your last comment, use $f \times g$ where I would use $\langle f, g \rangle$. This practice saves on LaTeX characters of course, but I find it confusing or ambiguous. Thus, I would say $(f \times g)(a, c) = (f(a), g(c))$ and $\langle f, g \rangle(a) = (f(a), g(a))$, and not $(f \times g)(a) = (f(a), g(a))$.

Pressing on now, let’s consider bicomodule composition, both in general monoidal categories and in cartesian monoidal categories. Let $X$, $Y$, and $Z$ be comonoids in a monoidal category $M$, and let $(C, \alpha: C \to X \otimes C, \alpha': C \to C \otimes Y)$ and $(D, \beta: D \to Y \otimes D, \beta': D \to D \otimes Z)$ be bicomodules (from $X$ to $Y$ and $Y$ to $Z$, respectively). To compose these, we just dualize the way we compose bimodules, which involves taking a tensor product $\otimes_Y$ over the “middle” monoid $Y$, which in turn involves a familiar coequalizer construction. Dualizing, we take the equalizer of a pair of maps

$E \to C \otimes D \stackrel{\overset{\alpha' \otimes 1_D}{\to}}{\underset{1_C \otimes \beta}{\to}} C \otimes Y \otimes D$

and on this equalizer, one can define canonical maps $\gamma: E \to X \otimes E$, $\gamma': E \to E \otimes Y$, provided that tensoring on either side – $X \otimes -$ or $- \otimes Z$ – preserves (reflexive) equalizers. In this way one gets a left-$X$ right $Z$ bicomodule $E$, and this is the bicomodule composite of $C$ and $D$.

Again, it is good to work out what is going on here in the cartesian monoidal case (which is much simpler). Here, the equalizer described above works out to be a construction of the pullback

$\array{ E & \to & C \\ \downarrow & & \downarrow \mathrlap{\alpha^{'}_{2}} \\ D & \underset{\beta_1}{\to} & Y }$

and we can also check that $X \times -$ and $- \times Z$ preserve reflexive equalizers, or more simply, pullbacks. In any case, this pullback construction is precisely how we compose spans from $X$ to $Y$ and from $Y$ to $Z$:

$\array{ & & C & & & & D && \\ & ^\mathllap{\alpha_1} \swarrow & & \searrow^\mathrlap{\alpha_2^'} & & ^\mathllap{\beta_1} \swarrow & & \searrow^\mathrlap{\beta_2^'} & \\ X & & & & Y & & & & Z }$

to get a span from $X$ to $Z$.

Now, an important insight (we are still in the monoidal case) is that the structure of a category $C$ can be given as a span

$\array{ & & C_1 & & \\ & ^\mathllap{dom} \swarrow & & \searrow^\mathrlap{cod} & \\ C_0 & & & & C_0 }$

equipped with a monad structure on this span, in the bicategory of spans. Here, a monad structure boils down to morphisms of spans from $C_0$ to $C_0$ of the form

$m: C_1 \times_{C_0} C_1 \to C_1, \qquad \qquad u: C_0 \to C_1$

where $C_1 \times_{C_0} C_1$ is the apex of the span-composite $C \circ C$, and $m$ is the function which maps a composable pair of arrows to its composite, and where $C_0$ is the apex of the span consisting of two identity arrows going down to $C_0$, and $u$ maps an element of $C_0$ (“an object”) to its identity arrow.

The point is that this notion of category can be carried out analogously in the bicategory of bicomodules in a monoidal category (again, provided the technical equalizer preservation assumption is satisfied). We can try it out for example in $Vect_{fd}$ (finite-dimensional vector spaces over some field), which is related to what you were asking about in comment 10. The nice thing about this case is that we have an equivalence $Vect_{fd}^{op} \simeq Vect_{fd}$, given by taking dual spaces. Thus, we have an option between considering monads in the bicategory of bicomodules between comonoids (called coalgebras in the $Vect_{fd}$-case) or, dually but equivalently, comonads $(C, \varepsilon: C \to R, \delta: C \to C \otimes_R C)$ in the bicategory of finite-dimensional bimodules between finite-dimensional algebras $R$. These are the same as coalgebras in the category of $R$-bimodules.

There are lots of these things around, and it might be good to collect a stockpile of them. One thing I should warn about, based on a quick glance at Vicary’s paper, is that his $R Q$ construction is (if I’m not mistaken) a cofree coalgebra construction, which takes us outside the realm of $Vect_{fd}$ (it’s an infinite-dimensional construction). There may be some workarounds, but it’s to say that I’m not sure where you’d like to go from here.

• CommentRowNumber35.
• CommentAuthorTobyBartels
• CommentTimeOct 27th 2011

There are good category-theoretic reasons, not just Todd’s personal preference, for using $f \times g$ and $\langle{f,g}\rangle$ as he does. (They may be extracted from pairing, although you don’t really have to read them.)

• CommentRowNumber36.
• CommentAuthorBen_Sprott
• CommentTimeNov 1st 2011

Hi,

I am not sure how to proceed. I have to try all this stuff out, first.