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Some generality is sacrificed in the theorems asserting equivalence of different notions of monoidal category: these are equivalences of the categories of monoidal categories, but we could have made them equivalences of 2categories in other words, included monoidal transformations. I have not aimed to exhaust the subject, or the reader. An unbiased monoidal category is the same, but with n allowed to be any natural number. Here we eliminate the bias. We require, of course, that the coherence isomorphisms satisfy all sensible axioms.
We will show in the next section that the unbiased and classical definitions are equivalent, in a strong sense. The unbiased definition seems much the more natural having, for instance, no devious coherence axioms , and much more useful for the purposes of theory. When verifying that a particular example of a category has a monoidal structure, it is sometimes easier to use one definition, sometimes the other; but the equivalence result means that we can take our pick.
Here is the definition of unbiased monoidal category. Definition 3. An unk1 kn 1 a1 , The bark of the associativity axiom is far worse than its bite. All it says is that any two ways of removing brackets are equal: for instance, that the diagram commutes. In an unbiased strict monoidal category, the coherence axioms naturality, associativity and identity hold automatically.
Clearly, unbiased strict monoidal categories are in one-to-one correspondence with ordinary strict monoidal categories. We have given a completely explicit definition of unbiased monoidal category, but a more abstract version is possible. See Kelly and Street, , for instance; terminology varies between authors. This may easily be verified. A different abstract way of defining unbiased monoidal category will be explored in Section 3.
The next step is to define maps between lax and unbiased monoidal categories. Again, we could use the language of 2-monads to do this, but opt instead for an explicit definition. We remarked in 3. There are also the evident identities. For all three categories in the bottom respectively, middle or top row, the objects are small strict respectively, unbiased or lax monoidal categories.
For all three categories in the left-hand respectively, middle or right-hand column, the maps are strict respectively, weak or lax monoidal functors. It is easy to check that the three categories in the bottom row are isomorphic to the corresponding three categories in the classical definition. Of the nine categories, the three on the bottom-left to top-right diagonal are the most conceptually natural: a level of strictness has been chosen and stuck to. In this chapter our focus is on the middle entry, UMonCatwk , where everything is weak.
There the top row is obscured, as there is no very satisfactory way to laxify the classical definition of monoidal category. To complete the picture, and to make possible the definition of equivalence of unbiased monoidal categories, we define transformations. This time, there is only one possible level of strictness. Monoidal transformations can be composed in the expected ways, so that the nine categories above become strict 2-categories. In particular, UMonCatwk is a 2-category, so 1. As we might expect from the case of classical monoidal categories 1.
Proposition 3. Proof Mutatis mutandis, this is the same as the proof of 1. Theorem 3. Proof Let A be an unbiased monoidal category. By the last proposition, this is enough. An object of st A is a finite sequence a1 ,. A map a1 ,. The tensor in st A is given on objects by concatenation: a11 ,. It is absolutely straightforward and not too arduous to check that st A with this tensor forms a strict monoidal category. P is certainly full and faithful. The one here is of the latter type. The proof above was adapted from Joyal and Street , p.
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There the st construction was done for classical monoidal categories A, for which the situation is totally different. What happens is that the n-fold tensor product used in the definition of both st A and P must be replaced by some derived, non-canonical, n-fold tensor product, such as a1 ,. In contrast, the proof that every unbiased monoidal category is equivalent to a strict one is easy, short, and needs no supporting results. This does not, however, provide a short cut to proving any kind of coherence result for classical monoidal categories.
In the next section we will see that unbiased and classical monoidal categories are essentially the same, and it then follows from Theorem 3. However, as with any serious undertaking involving classical monoidal categories, the proof that they are the same as unbiased ones is close to impossible without the use of a coherence theorem. But what happens if we take a notion of monoidal category in which there is an n-fold tensor 3 Notions of monoidal category product for each n lying in some other subset of N?
Just as long as we add in enough coherence isomorphisms to ensure that any two n-fold tensor products built up from the given ones are canonically isomorphic, this new notion of monoidal category ought to be essentially the same as the classical notion. This turns out to be the case. In the introduction to this chapter, I argued that comparing definitions of monoidal category is important for understanding higher-dimensional category theory. Consider the definition of monoid. Usually a monoid is defined as a set equipped with a binary operation and a nullary operation, satisfying associativity and unit equations.
However, this process of categorification is dependent on presentation. So different presentations of the same 0-dimensional theory monoids give, under this process of categorification, different 1-dimensional theories of monoidal category. Thus, our purpose is to show that in this particular situation, the presentation-sensitivity of categorification disappears when we work up to equivalence.
More generally, a fully-developed theory of weak n-categories might include a formal process of weakening, which would take as input a theory of strict structures and give as output a theory of weak structures. If the weakening process depended on how the theory of strict structures was presented, then 3. We might hope so; but who knows? Here it is in outline. Also recall from p. Coherence axioms for a weak map of R-algebras Now we can state the results. It takes almost no calculation to see that there is a - UMonCatlax.
To see that it is an isocanonical functor 1-MonCatlax morphism requires calculations using the coherence axioms for an unbiased monoidal category. Restricting to the weak and strict cases is simple. Proof See Appendix B. The strategy is just the same as in 3. This is why 3. The unbiased coherence theorem 3. So plausibility is an obvious minimal requirement. As can be seen from the explicit description of free operads on p.
Cat-Operad naturally has the structure of a 2-category, because Cat does; and if two objects of Cat-Operad are equivalent then so are their images under both Alglax and Algwk. The result follows. However, the 2-categorical details are rather tiresome to check and the reader may prefer to avoid them. The main reason for including them below is that they reveal why the theorem holds at the lax and weak levels but not at the strict level.
To make the necessary distinctions, superscripts have been added to each actn naming the algebra concerned. Proof 3. All of the results above can be repeated with monoidal transformations brought into the picture.
That we did not need to mention monoidal transformations in order to prove the equivalence of the various notions of monoidal category says something about the strength of our equivalence result. So Theorem 3. Some people would, therefore, like to create a world where there are no coherence axioms at all — or anyway, as few as possible.
Whatever the merits of this aspiration, it is a fact that it can be achieved in some measure; that is, there exist approaches to various higher categorical structures that involve almost no coherence axioms. In this chapter we look at two different such approaches for monoidal categories. The first exploits the relation between monoidal categories and multicategories.
I will describe it in some detail. The second is based on the idea of the nerve of a category, and has its historical roots in the homotopy-algebraic structures known as -spaces. Since it has less to do with the main themes of this book, I will explain it more sketchily. So, we start by looking at monoidal categories vs. One might argue that multicategories are conceptually more primitive than monoidal categories: that an operation taking several inputs and producing one output is a more basic idea than a set whose elements are ordered tuples. Now, the monoidal category of vector spaces contains no more or less information than the multicategory of vector spaces; given either one, the other can be derived in its entirety.
And this is what we do. Formally, we have a functor V assigning to each monoidal category its underlying multicategory; we want to show that V gives an equivalence between monoidal categories and some subcategory R of Multicat; and we want, moreover, to describe R. Before starting we have to make precise something that has so far been left vague. In Example 2. We answer it now, and so obtain a precise definition of the functor V. If A is a strict monoidal category then the expression makes perfect sense.
Bringing into play the coherence maps of A, we can also define composition in C, and so obtain the entire multicategory structure of C without trouble; we arrive at a functor V : UMonCatwk - Multicat. What if we insist on starting from a classical monoidal category? We can certainly obtain a multicategory by passing first from classical to unbiased and then applying the functor V just mentioned. But there is also a way of passing from classical monoidal categories to multicategories without making any arbitrary choices. Let A be a classical monoidal category.
Now define a multicategory C by taking an object to - a to be a fambe, as usual, just an object of A, and a map a1 ,. This yields another functor - Multicat. The results for which we hoped, exhibiting monoidal categories as special multicategories, can be phrased in various different ways. A multicategory is representable if it admits a representation.
See Section 2. Under these equivalent conditions, a map in C is universal if and only if it is pre-universal. The functor V is faithful, and therefore provides an equivalence between UMonCatwk or MonCatwk as you prefer and the subcategory RepMulti of Multicat consisting of the representable multicategories and the universalpreserving maps. The only subtle point here is that the existence of a pre-universal map for every given domain is not enough to ensure that the multicategory comes from a monoidal category.
A specific example appears in Leinster , but the point can be explained here in the familiar context of vector spaces. Then it does not follow for purely formal reasons that the tensor product is associative up to isomorphism : one has to use some actual properties of vector spaces.
The energetic reader with plenty of time on her hands will have no difficulty in proving Theorem 3. So in a sense that is an end to the matter: monoidal categories can be recognized as multicategories with a certain property, and monoidal functors similarly, all as hoped for originally. We can, however, take things further. With just a little more work than a direct proof would involve, Theorem 3. This theory is a fairly predictable extension of the theory of fibrations of ordinary categories, and the result of which 3.
The basic theory of fibrations of multicategories is laid out in Leinster , which culminates in the deduction of 3. Here is the short story. For any category D, a fibration or really, opfibration over D - Cat. We looked at is essentially the same thing as a weak functor D the case of discrete fibrations in Section 1. With appropriate definitions, a similar statement can be made for multicategories. Taking D to be the terminal - 1 is a fibration exactly multicategory 1, we find that the unique map C when C is representable, and that weak functors 1 - Cat are exactly unbiased monoidal categories.
Universal and pre-universal maps in C correspond to what are usually called cartesian and pre-cartesian maps. So a representable multicategory is essentially the same thing as a monoidal category. The idea can be explained as follows. Recall that every small category has a nerve, and that this allows categories to be described as simplicial sets satisfying certain conditions.
A functor op - Set is called a simplicial set. More generally, a functor op - E is called a simplicial object in E. This applies in particular to the ordered sets [n], and so we may define the nerve N A of a small category A as the simplicial set NA : op [n] - Set Cat [n], A. Hence Cat is equivalent to the full subcategory of [ op , Set] whose objects are those simplicial sets X isomorphic to N A for some small category A. There are various intrinsic characterizations of such simplicial sets X. We do not need to think about the general case for now, only the special case of one-object categories, that is, monoids.
Proof Straightforward. Monoids are, therefore, the same thing as simplicial sets satisfying any of the conditions a — c. The proposition can be generalized: replace Set by any category E possessing finite products to give a description of monoids in E as certain simplicial objects in E. Their converses are straightforward inductions. HMonCat op is the category of homotopy monoidal categories and natural transformations between them. It can be shown that any loop space provides an example. Before I say anything about the comparison with ordinary monoidal categories, let me explain another route to the notion of homotopy monoidal category.
Let D be the augmented simplex category 1. The fact that is D with the object 0 removed is a red herring: we will not use this connection between and D. For given a weak monoidal functor D - E, the image of any monoid in D is a monoid in E, and in particular, the object 1 of D has a unique monoid structure, giving a monoid in E. You might object that this is useless as a definition of monoidal category, depending as it does on pre-existing concepts of monoidal category and colax monoidal functor.
There are several responses. One is that we could give an explicit description of what a colax monoidal functor D - Cat is, along the lines of the traditional description of cosimplicial objects by face and degeneracy maps, and this would eliminate the dependence. A third is that while 3. This exhibits an advantage of the D approach over the approach: we can use it to discuss homotopy monoids in monoidal categories where the tensor is not cartesian product.
Much more on this can be found in my b and a.
It can also be proved directly in the following way. This fails when E is a monoidal category whose tensor product is not the cartesian product. It could be argued that in this situation, it would be better to define a simplicial object in E not as a functor op - E, but rather as a colax monoidal func- E.
For example, it was the colax monoidal version that made tor D possible the definition of homotopy differential graded algebra referred to above. In Proposition 3. So by using condition b of Proposition 3. Our two ways of approaching homotopy monoidal categories, via either nerves in a finite product category or monoids in a monoidal category, are 3 Notions of monoidal category therefore equivalent in a strong sense. The next question is: are they also equivalent to the standard notion of monoidal category?
To define X we will need to use unbiased bicategories defined in the next section. Also, the unbiased monoidal category A may be regarded as an unbiased bicategory with only one object. The homotopy monoidal category X is defined by taking X [n] to be the category whose objects are all weak functors [n] - A of unbiased bicategories and whose maps are transformations of a suitably-chosen kind. This is just a categorification of the usual nerve construction. The category A is X . The coherence axioms follow from the fact that we chose adjoint equivalences — that is, they follow from the triangle identities.
It is fairly easy to see that composing one functor with the other does not yield a functor isomorphic to the identity, either way round. I believe, however, that if HMonCat is made into a 2-category in a suitable way then each composite functor is equivalent to the identity.
This would mean that the notion of homotopy monoidal category is essentially the same as the other notions of monoidal category that we have discussed. To summarize the chapter so far: we have formalized the idea of nonstrict monoidal category in various ways, and, with the exception of homotopy 3. There are still more notions of monoidal category that we might contemplate — for instance, the anamonoidal categories of Makkai — but we leave it at that.
This is usually at the expense of setting up some slightly more sophisticated language, which is why things so far have been done for monoidal categories only. Here we run through what we have done for monoidal categories and generalize it to bicategories, noting any wrinkles. Unbiased bicategories Definition 3. Remarks 3. A lax bicategory with exactly one object is, of course, just a lax monoidal category, and similarly for the weak and strict versions. Unbiased strict 2-categories are in one-to-one correspondence with ordinary strict 2-categories, easily.
As in the case of monoidal categories, there is an abstract version of the definition of unbiased bicategory phrased in the language of 2-monads. We met the ordinary category Cat-Gph earlier 1. An unbiased bicategory is exactly a weak algebra for this 2-monad, and the same applies in the lax and strict cases. Although we could, say, consistently choose the first option and so arrive at a weak functor Hom, this is an arbitrary choice.
So there is no canonical Hom-functor in the classical world. For instance, UBicatwk is the category of unbiased bicategories and weak functors. Differences from the theory of unbiased monoidal categories emerge when we try to define transformations between functors between unbiased bicategories. This should not come as a surprise given what we already know in the classical case about transformations and modifications of bicategories vs.
More mysteriously, there seems to be no satisfactorily unbiased way to formulate a definition of transformation or modification for unbiased bicategories; it seems that we are forced to grit our teeth and write down biased-looking definitions. This done, we obtain a notion of biequivalence of unbiased bicategories.
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Just as in the classical case 1. The st construction for monoidal categories 3. Algebraic notions of bicategory Here we take the very general family of algebraic notions of monoidal category considered in Section 3. It is therefore only the third step that we need to change here. This defines a category CatAlgwk R. CatAlgwk The lax and strict cases are, of course, done similarly. The proofs are as in the monoidal case 3. Then, there is the irrelevance of signature theorem, analogous to 3.
Again, the proof is essentially unchanged. We saw on p. In the case of bicategories it is two levels better, because modifications are not needed either. Non-algebraic notions of bicategory I will say much less about these. Take representable multicategories first. All the results on monoidal categories as representable multicategories can be extended unproblematically to bicategories, and the same goes for the theory of fibrations of multicategories Leinster, Consider, finally, homotopy monoidal categories.
A homotopy bicategory - Cat satisfying conditions similar to can be defined as a functor op but, of course, looser than those in Proposition 3. The new conditions involve pullbacks rather than products, and can be found by considering nerves of categories in general instead of just nerves of monoids. See Section Notes The notion of unbiased monoidal category has been part of the collective consciousness for a long while Kelly, , and 9. Around 30 years ago Kelly and his collaborators began investigating 2-monads and 2-dimensional algebraic theories see Blackwell, Kelly and Power, , for instance , and they surely knew that unbiased monoidal categories were equivalent to classical monoidal categories in the way described above, although I have not been able to find anywhere this is made explicit before my own b.
If I am interpreting the somewhat daunting literature correctly, one can also deduce from it some of the more general results in Section 3.
Here, in contrast, we have a quick and natural route to the results. Essentially, operads take the place of 2-monads as the means of describing an algebraic theory on Cat. In order even to state the problem at hand, in any approach, the explicit or implicit use of operads seems inevitable; an advantage of our approach is that we use nothing more. Housman In a category, an arrow has a single object as its domain and a single object as its codomain.
In a multicategory, an arrow has a finite sequence of objects as its domain and a single object as its codomain. What other things could we have for the domain of an arrow, keeping a single object as the codomain? Could we, for instance, have a tree or a many-dimensional array of objects? In different terms logic or computer science : what can the input type of an operation be?
In this chapter — the central chapter of the book — we answer these questions. We formalize the idea of an input type, and for each input type we define a corresponding theory of operads and multicategories. From now on, operads and multicategories as defined in Chapter 2 will be called plain operads and plain multicategories.
The formal strategy is as follows. Here C0 is the set of all objects, C1 is the set of all arrows, dom assigns to an arrow the sequence of objects that is its domain, and cod assigns to an arrow the single object that is its codomain. The crucial point is that this formalism works for any monad T on any category E, as long as E and T satisfy some simple conditions concerning pullbacks. So when T is the identity monad on Set, a T -multicategory is an ordinary category, and when T is the free-monoid monad on Set, a T multicategory is a plain multicategory.
We also define T -operads. As in the plain case, these are simply T multicategories C with only one object — or formally, those in which C 0 is a terminal object of E. There is a canonical notion of an algebra for a T -multicategory. Like their plain counterparts, generalized operads and multicategories can be regarded as algebraic theories single- and multi-sorted, respectively ; algebras are the accompanying notion of model. We start Section 4.
Next Section 4. There are many examples throughout, but some of the most important ones are done only very briefly; we do them in detail in later chapters. A category E is cartesian if it has all pullbacks. A functor E - F is cartesian if it preserves pullbacks. TA - TB Tf is a pullback. Remarks 4. All of our examples of cartesian categories will have a terminal object, hence all finite limits. Cartesian categories, cartesian functors and cartesian natural transformations form a subcategory CartCat of Cat, and a cartesian monad is exactly a monad in CartCat.
See p. The rest of the section is examples. Example 4. Certainly E is cartesian. An easy calculation shows that the monad T is cartesian too: Leinster a, 1. Let us show the latter in detail, using an argument of Weber , 2. Write [a1 ,. Hence the image of the square under T is not a pullback. Carboni and Johnstone proved this first ; an alternative proof is in Appendix C. The result implies that the free monoid monad 4. Further examples appear below. Now, this is not a strongly regular presentation of the theory, and in fact there is no strongly regular presentation, but T is a cartesian monad.
So not all cartesian monads on Set arise from strongly regular theories. These assertions are proved in Example C. As we saw in Section 2. That T is cartesian follows from theory we develop later 6. If we had taken strict symmetric monoidal categories instead — in other words, insisted that the symmetries were strict — then we would just be looking at commutative monoids in Cat, and the monad would fail to be cartesian by the argument of Example 4. It follows from later theory 6. All of these monads are cartesian. In particular, we saw that a strict n- Set with extra tuple category could be described as a functor Hn op structure, where H is as in 4.
The forgetful functor from the category of strict n-tuple categories and strict maps between them to the functor category [ Hn op , Set] has a left adjoint, the adjunction is monadic, and the induced monad on [ Hn op , Set] is cartesian. This can be shown by a similar method to that used for strict n-categories in Appendix F. Alert readers may have noticed that nearly every one of the above examples of a cartesian monad on Set is, in fact, the free-algebra monad for a certain plain operad. This is no coincidence, as we discover in Section 6. The strategy for making these definitions is as described in the introduction to this chapter, dressed up a little: instead of handling the data and axioms for a T -multicategory directly, we introduce a bicategory E T and define a T multicategory as a monad in E T.
This amounts to the same thing, as we shall see. All we are doing is generalizing the description of a small category as a monad in the bicategory of spans: 5. Since the choice of pullbacks in E was arbitrary, it is inevitable that composition of 1-cells in E T does not obey strict associativity or unit laws. That it obeys them up to isomorphism is a consequence of T being cartesian. Changing the choice of pullbacks in E only changes the bicategory E T 4 Generalized operads and multicategories: basics up to isomorphism in the category of bicategories and weak functors : see p.
Here is the most important definition in this book. It is due to Burroni , and in the form presented here uses the notion of monad in a bicategory p. Definition 4. A T -multicategory is a monad in the bicategory E T. Just as a T -multicategory is a generalized category, a T -operad is a generalized monoid. The category of T -graphs is written T -Graph. The category of T -multicategories and maps between them is written T -Multicat. The full subcategory consisting of T -operads is written T -Operad.
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When the extra clarity is needed, we will refer to T -multicategories as E, T multicategories and to the category they form as E, T -Multicat; similarly for operads and graphs. We now look at some examples of generalized multicategories. These examples are only described briefly here, with proper discussions postponed to later chapters. We start with the two motivating cases. More generally, if E is any cartesian category then E, id -multicategories are categories in E and E, id -operads are monoids in E.
A T -multicategory structure on a T -graph C therefore consists of a function comp as in Fig. So a T -multicategory is just a plain multicategory; indeed, there are equivalences of categories T -Multicat Multicat, T -Operad Operad. A first attempt might be to take the free commutative monoid monad T on Set. A better idea is to take the free symmetric strict monoidal category monad on Cat, thus replacing identities by isomorphisms: see 4.
For instance: a. Composition in Cat is usual composition of functors, taking opposites where necessary. That Cat does form a T -multicategory and not just a plain multicategory is a statement about the behaviour of contravariance with respect to products and functors. T -multicategories are the same as in the previous example except that substitution reverses order. Loop spaces give an example. Fix a space X with a basepoint x. So there is a resulting T -multicategory whose objects are loops and whose maps encode all the information about concatenation of loops, homotopy classes of homotopies between loops, and reversal of loops.
A T dom codC of sets and functions. The unary arrows form a category D, and the nullary arrows define a functor Y : D - Set in which Y b is the set of nullary arrows with codomain b. So a T -multicategory is the same thing as a small category D together with a functor Y : D - Set, and in particular, a T -operad is a monoid acting on a set.
In fact, T -Multicat is equivalent to the category whose objects are discrete opfibrations between small categories and whose morphisms are commutative squares. Then a T -multicategory is exactly a plain multicategory with no nullary arrows. Some authors prefer to exclude the possibility of nullary operations: see the Notes to Chapter 2. Here we regard a monoid as a one-object category: p. For a specific example, let M be the large monoid of all cardinals under multiplication.
Let C be the large category of fields and homomorphisms between them which are, of course, all injective. We have seen that when T is the identity monad on Set, a T -operad is exactly a monoid. We have seen that when T is the free monoid monad, a T operad is exactly a plain operad. We have seen that when T is the free plain operad monad, a T -operad is as just described. The T -operads of the present example are described in Section 7. In this context labels appear on leaves rather than vertices, and trees are amalgamated by grafting leaves to roots rather than by substituting trees into vertices.
A T -multicategory consists 4. Put another way, a T -multicategory C is a plain multicategory in which the hom-sets are graded by trees: to each a1 ,. However, T -multicategories are not the same thing as multicategories enriched in Top, as in a T -multicategory C there is a topology on the set of objects. A multicategory enriched in Top is a T -multicategory in which the set of objects has the discrete topology. This difference should not be found surprising or disappointing: it exhibits the tension between internal and enriched category theory, previously discussed in Section 1.
We saw some examples of these in Section 3. Any symmetric multicategory A gives rise to a T multicategory C as follows. The category C0 is discrete, with the same objects as A. The rest of the structure of C is obvious. In particular, symmetric operads are special T -operads. See also the comments on p. This is the subject of Chapter 5. In Section A weak n-tuple category might be defined as an algebra for a certain T -operad.
We look at weak double categories in Section 5. In the situation of ordinary categories rather than generalized multicategories, not only is there the concept of a functor between categories, but also there are the concepts of a module between categories p. The same goes for plain multicategories, as we saw in Section 2. In fact, both of these concepts make sense for T multicategories in general.
We meet the definitions and see the connection between them in Section 5. We prove this in Section 6. That the category E, T -Multicat is independent up to isomorphism of the choice of pullbacks in E follows from this functoriality by considering the identity map from E with one choice of pullbacks to E with another choice of pullbacks.
Here we meet algebras for generalized operads and multicategories. There are several ways of framing the definition. I have chosen the one that seems most useful in practice; two alternatives are discussed in Sections 6. Let us begin by considering algebras for a plain multicategory C. These are maps from C into the multicategory of sets, but this is not much use for generalization as there is not necessarily a sensible T -multicategory of sets for arbitrary cartesian T. However, as we saw in Chapter 2, algebras for plain multicategories can be described without explicit reference to the multicategory of sets.