Research Article | Open Access
A Morphism Double Category and Monoidal Structure
We provide a recipe for “fattening” a category that leads to the construction of a double category. Motivated by an example where the underlying category has vector spaces as objects, we show how a monoidal category leads to a law of composition, satisfying certain coherence properties, on the object set of the fattened category.
1. Introduction and Geometric Background
The interaction of point particles through a gauge field can be encoded by means of Feynman diagrams, with nodes representing particles and directed edges carrying an element of the gauge group representing parallel transport along that edge. If the point particles are replaced by extended one-dimensional string-like objects, then the interaction between such objects can be encoded through diagrams of the form (1) where the labels and describe classical parallel transport and , which may take values in a different gauge group, describes parallel transport over a space of paths.
We will now give a rapid account of some of the geometric background. We refer to our previous work  for further details. This material is not logically necessary for reading the rest of this paper but is presented to indicate the context and motivation for some of the ideas of this paper.
Consider a principal -bundle , where is a smooth finite dimensional manifold and a Lie group, and a connection on this bundle. In the physical context, may be spacetime, and describes a gauge field. Now consider the set of piecewise smooth paths on , equipped with a suitable smooth structure. Then, the space of -horizontal paths in forms a principal -bundle over . We also use a second gauge group (that governs parallel transport over path space), which is a Lie group along with a fixed smooth homomorphism and a smooth map such that each is an automorphism of , such that for all and . We denote the derivative by , viewed as a map , and denote by , to avoid notational complexity. Given also a second connection form on and a smooth -equivariant vertical -valued -form on , it is possible to construct a connection form on the bundle where is the -valued -form on specified by which is a Chen integral.
Consider a path of paths in specified through a smooth map where each is -horizontal and the path is -horizontal. Let . The bi-holonomy is specified as follows: parallel translate along by , then up the path by , back along -reversed by and then down by , then the resulting point is
The following result is proved in .
Theorem 1. Suppose that is smooth, with each being -horizontal and the path being -horizontal. Then, the parallel translate of by the connection along the path , where , results in with being the “bi-holonomy” specified as in (7), and solving the differential equation with initial condition being the identity in .
Consider the category whose objects are fibers of a given vector bundle over and whose arrows are piecewise smooth paths in (up to “backtrack equivalence”; for more on this notion see ) along with parallel transport operators, by a connection , along such paths. Note that all arrows are invertible. In Figure 1, is the vector space which is the fiber over the corresponding point . For the path , there is a parallel transport operator . Next, if is a path from the base of the fiber to the base of , then there is a corresponding parallel transport operator .
A “higher” morphism is obtained from any suitably smooth path of paths, starting with the initial path and ending with (again backtracks need to be erased). Using the connection , this produces parallel transport operators and paths and . Moreover, another connection and -form , along with a path of paths lead to a linear map , where is the vector space of all linear maps . We view this, in a “first approximation,” as a morphism from the object to the object (say, mapping all paths from to to the path ). In this paper, we will not develop this framework in full detail (that would build on the theory from our earlier work ) but focus on more algebraic aspects and other purely algebraic issues (such as monoidal structures).
Instead of vector bundles, one could also work with the principal bundle itself, taking as objects of a category all the fibers of the bundle and as morphisms the -equivariant bijections , where and are fibers of , over points and , and paths running from to .
The interface between gauge theory and category theory, in various forms and cases, has been studied in many works, for instance [1, 3–7]. In the present paper, we extract the abstract essence of some of these structures in a category theory setting, leaving the differential geometry behind as the concrete context. We abstract the process of passing from the point-particle picture to a string-like picture to a functor which generates a category from a category . Proposition 5 describes properties of a natural product operation on the objects of when is a monoidal category. An excellent review of monoidal categories in relation to topological quantum field theory can be found in . Symmetric monoidal bicategories are discussed in  in a context different from ours.
2. The Fat Category
Let be a category. We define a new category as follows. The objects of are the morphisms of . A morphism in from the object to the object consists of morphisms and in , along with a set-mapping which maps to as follows: (In a later section we require that the hom-sets themselves also have algebraic structure that should be preserved by such .) Here is a diagram displaying a morphism of : (13) It is clear that this does specify a category, which we call the fat category for (composition is “vertical,” with successive s composed). Sometimes it will be easier on the eye to write for . Thus, diagram (13) can also be displayed as (15) The composition of morphisms in is defined “vertically” by drawing the diagram of below that of and composing vertically downward.
Commutative diagrams in lead to morphisms of in a natural way and yield a subcategory of that is recognizable as the “category of arrows” [10, §I.4], sometimes denoted as .
Lemma 2. Any commutative diagram (16) in , in which is an isomorphism, generates a morphism in , (18) where Moreover, if (20) is a commutative diagram in , where and are isomorphisms, then the composite of the induced morphisms, is the morphism in induced by the commutative diagram (22)
3. A Double Category of Isomorphisms
Let be the category whose objects are the invertible arrows of and whose arrows are the arrows (23) in in which the verticals and are also isomorphisms in . This is, for all purposes here, as good as assuming that all arrows of are invertible, since we will only work with such arrows. In the geometric context, the arrows represent parallel transports and so the invertibility assumption is natural. The mapping is motivated by the “surface” parallel transport mentioned briefly in (10).
Let us define horizontal composition of morphisms in as follows: (24) where the composition is defined only when , and is given by Note that satisfies Consider now the following diagram: (27)
Proposition 3. The morphisms of form a double category under the laws of composition and in the sense that for diagram (27), with notation as explained above, for all morphisms , , , in for which the compositions on both sides of (28) are meaningful.
Proof. Denote by the morphism of specified by the upper left square in (27), by the morphism specified by the upper right square, by the morphism specified by the lower left square, and, lastly, by the morphism specified by the lower right square.
Let . Then, Comparing (29) and (30), we have the claimed equality (28).
Then, equipped with both laws of composition and is a double category . In the geometric context, this is expressed as a flatness condition for the connection described in the Introduction; for more, see, for instance, [1, 3].
4. Enrichment for Morphisms
We continue with the notation and structures as before; is a category and is the “fat” category described in Section 2. Now let be a subcategory of , having the same objects but possibly fewer morphisms. The idea is that the hom-sets in could have additional structure; for example, if has only one object , a fiber of a vector bundle, then is a group under composition. The morphisms of could be required to be group automorphisms. We require that for any objects of and isomorphism , the map is a morphism of .
Proposition 4. Let be any subcategory of having the same objects as , and satisfying the condition (31) as explained above. Both horizontal and vertical composites of morphisms in are in . Thus, is a double category.
Proof. The consistency condition between horizontal and vertical compositions has already been checked in Proposition 3. Thus, we need only to check that horizontal composition, specified in (25) as is a morphism of , for all invertible and all , morphisms in . Observe that where the notation is as in (31). Thus, is a composite of morphisms in .
5. Monoidal Structures
In this section we will explore some algebraic structural enhancements of the fattened category . The discussion is motivated by intrinsic algebraic considerations, but we discuss briefly now the relationship with the geometric context.
Consider the very special case where is the category with only one object , the fiber over a fixed point in a vector bundle, and a morphism is a an ordered pair as follow: consisting of a piecewise smooth loop based at (with backtracks erased) along with a linear map representing parallel transport around the loop. For in this special case, a morphism arises from paths of paths along with a linear map , where is the vector space of all linear maps . Each hom-set is a monoid: composition is given by concatenation of loops along with ordinary composition of linear maps in : where is the loop followed by the loop . (Since this discussion is primarily for motivation, we leave out technical details of “backtrack erasure.”)
Turning to the abstract setting, we assume henceforth that is a monoidal category. This means that there is a bifunctor and there is an identity object in for which certain natural coherence conditions hold as we now describe. In addition, there exists a natural isomorphism , the associator, which associates to any of the objects , , of an isomorphism such that the following diagram commutes:(39)
There are also natural isomorphisms and , the left and right unitors, associating to each object in morphisms such that (41) commutes for all objects and in .
Note that naturality means there are certain other conditions as well. For example, that the left unitor is a natural transformation means that for any morphism in the diagram (42) commutes; here, in the upper horizontal arrow, is the unique morphism in .
We now define a product on the objects of as follows:
In the fat category , we then have associators and unitors as follows. First, the unit is where denotes the identity object in and the identity map on . We will often denote also simply as , the meaning being clear from context. For any object , there is the left unitor (46) where the mapping takes to , as follows from the remarks made above for (42). The right unitor is (48) where Again, this is indeed a morphism in by essentially the same argument that was used above in (46) for the left unitor.
The associator in is given as follows. Consider objects in , for . The fact that is a natural transformation means that the diagram (50) is commutative. Hence, by the first half of Lemma 2, this induces a morphism(51)in . In fact, is an isomorphism since the vertical arrows in (50) are isomorphisms.
We prove the coherence condition for unitors. For this we have the following diagram: (52) The two triangles at the two ends of this “trough” commute because of coherence in , the top rectangle also commutes because of the naturality of . Then, it is entertaining to check that the two rectangular “slanted sides” are also commutative. In fact, the slant side on the left is as a morphism in , and the slant side on the right is Thus, viewed as a diagram in , the “trough” looks like(55)Since the trough commutes in , so does its avatar (55) in , thanks to the second half of Lemma 2. This verifies the coherence property in involving the unitors.
Now, we turn to coherence for the associators. In the following diagram, where we leave out the products for ease of viewing, the slant arrows are all tensor products of the and the horizontal and vertical arrows are various associators:(56)
Coherence in the monoidal category implies that the two rectangles at the end of this box are commutative, as mentioned earlier. Naturality of the associator implies that the top, bottom, and sides are also commutative. Thus, the entire diagram is commutative. If we abbreviate the objects in as for , we can read the full diagram as a diagram in the category as follows: (58) As a diagram in , this is commutative, by Lemma 2. This establishes coherence of the associator in .
We have completed the proof of Proposition 5.
Proposition 5. Suppose that is a monoidal category and let be the category specified above in the context of (11). Then, with tensor product as defined in (44), satisfies all conditions of a monoidal category at the level of objects.
6. Concluding Remarks
In this paper, we have presented certain “fattened” categories , , and constructed out of a given category ; the morphisms of form a double category. It is shown how a monoidal structure on induces a multiplication on the objects of that satisfies certain coherence properties.
A. Lahiri acknowledges research support from Department of Science and Technology, India, under Project no. SR/S2/HEP-0006/2008. A. N. Sengupta acknowledges research support at an earlier stage of this work from US NSF Grant DMS-0601141. The authors have used macros developed by Paul Taylor to draw all diagrams.
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