A Morphism Double Category and Monoidal Structure

A double category is constructed from a `fattened' version of a given category, motivated in part by a context of parallel transport. We also study monoidal structures on the underlying category and on the fattened category.


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 onedimensional string-like objects, then the interaction between such objects can be encoded through diagrams of the form where the labels and describe classical parallel transport and ℎ, 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 [1] 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 P of piecewise smooth paths on , equipped with a suitable smooth structure. Then, the space P ofhorizontal paths in forms a principal -bundle over P . 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 ℎ, ℎ ∈ . 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 2-form on , it is possible to construct a connection form ( , ) on the bundle P Algebra where is the -valued 1-form on P specified by which is a Chen integral. Consider a path of paths in specified through a smooth mapΓ where eachΓ is -horizontal and the path →Γ(0, ) is -horizontal. Let Γ = ∘Γ. The bi-holonomy ( , ) ∈ is specified as follows: parallel translateΓ(0, 0) along Γ 0 | [0, ] by , then up the path Γ | [0, ] by , back along Γ -reversed by and then down Γ 0 | [0, ] by , then the resulting point isΓ The following result is proved in [1].
Consider the category C 0 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 [2]) along with parallel transport operators, by a connection , along such paths. Note that all arrows are invertible. In Figure 1, 1 is the vector space which is the fiber over the corresponding point 1 . For the path 1 , there is a parallel transport operator 1 : 1 → 1 . Next, if 2 is a path from the base of the fiber 2 to the base of 2 , then there is a corresponding parallel transport operator 2 : 2 → 2 . A "higher" morphism 1 → 2 is obtained from any suitably smooth path of paths, starting with the initial path 1 and ending with 2 (again backtracks need to be erased). Using the connection , this produces parallel transport . We view this, in a "first approximation, " as a morphism from the object Mor( 1 , 1 ) to the object Mor( 2 , 2 ) (say, mapping all paths from 1 to 1 to the path 2 ). In this paper, we will not develop this framework in full detail (that would build on the theory from our earlier work [1]) 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 C 0 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][4][5][6][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 F(C) from a category C. Proposition 5 describes properties of a natural product operation on the objects of F(C) when C is a monoidal category. An excellent review of monoidal categories in relation to topological quantum field theory can be found in [8]. Symmetric monoidal bicategories are discussed in [9] in a context different from ours.

The Fat Category
Let C be a category. We define a new category F(C) as follows. The objects of F(C) are the morphisms of C. A morphism in which maps 1 to 2 as follows: Algebra 3 (In a later section we require that the hom-sets Mor( , ) themselves also have algebraic structure that should be preserved by such ℎ.) Here is a diagram displaying a morphism of F(C): It is clear that this does specify a category, which we call the fat category for C (composition is "vertical, " with successive ℎs composed). Sometimes it will be easier on the eye to write for → . Thus, diagram (13) can also be displayed as The composition V∘ of morphisms in F(C) is defined "vertically" by drawing the diagram of V below that of and composing vertically downward.
Commutative diagrams in C lead to morphisms of F(C) in a natural way and yield a subcategory of F(C) that is recognizable as the "category of arrows" [10, §I.4], sometimes denoted as Arr(C).

Lemma 2. Any commutative diagram
in C, in which 1 is an isomorphism, generates a morphism in F(C), (20) is a commutative diagram in C, where 1 and 1 are isomorphisms, then the composite of the induced morphisms, is the morphism in F(C) induced by the commutative diagram

A Double Category of Isomorphisms
Let F(C) 0 be the category whose objects are the invertible arrows of C and whose arrows are the arrows in F(C) in which the verticals 1 and 2 are also isomorphisms in C. This is, for all purposes here, as good as assuming that all arrows of C 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 ℎ is motivated by the "surface" parallel transport mentioned briefly in (10).
where the composition is defined only when 1 = 2 , and ℎ is given by Note that ℎ satisfies Consider now the following diagram: The morphisms of F(C) 0 thus have two laws of composition: ∘ and ∘ . As we see below, these compositions obey a consistency condition (28), which thereby specifies a double category [10, 11, §I.5].
Proposition 3. The morphisms of F(C) 0 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 , , ℎ , ℎ in Mor (F(C) 0 ) for which the compositions on both sides of (28) are meaningful.
Proof. Denote by ℎ the morphism of F(C) 0 specified by the upper left square in (27), by ℎ 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.
Then, F(C) 0 equipped with both laws of composition ∘ and ∘ is a double category [11]. 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].

Enrichment for Morphisms
We continue with the notation and structures as before; C is a category and F(C) is the "fat" category described in Section 2. Now let F(C) 1 be a subcategory of F(C) 0 , having the same objects but possibly fewer morphisms. The idea is that the hom-sets in F(C) could have additional structure; for example, if C has only one object , a fiber of a vector bundle, then Mor( , ) is a group under composition. The morphisms of F(C) 1 could be required to be group automorphisms. We require that for any objects , , of C and isomorphism : → , the map 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 F(C) 1 , for all invertible 1 ∈ Mor( 1 , 1 ) and all ℎ : where the notation is as in (31). Thus, ℎ is a composite of morphisms in F(C) 1 . Algebra 5

Monoidal Structures
In this section we will explore some algebraic structural enhancements of the fattened category F(C) 0 . 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 C 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: 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 C is a monoidal category. This means that there is a bifunctor and there is an identity object 1 in C 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 C an isomorphism such that the following diagram commutes: There are also natural isomorphisms and , the left and right unitors, associating to each object in C morphisms such that commutes for all objects and in C.
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 C the diagram commutes; here, in the upper horizontal arrow, 1 is the unique morphism 1 : 1 → 1 in C.
We now define a product on the objects of F(C) Obj (F (C)) × Obj (F (C)) → Obj (F (C)) : 6 Algebra as follows: In the fat category F(C), we then have associators and unitors as follows. First, the unit is where 1 denotes the identity object in C and 1 the identity map on 1. We will often denote 1 also simply as 1, the meaning being clear from context. For any object → , there is the left unitor takes 1 ⊗ to , as follows from the remarks made above for (42). The right unitor is where ( , , ) : Again, this is indeed a morphism in F(C) by essentially the same argument that was used above in (46) The two triangles at the two ends of this "trough" commute because of coherence in C, 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 F(C), and the slant side on the right is Since the trough commutes in C, so does its avatar (55) in F(C), thanks to the second half of Lemma 2. This verifies the coherence property in F(C) 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: Coherence in the monoidal category C 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 F(C) as = ( , , ) , for ∈ {1, 2, 3, 4}, we can read the full diagram as a diagram in the category F(C) as follows: ( ( 1 2 ) As a diagram in F(C), this is commutative, by Lemma 2. This establishes coherence of the associator in F(C).
We have completed the proof of Proposition 5.

Proposition 5.
Suppose that C is a monoidal category and let F(C) be the category specified above in the context of (11). Then, with tensor product as defined in (44), F(C) satisfies all conditions of a monoidal category at the level of objects.

Concluding Remarks
In this paper, we have presented certain "fattened" categories F(C), F(C) 0 , and F(C) 1 constructed out of a given category C; the morphisms of F(C) 0 form a double category. It is shown how a monoidal structure on C induces a multiplication on the objects of F(C) that satisfies certain coherence properties.