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The Uniformity Principle on Traced Monoidal Categories†

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The Uniformity Principle on Traced Monoidal Categories† Publ. RIMS, Kyoto Univ. 40 (2004), 991–1014 The Uniformity Principle on Traced Monoidal Categories† By Masahito Hasegawa∗ Abstract The uniformity principle for traced monoidal categories has been introduced as a natural generalization of the uniformity principle (Plotkin’s principle) for fixpoint operators in domain theory. We show that this notion can be used for constructing new traced monoidal categories from known ones. Some classical examples like the Scott induction principle are shown to be instances of these constructions. We also characterize some specific cases of our constructions as suitable enriched limits. §1. Introduction Traced monoidal categories were introduced by Joyal, Street and Verity [16] as the categorical structure for cyclic phenomena arising from various areas of mathematics, most notably knot theory [25]. They are (balanced) monoidal categories [15] enriched with a trace, which is a natural generalization of the traditional notion of trace in linear algebra, and can be regarded as an operator to create a “loop”. For example, the braid closure operation — e.g. the con- struction of the trefoil knot from a braid depicted below — can be considered as the trace operator on the category of tangles. ⇒ = Communicated by R. Nakajima. Received December 16, 2003. 2000 Mathematics Subject Classification(s): 18D10, 19D23 †This article is an invited contribution to a special issue of Publications of RIMS com- memorating the fortieth anniversary of the founding of the Research Institute for Math- ematical Sciences. ∗Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan; PRESTO, Japan Science and Technology Agency, Japan c 2004 Research Institute for Mathematical Sciences, Kyoto University. All rights reserved. 992 Masahito Hasegawa In computer science, specialized versions of traced monoidal categories natu- rally arise from considerations on recursion/feedback operators (in the fairly general sense) as well as cyclic data structures. Here is a graphical account of the recursive (fixpoint) computation created from a circularly shared structure: f f ⇒ f = f In the middle of 90’s, Martin Hyland and the author independently ob- served a bijective correspondence between Conway (Bekiˇc, or dinatural diago- nal) fixpoint operators [3, 23] and traces on categories with finite products [8, 9]. Thus, in this setting, the notion of trace precisely captures the well-behaved fixpoint operators commonly used in computer science. An extra bonus of this trace-fixpoint correspondence is that the uniformity principle (`a la Plotkin) on a fixpoint operator [20, 6, 23] precisely amounts to a uniformity principle on the corresponding trace, as introduced in the previous work [8, 9] (see also historical remarks at the end of Sec. 3). This uniformity principle is general enough to make sense for arbitrary traced monoidal cate- gories. An application of this concept is found in Selinger’s work on categorical models of asynchronous communications [21]. In the present paper we demonstrate that this uniformity principle on traced monoidal categories can be used for constructing new traced monoidal categories (and categories with fixpoint operators) from known ones. The con- struction is very simple and in some sense old – its origin can be traced back to the Scott induction principle. We must admit that, by its general nature, our constructions do not im- mediately provide mathematically strong results. However, we think that our development is a natural one, in that it adds a new view to the relationship between model construction techniques and proof principles; in particular, we expect that results of this paper can be combined with the well-established relationship between categorical glueing and logical relations. Moreover, our constructions seem to enjoy characterizations by some uni- versal property, as certain limits in an enriched sense. We study this issue for some specific cases, point out the difficulties in this direction, and propose a solution for which our leading example (with a slight modification) is charac- terized as a Graph-enriched limit. Uniformity on Traced Categories 993 The rest of this paper is organized as follows. In Sec. 2, we explain the graphical notations for symmetric monoidal categories which are used through- out this paper, and recall the notion of traced symmetric monoidal categories. We then introduce the notion of uniformity and strict maps for traced monoidal categories in Sec. 3. Sec. 4 is devoted for some basic results on the uniformity principle. In Sec. 5 we recall the correspondence between traces and fixpoint operators and give the relationship between the uniformity principle for traces and that for Conway operators. Sec. 6 and 7 form the central part of this paper, where we show how the uniformity principle can be used for constructing new traced monoidal categories. Sec. 8 gives some observations on the uniformity principle and the Int construction. This paper is a revised and extended version of the work presented at the Category Theory and Computer Science (CTCS’02) conference [10]. Remark 1.1. Although many of the notions and results in this work ap- ply to general traced balanced monoidal categories (where we may have non- trivial intertwiners and twists [15, 22] which are essential for interpreting the ribbon diagrams [25]), in this paper we restrict our attention only to traced symmetric monoidal categories, firstly because for ease of presentation, and secondly because most of examples relevant to computer science seem to be instances of symmetric monoidal categories. So by a traced monoidal category, in this paper we mean a traced symmetric monoidal category unless otherwise stated. §2. Preliminaries §2.1. Graphic presentation of monoidal categories In this paper, an arrow f : A1 ⊗ A2 ⊗···⊗Am → B1 ⊗ B2 ⊗···⊗Bn in a monoidal category [19, 15] (pretended as if the tensor product is strictly associative for brevity) is drawn as (from left to right) Am Bn : : A2 f B2 A1 B1 The identity morphism is drawn just as a straight arrow. The composition of f : X → Y and g : Y → Z is represented as a sequential composition 994 Masahito Hasegawa X f Y Y g Z ⇒ X f Y g Z while the tensor f ⊗ g : A ⊗ C → B ⊗ D of f : A → B and g : C → D is drawn as a parallel composition C D g C g D ⇒ A B A f B f The symmetry cX,Y : X ⊗ Y → Y ⊗ X in a symmetric monoidal category is represented by a cross wiring: Y X X Y A formal justification of these graphical presentations has been given by Joyal and Street [14]. §2.2. Traced monoidal categories C X A trace on a symmetric monoidal category is a family of functions TrA,B : C(A ⊗ X, B ⊗ X) →C(A, B) subject to the following conditions: • it is natural in A and B (left/right tightening), and dinatural in X (sliding) • I X⊗Y X Y vanishing: TrA,B(f)=f and TrA,B (f)=TrA,B(TrA⊗X,B⊗X (f)) • X ⊗ ⊗ X superposing: TrC⊗A,C⊗B(idC f)=idC TrA,B(f) • X yanking: TrX,X(cX,X)=idX where, again for brevity, we pretend as if the tensor is strictly associative. A traced symmetric monoidal category is a symmetric monoidal category equipped with a (specified) trace – note that there can be many ways of giving traces. Trace admits a natural graphical presentation as a “feedback”: for f : ⊗ → ⊗ X → A X B X, its trace TrA,B(f):A B can be drawn as A f B Using this notation, the axioms for trace given above can be graphically pre- sented as follows. Uniformity on Traced Categories 995 Naturality (Left Tightening) Naturality (Right Tightening) f = f f = f h h h h Dinaturality (Sliding) h f = f h Vanishing (I) Vanishing (⊗) I f = f f f = Superposing Yanking f f = = §3. Uniformity for Traces Definition 3.1. Consider a traced monoidal category C with trace Tr. We say h : X → Y is strict in C (with respect to the trace Tr) if the following condition holds: For any f : A ⊗ X → B ⊗ X and g : A ⊗ Y → B ⊗ Y , (idB ⊗ h) ◦ f = g ◦ (idA ⊗ h):A ⊗ X → B ⊗ Y f A ⊗ XB⊗ X idA⊗h idB ⊗h ⊗ ⊗ A YBg Y X Y → implies TrA,B(f)=TrA,B(g):A B. In terms of the graphic notation: X X Y X Y Y h = h ⇒ = A f B A g B A f B A g B 996 Masahito Hasegawa Definition 3.2. Let C be a traced monoidal category with trace Tr, and S be a class of arrows of C.WesayTr is uniform (or: Tr satisfies the uni- formity principle) for S if, for any h : X → Y of S, the condition in Def. 3.1 holds. Thus the class of strict maps is the largest class of arrows for which the trace satisfies the uniformity principle. As noted by Selinger [21], the uniformity principle can be seen a proof principle for showing two traces are equivalent: to prove TrX (f)=TrY (g), it suffices to find h : X → Y of S such that (id⊗h)◦f = g◦(id⊗h) holds. In such applications, it is often convenient to give S a priori as a suitable subcategory containing reasonably rich class of arrows; see ibid. for several good examples. However, we note that there is no reason to expect that the class of strict maps form a category – in fact there are counterexamples, as we will see shortly. For now, we shall recall some popular examples, where strict maps actually form categories. Example 3.1. • The category of finite dimensional vector spaces over a field and linear maps, where the trace is a natural generalization of the standard trace ⊗ → ⊗ W → [16]: for a linear map f : U K W V K W , its trace TrU,V (f):U V is given by W (TrU,V (f))i,j =Σkfi⊗k,j⊗k W —if U = V = K,wehaveTr (f)=Σkfk,k as expected.
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