G.M. Kelly and S. Mac Lane, "Coherence in closed categories", J. Pure Appl. Alg. 1 no. 1, (1971), 97--140; erratum, ibid. no. 2, 219.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....diagram will be non commutative in the category of vector spaces. Clearly, such a proof would rely on the description of commutative (and non commutative) diagrams in a free category. This description constitutes a coherence problem. A partial coherence theorem was proved by Kelly and Mac Lane [9] (1971) every diagram, that does not contain essential occurrences of tensor unit I (i.e. such that all occurrences of I can be eliminated by canonical isomorphisms) is commutative. An important contribution was made by R. Voreadou [18] She proposed a new and very fruitful idea: to give an ....

....complete it during various visits (G. Longo, ENS, Paris, H. Schwichtenberg, Ludwig Maximilan University, It is easy to compare with our paper, since in [17] the proof theoretical approach is used. For example, he considered the arrows that have in fact different graphs of naturality conditions [9], 18] It can be shown, that already the instances of such arrows in 2 dimensional vector spaces are distinct Munchen, I. Moerdijk, Utrect University, P. Damphousse, Tours University, and, finally, Zhaohui Luo and the University of Durham) I would thank Michael Barr for important information, ....

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G.M. Kelly and S. Mac Lane. Coherence in Closed Categories. Journal of Pure and Applied Algebra, 1(1):97--140, 1971.

....The underlying logic of (Lambek 1969) may be seen now as noncommutative intuitionistic multiplicative linear logic. An important application of Lambek s idea came about in the early 70 s in an article by Kelly and Mac Lane on the coherence problem for symmetric monoidal closed categories, see (Kelly and MacLane 1971). A coherence theorem is a result which tells when diagrams commute in a free category of a given kind. Basically what is desired is a criterion for determining whether a diagram consisting of structural morphisms from (in this case) the theory of symmetric monoidal closed categories commutes. ....

....(Blute 1993) calls its Kelly Mac Lane graph. The authors show that two morphisms are equal i they have the same Kelly Mac Lane graph. Today, we understand that Kelly Mac Lane graphs are really the axiom links of proof nets in (unitfree) intuitionistic multiplicative linear logic. In particular, (Kelly and MacLane 1971) shows how to compose Kelly Mac Lane graphs, and proves that composition is always compatible, in the sense that no loops are made when forming the composite, which we see now as a consequence of the correctness criterion for proof nets. See also (Kelly and Laplaza 1980) for a similar work on ....

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G. M. Kelly, S. Mac Lane, Coherence in closed categories, J. Pure and Applied Algebra 1, pp. 97-140, 1971.

....Linear Logic is a looking glass through which fundamental properties of functional computation appear symetrized and simplified. It should also be remarked that the axiom links from Linear Logic play a similar role to the formula links originally introduced by Lambek in his study of SMC objects [14], and that proof nets were really the missing tool to understand linearity. In proving the result, we used essentially the topological properties of the proof nets of linear logic, which simplified enormously our task (for example, the reduction to non ambiguous formulae when working directly with ....

S. Mac Lane and G. M. Kelly. Coherence in Closed Categories. Journal of Pure and Applied Algebra, 1(1):97--140, 1971.

....later confusion. A monoidal category is a category D together with a product = D : D Theta D Gamma D and a unit object u = uD such that is associative and unital up to coherent natural isomorphism; D is symmetric monoidal if is also commutative up to coherent natural isomorphism. See [9, 10, 15] for discussions of the precise meaning of coherence. A symmetric monoidal category D is closed if it has internal hom objects F (d; e) with adjunction DIAGRAM SPACES, DIAGRAM SPECTRA, AND FSP S 3 isomorphisms D(de; f) D(d; F (e; f) There are evident notions of monoids in monoidal ....

....unit, and commutativity isomorphisms of A and B commute. The functor F is strong monoidal or strong symmetric monoidal if and OE are isomorphisms. The definition is incomplete in that we have not specified the relevant coherence diagrams , but the intuition should be clear enough. See [9, 10] for details. The direction of the arrows and OE in the definition leads to the following conclusion. Lemma 1.2. If F : A Gamma B is lax monoidal and M is a monoid in A with unit j : uA Gamma M and product : M A M Gamma M , then F (M) is a monoid in B with unit F (j) ffi : uB Gamma F ....

G. M. Kelly and S. Mac Lane. Coherence in closed categories. J. Pure and Applied Algebra. 1(1971), 97-140.

....14 Fact 2.3. i) In a free symmetric monoidal closed category, the diagram commutes if and only if A = X = In particular, it does not commute if X = and A 6= ii) In a free autonomous category, the diagram commutes if and only if X = or A = X = The result (i) above is stated in [9]. We recast the problem in our type theory and give a syntactic proof. The two terms inhabiting ( A ( X) X) X ( A ( X) X) X are (we write X(A) as a shorthand for A ( X) i) z X(X(A) h ; z=x X ix (ii) z X(X(A) hz ; x A :h ; w X(A) hw ; x=v X 3 iv 3 =v X 2 iv 2 ....

....rather than types. Interpretation in Autonomous and Autonomous Categories 19 We shall extend the structure M further to an interpretation of (provable) typing judgements by assigning a map [ M [ A ] M in C to each s : A, with the help of the Kelly MacLane Coherence Theorem [9] for symmetric monoidal categories. Note that we do not need the Coherence Theorem for symmetric monoidal closed categories to prove the soundness of interpretation. The interpretation is de ned by induction on the derivation of judgement. Since the derivation of judgement is not unique, the de ....

G. M. Kelly and S. MacLane. Coherence in closed categories. Journal of Pure and Applied Algebra, 1:97-140, 1971.

.... autonomous categories: in any free such category, if A ( B, regarded as a type, is linearly balanced and trim then there can be at most one map from A to B. The argument cuts down to a Coherence Theorem for symmetric monoidal closed categories, and is a new proof of essentially the same result in [13]. Key Words: symmetric monoidal closed and autonomous categories, internal language, Coherence Problem, calculus. Oxford University Computing Laboratory, Wolfson Building, Parks Road, Oxford OX1 3QD, ENGLAND. twk comlab.ox.ac.uk y Luke.Ong comlab.ox.ac.uk. Tel: 44 1865 273840; Fax: 44 ....

....University Computing Laboratory, Wolfson Building, Parks Road, Oxford OX1 3QD, ENGLAND. twk comlab.ox.ac.uk y Luke.Ong comlab.ox.ac.uk. Tel: 44 1865 273840; Fax: 44 1865 273839. 1 A statement of the form: all diagrams of a certain class (e.g. composed of canonical maps) commute , as in [18, 13]. CONTENTS 2 Contents 1 Introduction 3 1.1 An Overview . 3 1.2 A Survey of Related Work . 9 2 Autonomous and Autonomous Type Theories 12 2.1 ....

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G. M. Kelly and S. MacLane. Coherence in closed categories. Journal of Pure and Applied Algebra, 1:97--140, 1971.

....Fact 2.2. 1) In a free symmetric monoidal closed category, the diagram commutes if and only if A = X = In particular, it does not commute if X = and A 6= 2) In a free autonomous category, the diagram commutes if and only if X = or A = X = The result (i) above is stated in [9]. We recast the problem in our type theory and give a syntactic proof. The two terms inhabiting ( A ( X) X) X ( A ( X) X) X are (we write X(A) as a shorthand for A ( X) 1) z X(X(A) h ; z=x X ix (2) z X(X(A) hz ; x A :h ; w X(A) hw ; x=v X 3 iv 3 =v X 2 iv 2 =v ....

....they are understood to be type denotations rather than types. We shall extend the structure M further to an interpretation of (provable) typing judgements by assigning a map [ Gamma ] M Gamma [ A ] M in C to each Gamma s : A, with the help of the Kelly MacLane Coherence Theorem [9] for symmetric monoidal categories. Note that we do not need the Coherence Theorem for symmetric monoidal closed categories to prove the soundness of interpretation. The interpretation is defined by induction on the derivation of judgement. Since the derivation of judgement is not unique, the ....

G. M. Kelly and S. MacLane. Coherence in closed categories. Journal of Pure and Applied Algebra, 1:97--140, 1971.

....Fact 2. 3 (i) In a free symmetric monoidal closed category, the diagram commutes if and only if A = X = In particular, it does not commute if X = and A 6= ii) In a free autonomous category, the diagram commutes if and only if X = or A = X = The result (i) above is stated in [8]. We shall recast the problem in our type theory and give a syntactic proof. The two terms inhabiting ( A ( X) X) X ( A ( X) X) X are (we write X(A) as a shorthand for A ( X) 1) X(X(X(A) z X(X(A) h ; z=x X ix : X(X(X(A) 2) X(X(X(A) z X(X(A) hz ; x A ....

....they are understood to be type denotations rather than types. We shall extend the structure M further to an interpretation of (provable) typing judgements by assigning a map [ Gamma ] M Gamma [ A ] M in C to each Gamma s : A, with the help of the Kelly MacLane Coherence Theorem [8] for symmetric monoidal categories. Note that we do not need the Coherence Theorem for symmetric monoidal closed categories to prove the soundness of interpretation. The interpretation is defined by induction on the derivation of judgement. Since the derivation of judgement is not unique, the ....

G. M. Kelly and S. MacLane. Coherence in closed categories. Journal of Pure and Applied Algebra, 1:97--140, 1971.

.... into a MacLane style coherence result for monoidal categories (i.e. a statement of the form all diagrams of a certain class composed of canonical maps commute ) The result may seem to extend the classic Coherence Theorem for (symmetric) monoidal closed categories due to Kelly and MacLane [7], but it does not really (as every such sequent is trim in the sense of [8] However, we do have a slightly stronger result later. 7 Joker Moves and Conditionally Exhausting Strategies We can now consider the game interpretation of the tensor unit. We define the atomic game G = h fg; f( O)g; ....

....is linearly balanced and contains only two negative occurrences of the tensor unit, then there is at most one = equivalence class of proofs of Gamma. Triple Unit Problem There are sequents for which we cannot hope to prove coherence . The standard example is the Triple Unit Problem (see e.g. [7]) but here we recast it using the language of IMLL, via the Categorical Type Theory Correspondence. If A is an atomic type not equal to , it is known that there are exactly two inequivalent ways of proving ( A ( A ( in IMLL. See [9] for a proof. The Tower of Units ....

G. M. Kelly and S. MacLane. Coherence in closed categories. Journal of Pure and Applied Algebra, 1:97--140, 1971.

....derivation systems on sequent calculi with exchange and product are not Church Rosser. Thus his coherence results for categories having a symmetric product (either monoidal or cartesian) are false. 1 Introduction Gentzen s sequent calculi [9] have been applied extensively in category theory, e. g [2, 3, 4, 6, 7, 8]. Sequents correspond to morphisms of a category, and the rules of the calculus correspond to categorical structures (e.g. having an associative tensor product) Cut elimination was then used to put bounds on the complexity of these structures, e.g. to produce exhaustive lists (perhaps with ....

G.M. Kelly and S. Mac Lane, "Coherence in closed categories", J. Pure Appl. Alg. 1 no. 1, (1971), 97--140; erratum, ibid. no. 2, 219.

....(instead of b we may take, e.g, b i , according to the order of leaves in the tree of derivation) It is easily checked, that d 0 will be a derivation too and all the sequents in d will be balanced. With every derivation d : Delta A in L its graph (d) in the sense of Kelly Mac Lane [12] can be connected. Its vertices are the occurrences of variables in Delta A and its edges (or linkages) connect the occurrences of the same variable having opposite signs in Delta A. Another special property of this graph is that all its edges are disjoint (have no common vertices) ....

....property of this graph is that all its edges are disjoint (have no common vertices) Remark.The edges of the graph of an axiom A A connect the vertices generated by the same occurrence of a variable in A. We shall formulate here some necessary results about these graphs (for the details see [12]) Lemma 4. Let d; d 0 : Delta A and d j d 0 . Then (d) d 0 ) Lemma 5. If d is a cut free atomic derivation, then the vertices linked in (d) are exactly the descendants of (the left and right side of) the same axiom. If we have two derivations f : A B and g : B C, the ....

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G.M.Kelly and S.Mac Lane. Coherence in Closed Categories. Journal of Pure and Applied Algebra,1(1):97- 140, 1971.

....both for propositions of linear logic and for their denotations in our monoidal category. The idea then is that a sequent of form C 1 ; C 2 ; Cn A will be interpreted as a map C 1 fflC 2 ffl. fflC n A from the tensor product of the C i to A. Thus a coherence result is assumed [11]. When Gamma is the sequence C 1 ; C 2 ; Cn , we write Gamma A for this map. We seek to enrich the sequent judgement to a term assignment judgement of the form x 1 : C 1 ; x 2 : C 2 ; xn : Cn e : A where the x i are (distinct) variables and e is a term; usually we ....

G.M. Kelly and S. Mac Lane. Coherence in closed categories. Journal of Pure and Applied Algebra, 1:97--140, 1971.

....N ) and we have an isomorphism DA (M L Omega AN; P ) DA (M; RHomA (N; P ) 5. 3) The standard coherence isomorphisms ( associativity and commutativity constraints) on the tensor product pass to the derived category, which is thus a symmetric monoidal closed category in the sense of [43, 36]. There are general accounts of duality theory in such a context in the literature of both algebraic geometry [21, x1] 19] and algebraic topology [23] we follow [40, III xx1 2] Observe first that, by an easy direct inspection of definitions, the functor HomA (M; N) preserves cofiber ....

G. M. Kelly and S. MacLane. Coherence in closed categories. J. Pure and Applied Algebra, 1(1971), 97--140. 136 IGOR KRIZ AND J. P. MAY

....one must prove that derivations yielding the same judgement are given the same meaning. This idea has also appeared in the context of category theory and our use of the term coherence is partially inspired by its use there, where it means the uniqueness of certain canonical morphisms (see e.g. [KL71] and [LP85] Although we have not attempted a rigorous connection in this paper, the possibility of unifying coherence results for a variety of different calculi offers an interesting direction of investigation. In the case of Fun, we show the coherence of our semantic approach by proving that ....

G. M. Kelly and S. Mac Lane. Coherence in closed categories. J. Pure Appl. Algebra, 1:97--140, 1971. Erratum ibid. 2(1971), p. 219.

....B, j iff j j J = j j J for all assignments J in all possible SMC categories K. For every SMC category K the category N(K) with superpositions of functors Omega and Gamma , as objects and natural transformations of these functors as morphisms has a canonical structure of SMC category (cf. [9]) Standard interpretation k Gamma kK : F(A) N(K) is defined by the assignment of 1 : K K (identity functor) to every atom. Mac Lane conjecture for an SMC category K can be formulated in two equivalent forms (with F(A) as free SMC category) 1) Let ; A B be canonical maps. Then j ....

....Definition 3.1 Canonical maps, obtained from aABC ; a Gamma1 ABC ; bA ; b Gamma1 A ; cAB ; id A by ( Omega ) and composition are called central isomorphisms. Theorem 2 All central isomorphisms ; A B with balanced A B are equivalent. Theorem 3 (Coherence for closed categories, [9]] Let ; A B be canonical maps and the sequent A B be balanced and do not contain occurrences of the subformulas of the form C Gamma , D with D constant and C non constant. Then j . There is also a kind of cut elimination theorem, proved by Kelly and Mac Lane ( 9] Theorem 6.5, ....

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G.M. Kelly and S. Mac Lane. Coherence in Closed Categories. Journal of Pure and Applied Algebra, 1(1):97--140, 1971.

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G.M. Kelly and S. Mac Lane, "Coherence in closed categories", J. Pure Appl. Alg. 1 no. 1, (1971), 97--140; erratum, ibid. no. 2, 219.

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KELLY G.M. & S. MAC LANE. Coherence in closed categories. Journal of Pure and Applied Algebra 1 1971 97--140.

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G. M. Kelly, S. Mac Lane. Coherence in closed categories. In J. Pure and Applied Algebra 1, pp. 97-140, 1971.

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Kelly, G.M. and Mac Lane, S. (1971) Coherence in closed categories. J. Pure Appl. Algebra 1(1):97-140.

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Kelly, G.M. and Mac Lane, S., Coherence in closed categories, J. Pure and Applied Algebra 1 (1971) 97--140.

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G. M. Kelly and S. Mac Lane. Coherence in closed categories, J. Pure Appl. Alg. 1 (1971), pp. 97-140.

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G. M. Kelly and S. Mac Lane. Coherence in closed categories, J. Pure Appl. Alg. 1 (1971), pp. 97-140.

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Kelly, G.M. and S. Mac Lane "Coherence in closed categories", Journal of Pure and Applied Algebra 1 (1972) 97--140.

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G.M. Kelly, S. Mac Lane, Coherence in Closed Categories, J. Pure and Applied Algebra 1, p. 97-140, (1971).

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