G. M. Kelly. On Mac Lane's conditions for coherence of natural associativities, commutativities, etc. J. Algebra, 1:397--402, 1964.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....structural natural transformations in the definition of a premonoidal category commutes. Proof. Since the centre of a premonoidal category is a monoidal category and all the structural maps are central, the result follows immediately from coherence for a monoidal category as in Kelly s refinement [10] of Mac Lane s proof. All of the premonoidal categories of primary interest to us are symmetric in some reasonable sense, and we require that symmetry for a soundness proof for models of the c calculus, so we make precise the notion of a symmetry for a premonoidal category. 70 Definition 3.7 ....

Kelly, G.M.: On Mac Lane's conditions for coherence of natural associativities, commutativities, etc. J. Algebra 1 (1964) 397--402. 86

....equal when realized. If a = b = c = d = 1 then asking for commutativity is precisely asking for Mac Lane s pentagonal coherence of associativity condition to hold. If a = d = 1 and b = c = 0 then equality of the legs of (4. 3) when realized is, up to choice of orientation, Kelly s refinement [7] of Mac Lane s conditions [15] for the coherence of an identity isomorphism. We have proved the usual coherence theorem for monoidal categories. Example 4.10 Coherence for Commutativity Isomorphisms: The theory of coherent situations presented so far is quite sufficient for our main purpose the ....

Kelly, G.M., On Mac Lane's conditions for coherence of natural associativities, commutativities, etc., J. Algebra 1 (1964) 397--402.

.... consists of, apart from three naturality squares, just one axiom: Mac Lane s pentagon [13] For monoidal categories, the minimal list consists of two more naturality squares and one more axiom; Kelly proved that three further axioms included by Mac Lane are actually consequences of the other ones [12], and that the list is indeed minimal. So there are four steps to a contractibility theorem: establishing a list, proving this list implies contractibility, reducing the list if possible, and proving that the nal list is minimal. In this paper I do part of the rst step for two particular ....

G. M. Kelly, On Mac Lane's conditions for coherence of natural associativities, commutativities, etc., J. Algebra 1 (1964), 397-402.

....in the definition of a premonoidal category commutes. John Power and Edmund Robinson 6 Proof. Since the centre of a premonoidal category is a monoidal category and all the structural maps are central, the result follows immediately from coherence for a monoidal category as in Kelly s refinement (Kelly 1964) of Mac Lane s proof. We now turn to the definition of a symmetric premonoidal category. Definition 3.7. A symmetry for a premonoidal category is a central natural isomorphism with components c : x Omega y Gamma y Omega x, satisfying the two conditions c 2 = 1 and equality of the evident ....

Kelly, G.M. (1964) On Mac Lane's conditions for coherence of natural associativities, commutativities, etc. J. Algebra 1, 397--402.

....to the triangle that has the distinguished object I in the middle. It is well known that in this case the commutativity of these diagrams implies the commutativity of the two triangles that have I on one extreme or the other, and that the right and left arrows I Omega I Gamma I coincide [6]. This in turn implies the commutativity of all the diagrams [11] Results like those of [6] can be shown in the present context. Theory and Applications of Categories, Vol. 3, No. 2 37 8.1. Proposition. If D = D; d; m; fi; j; is a pseudomonad on an object K, then we have the following ....

.... this case the commutativity of these diagrams implies the commutativity of the two triangles that have I on one extreme or the other, and that the right and left arrows I Omega I Gamma I coincide [6] This in turn implies the commutativity of all the diagrams [11] Results like those of [6] can be shown in the present context. Theory and Applications of Categories, Vol. 3, No. 2 37 8.1. Proposition. If D = D; d; m; fi; j; is a pseudomonad on an object K, then we have the following equalities: 1: 1K D D DD DD fi j d : dD t t t t t t Dd J J J J J J Id D ....

G. M. Kelly. On MacLane's conditions for coherence of natural associativities, commutativities, etc. Journal of Algebra 1, 1964, pp. 397-402.

.... Omega Omega Omega Omega Omega r C C Omega D= C Omega D J J J J J J s Omega Omega Omega Omega Omega Omega s OE D Omega C 77 The commutative diagrams in definitions 3.3.2 and 3.3.3 are known collectively as the MacLane Kelly equations. It follows from a coherence theorem [31, 29] that any diagram made up of instances of l, r, s and a will commute. Note that any category with finite products has a cartesian symmetric monoidal structure, with Omega given by the binary product and O by the terminal object. The isomorphisms l, r, s and a are given by the universal properties ....

G. M. Kelly. On MacLane's conditions for coherence of natural associativities, commutativities, etc. J. Algebra 1 (1964), 397-402

....monoidal subcategory, called the center of P, and denoted by P ffl . Clearly, the center is the largest subcategory on which # restricts to a proper (bifunctorial) tensor product. Coherence for premonoidal categories follows easily from Mac Lane s result for monoidal categories (Mac Lane 1963; Kelly 1964), since all the relevant coherence diagrams are contained in the center. Premonoidal categories share many properties of monoidal categories, but some special care is necessary when manipulating them. For instance, one should keep in mind that there are innocent looking expressions, such as A # A, ....

G. M. Kelly. On Mac Lane's conditions for coherence of natural associativities, commutativities, etc. J. Algebra, 1:397--402, 1964.

.... C is the constant functor which associate e and id e respectively to each object and each morphism of C, # , # is the pairing of functors induced by the cartesian product, and #: 1# 2 # # 2# 1 is a natural isomorphism, called the symmetry of C, subject to the Kelly MacLane coherence axioms [9, 7]: # x,z# id y ) # (id x# # y,z ) # x# y,z , 4) # y,x # # x,y = id x# y . 5) Equation (1) states that the tensor is associative on both objects and arrows, while (2) and (3) state that e and id e are, respectively, the unit object and the unit arrow for# . Concerning the coherence ....

G.M. Kelly. On MacLane's Conditions for Coherence of Natural Associativities, Commutativities, etc. Journal of Algebra, n. #, pp. ###--###, ####.

.... associate e and id e respectively to each object and each morphism of C, h ; i is the pairing of functors induced by the cartesian product, and 2 fl: 1 Omega 2 Gamma 2 Omega 1 is a natural isomorphism, called the symmetry of C, subject to the following Kelly MacLane coherence axioms [9, 7]: fl x;z Omega id y ) ffi (id x Omega fl y;z ) fl x Omega y;z ; 4) fl y;x ffi fl x;y = id x Omega y : 5) Clearly, equation (1) states that the tensor is associative on both objects and arrows, while (2) and (3) state that e and id e are, respectively, the unit object and the unit arrow ....

G.M. Kelly. On MacLane's Conditions for Coherence of Natural Associativities, Commutativities, etc. Journal of Algebra, n. 1, pp. 397--402, 1964.

.... (id A Omega ff B;C;D ) ae A Omega id C ) ffi ff A;II;C = id A Omega C II = ae II fl A;B ffi fl B;A = id B Omega A ae B = B ffi fl B;II ff C;A;B ffi fl (A Omega B) C ffi ff A;B;C = fl A;C Omega id B ) ffi ff A;C;B ffi (id A Omega fl B;C ) in order the coherence theorem holds (see [8]) namely: in a monoidal symmetric category, any two isomorphisms built out of ff, ae, id and fl A;B with A 6j B, using Omega and composition, coincide. Given a monoidal symmetric category C, an endofunctor T : C C is monoidal if, for each A; B 2 Obj C , there are a natural ....

G. M. Kelly. On MacLane's conditions for coherence of natural associativities. Journal of Algebra, 1:397 -- 402, 1964.

....to the weakly distributive category thus generated. In order to carry out this we must first introduce the rules of surgery for these nets. We have enumerated a more than full set of rules 2 A minimal set is indicated and it is left to the reader to mimic the categorical techniques of Kelly [K64] in diagrammatic form to prove that this is indeed a minimal set. The reason for enumerating the larger set is that it is this set which forms the basis for the expansion reduction rewriting we describe in the next section which is used to establish the coherence result. 2 We give only a ....

.... all the unit and counit manipulations rules (15) 48) can be reduced to just six rules: 15) 20) 21) 24) 29) and (30) This is a challenging exercise for the reader It is useful to realize that many of these follow quite easily from the coherence of the tensors and the tricks in [K64]. We end with some examples of the unit rewiring steps. First an illustration of an illegal application of Equation (22) it is easy to check that the net criterion is not preserved in the following rewrite. j Phi j Phi A Phi A Phi j Phi j Phi A Phi A Phi j j j ....

Kelly,G.M. "On Mac Lane's conditions for coherence of natural associativities, commutativities, etc.", Journal of Algebra 1, 4 (1964) 397--402.

....to the triangle that has the distinguished object I in the middle. It is well known that in this case the commutativity of these diagrams implies the commutativity of the two triangles that have I on one extreme or the other, and that the right and left arrows I Omega I Gamma I coincide [6]. This in turn implies the commutativity of all the diagrams [11] Results like those of [6] can be shown in the present context. Theory and Applications of Categories, Vol. 3, No. 2 37 8.1. Proposition. If D = D; d; m; fi; j; is a pseudomonad on an object K, then we have the following ....

.... this case the commutativity of these diagrams implies the commutativity of the two triangles that have I on one extreme or the other, and that the right and left arrows I Omega I Gamma I coincide [6] This in turn implies the commutativity of all the diagrams [11] Results like those of [6] can be shown in the present context. Theory and Applications of Categories, Vol. 3, No. 2 37 8.1. Proposition. If D = D; d; m; fi; j; is a pseudomonad on an object K, then we have the following equalities: 1: 1K D D DD DD fi j F NaN F NaN d F NaN F NaN : dD t t t t t t F ....

G. M. Kelly. On MacLane's conditions for coherence of natural associativities, commutativities, etc. Journal of Algebra 1, 1964, pp. 397-402.

....on V. Thus an action of V on A can equally be described by giving the functor : V A A along with the natural isomorphisms and satisfying (1.1) 1.3) fIn fact, 1.2) is a consequence of (1.1) and (1. 3) the proof of the corresponding result for a monoidal category given by Kelly in [KM] extends unchanged to the bicategory case, while the data (V; I; a; r; satisfying the monoidal category axioms and (1.1) 1.3) can be seen as describing a two object bicategory.g Of course any monoidal category V = V; I) as above has a canonical action on (the underlying ....

G.M. Kelly, On Mac Lane's conditions for coherence of natural associativities, commutativities, etc., J. Algebra 1 (1964), 397-402.

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G. M. Kelly. On Mac Lane's conditions for coherence of natural associativities, commutativities, etc. J. Algebra, 1:397--402, 1964.

No context found.

G.M. Kelly (1964), On MacLane's Conditions for Coherence of Natural Associativities, Commutativities, etc. Journal of Algebra, n. 1, 397--402, Acamedic Press.

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G.M. Kelly (1964), On MacLane's Conditions for Coherence of Natural Associativities, Commutativities, etc. Journal of Algebra, n. 1, 397--402, Acamedic Press.

No context found.

G.M. KELLY (1964), On MacLane's Conditions for Coherence of Natural Associativities, Commutativities, etc. Journal of Algebra 1, 397--402.

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