Saunders Mac Lane. Natural associativity and commutativity. Rice University Studies, 49:28--46, 1963.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....application of associativity comes from the action of an element of F . Thus, F can be called the geometry group of associativity, as it captures a number of specific geometrical properties of that identity, in particular those expressed in the well known MacLane Stashe# pentagon relation [6] [16]. When we replace the associativity identity x(yz) xy)z with the left self distributivity identity x(yz) xy) xz) Thompson s group F is no longer relevant, but there exists another group G LD that similarly captures the geometrical aspects of the identity. The group G LD happens to be an ....

S. Mac Lane, Natural associativity and commutativity, Rice Univ. Studies 49 (1963) 28--46

....applications in mathematical physics via compactifications of moduli spaces. Ah yes, I remember it well. Back in 1966 67, Boardman, Vogt and I overlapped at the University of Chicago; I was commuting from Notre Dame for the weekly Topology Seminar. Mac Lane ran a seminar on PACTs and PROPs [17, 18] we all attended. Boardman s linear isometries PROP (later reincarnated as an operad) was key to the developing understanding of infinite loop spaces, but today I d like to talk about a less acknowledged contribution of Mike s: the use of trees. For me, it all began with the study of homotopy ....

S. Mac Lane, Natural associativity and commutativity, Rice Univ. Studies 49 (1963), 28--46.

....Again generalizing work of Sugawara [13] I introduced the hexagon which in turn was generalized to higher dimensions in Milgram s permutahedra. From this point, the subject began to diverge. Another hexagon appears which together with the pentagon is crucial in Mac Lane s first coherence theorem [7] in category theory. My attempts to describe even two fold loop spaces by specific cell complexes became too complicated, although at this conference we have with hindsight seen that things could have been done that way [3] For infinite loop spaces, Boardman and Vogt [1] introduced what is now ....

S. Mac Lane. Natural associativity and commutativity. Rice Univ. Studies, 49:28--46, 1963.

....partial sum in a nice way. A diagram is a directed graph that lives in a category; i.e. the vertices are objects of the category and the arcs are morphisms. See, e.g. Mac Lane [7] a brief review is in section 3 below. We apply robustness to questions of diagram coherence related to Mac Lane [6] and Stasheff [11] They asked when the commutativity of all diagrams of a certain type is implied by that of a few particular cases just as associativity for n term expressions follows from that of the 3 term expressions. Instead, we study when the commutativity of a diagram can be inferred from ....

S. Mac Lane, Natural associativity and commutativity, Rice University Studies 49 1963 28--46.

.... simplex makes such cells easy to define. They turn out to be simply the well formed ( composable ) subpasting diagrams of the simplex, and the fact that they form a free category follows from generalities about pasting diagrams. The first coherence theorems were proved by Mac Lane [15] in 1963. Mac Lane has described coherence theorems as assertions that all of a certain class of diagrams commute, but in most of the classical examples the diagrams involved are diagrams of natural transformations, which are most naturally treated as pasting diagrams in the 2 (or even 3 ) ....

....the geometry of the free n category on the n cube , but his work did not achieve Walters hoped for simplification of Street s results. The relationship between orientals and higher coherence conditions has been particularly tantalizing. Mac Lane proved the first coherence theorems in 1963 [15] and his theorems have been important in the development of enriched category theory by Kelly [11] and others. The development of the theory of categories enriched over a bicategory led Street and Walters to realize that the associative law is obtainable from the free 3 category on the 3 simplex, ....

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Mac Lane, S., Natural associativity and commutativity, Rice University Studies 49 (1963) 28--46.

....two cycles from the basis [5] For 1 any 2 connected plane graph, the cycles determined by the boundaries of the finite regions constitute a 2 basis. This regional boundary basis is robust in the sense defined below. We apply robustness to questions of diagram coherence related to Mac Lane [6] and Stasheff [11] They asked when the commutativity of all diagrams of a certain type is implied by that of a few particular cases just as associativity for n term expressions follows from that of the 3 term expressions. Instead, we study when the commutativity of a diagram can be inferred from ....

S. Mac Lane, Natural associativity and commutativity, Rice University Studies 49 1963 28--46.

.... 1 Omega (x Omega 1) x Omega 1 x 1 Omega x l x Omega 1 a1;x;1 1 Omega rx rx oe l x 4 This definition raises an obvious question: how do we know we have found all the right coherence laws Indeed, what does right even mean in this context Mac Lane s coherence theorem [45] gives one answer to this question: the above coherence laws imply that any two isomorphisms built using a; l and r and having the same source and target must be equal. Further work along these lines allow us to make more precise the sense in which N is a decategorification of FinSet. For ....

S. Mac Lane, Natural associativity and commutativity, Rice U. Studies 49 (1963), 28-46.

....P form a 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, ....

S. Mac Lane. Natural associativity and commutativity. Rice University Studies, 49:28--46, 1963.

....One of these is the notion of a monad or triple on a category, which goes back to Godement [G] and was rst developed by Eilenberg, Moore, Beck and others. The other is that of a monoidal category or tensor category, which originates with B enabou [B e] and with Mac Lane s famous coherence theorem [MacL], and which pervades much of present day mathematics. For a monad S on a tensor category, there is a natural additional structure that one can impose, namely that of a comparison map S(X 1 Xn ) S(X 1 ) S(Xn ) n 0) compatible with the monad and tensor structures already ....

S. Mac Lane, Natural Associativity and Commutativity, Rice University Studies 49 (1963), 28-46.

....polytope. Before closing this section, we o#er some motivation for these results, and also contrast them with a recent construction of a signed associahedron by Simion [17] The classical associahedron makes its appearance in many di#erent places, such as coherence theorems for monoidal categories [14, 18], moduli spaces of pointed curves [10] spaces of Morse functions [11] and resolutions both for the associative law [1] and the Steinberg relations on elementary matrices [11] In many of these contexts, the sphericity of the boundary of the associahedron plays an important role. It is our hope ....

S. Mac Lane, Natural associativity and commutativity, Rice Univ. Studies 49 (1963), 28--46.

.... y;z;w ax;y;z Omega w a x;y;z Omega w a x;y Omega z;w 6 x Omega ay;z;w as well as the following diagram involving the unit laws: 1 Omega x) Omega 1 1 Omega (x Omega 1) x Omega 1 x 1 Omega x x Omega 1 a 1;x;1 1 Omega rx rx oe x Mac Lane s coherence theorem [22] says that every monoidal category is equivalent, as a monoidal category, to a strict monoidal category, that is, one for which the associators and unit laws are all identity morphisms. Sometimes we will use this to streamline formulas by not parenthesizing tensor products and not writing the ....

S. Mac Lane, Natural associativity and commutativity, Rice U. Studies 49 (1963), 28-46.

....would be between (homo)morphisms instead of between 2 functors, and one would probably get tricategories via some theory of weak enrichment . And in category theory, equality often is important, as can be seen from Kelly s body of work [24, 23, 4] and from the abundance of coherence theorems [27, 29, 30, 15]. No, the conceptual difference lies in the treatment of dimension. The cartesian product of 2 categories, and of categories, is basically set theoretical: C Theta D has as basic ingredient pairs (x; y) of dimension p for x 2 C p and y 2 D p , and functoriality then gives, more generally, pairs ....

S. Mac Lane, Natural associativity and commutativity, Rice University Studies 49 (1963), 28--46.

....monoid are replaced by isomorphisms satisfying coherence laws. The monoidal categories that arise in nature, such as nCob and Vect, are usually weak. People frequently ignore this fact, however (and the reader will note we did so in Section 1) The justification for doing so is Mac Lane s theorem [56] that any weak monoidal category is equivalent to a strict one. However, the sense of equivalence here is rather subtle and itself intimately connected with weakening. Following Kapranov and Voevodsky s principle, in addition to weakening algebraic structures, one should also weaken the sense in ....

S. Mac Lane, Natural associativity and commutativity, Rice U. Studies 49 (1963), 28-46.

.... Before closing this section, we offer some motivation for these results, and also contrast them with a recent construction of a signed associahedron by Simion [17] The classical associahedron makes its appearance in many different places, such as coherence theorems for monoidal categories [14, 18], moduli spaces of pointed curves [10] spaces of Morse functions [11] and resolutions both for the associative law [1] and the Steinberg relations on elementary matrices [11] In many of these contexts, the sphericity of the boundary of the associahedron plays an important role. It is our hope ....

S. Mac Lane, Natural associativity and commutativity, Rice Univ. Studies 49 (1963), 28--46.

....triequivalent to a Gray category [6] But there is also another coherence theorem for tricategories, stating that tricategories are (algebras for a) contractible (operad) 1] which roughly says that all diagrams in a tricategory commute . In the basic reference situation of monoidal categories [13, 14], the two coherence theorems are equivalent [8] A cursory inspection of this equivalence suggests that one could prove coherence for tricategories by rst proving contractibility and then constructing a functor st : Tricat Gray Cat adjoint to the inclusion Gray Cat Tricat. A closer ....

....theorem giving a minimal list of axioms from which all possible axioms on that set of data follow. In Mac Lane s original situation, of categories with a tensor product but no unit, the minimal list 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 ....

S. Mac Lane, Natural associativity and commutativity, Rice University Studies 49 (1963), 28-46.

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Saunders Mac Lane. Natural associativity and commutativity. Rice University Studies, 49:28--46, 1963.

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S. Mac Lane. Natural associativity and commutativity. Rice University Studies, 49:28--46, 1963.

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S. Mac Lane. Natural associativity and commutativity. Rice University Studies, 49:28--46, 1963.

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Saunders Mac Lane. Natural associativity and commutativity. Rice University Studies, 49:28--46, 1963.

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S. Mac Lane. Natural associativity and commutativity. Rice University Studies, 49:28--46, 1963.

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S. Mac Lane. Natural associativity and commutativity. Rice University Studies, 49:28--46, 1963.

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Saunders Mac Lane. Natural associativity and commutativity. Rice University Studies, 49:28--46, 1963.

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Saunders Mac Lane. Natural associativity and commutativity. Rice University Studies, 49:28--46, 1963.

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S. Mac Lane, Natural associativity and commutativity, Rice U. Studies 49 (1963), 28-46.

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S. Mac Lane. Natural associativity and commutativity. Rice Univ. Studies 49(1963), no. 4, 28-46.

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