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Wednesday, August 12, 2020 | History

3 edition of A coherence theorem for closed categories found in the catalog.

A coherence theorem for closed categories

A coherence theorem for closed categories

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  • 3 Currently reading

Published .
Written in English


Edition Notes

Statementby Rodiani Voreadou.
Classifications
LC ClassificationsMicrofilm 40971 (Q)
The Physical Object
FormatMicroform
Paginationii, 62 leaves.
Number of Pages62
ID Numbers
Open LibraryOL1827540M
LC Control Number89894627

Dimension three is an important test-bed for hypotheses in higher category theory and occupies something of a unique position in the categorical landscape. At the heart of matters is the coherence theorem, of which this book provides a definitive treatment, as well as covering related by: Extracting a proof of coherence for monoidal categories from a proof of normalization for monoids. I present a proof of this coherence theorem and also formalize it fully in the dependently.

Closed coherence for a natural transformation --Coherence for distributivity --Many-variable functorial calculus. I. --An abstract approach to coherence --Coherence for a closed functor --A cut-elimination theorem --A new result of coherence for distributivity. Series Title: Lecture notes in mathematics (Springer-Verlag), Responsibility. Analogous graphs occur in Kelly's and Mac Lane's coherence theorem for symmetric monoidal closed categories. A coherence theorem with respect to these graphs is proved for proof-net categories.

In particular, a cartesian closed category that has finite coproducts is a distributive category. The internal logic of cartesian closed categories is the typed lambda-calculus. Inheritance by reflective subcategories. In showing that a given category is cartesian closed, the following theorem is often useful (cf. A in the Elephant). Coherence and non-commutative diagrams in closed categories. [Rodiani Voreadou] -- With respect to closed categories, the allowable natural transformations are the composites of instances of the natural transformations which determine the closed structure.


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A coherence theorem for closed categories Download PDF EPUB FB2

Closed coherence for a natural transformation.- Coherence for distributivity.- Many-variable functorial calculus. I An abstract approach to coherence.- Coherence for a closed functor.- A cut-elimination theorem.- A new result of coherence for : Saunders Mac Lane.

The closed category k 67 85; B. The proof of Theorem 6 71 89; C. The proof of Theorem 7 73 91; PART THREE: A FINAL COHERENCE THEOREM FOR CLOSED CATEGORIES 88 ; References 90 ; Index of Terminology and Notations 91 free. Abstract. A coherence theorem states that the arrows between two particular objects in free categories are unique.

In this paper, we give a direct proof for cartesian closed categories (CCC's) without passing to typed lambda : Akira Mori, Yoshihiro Matsumoto. A kind of coherence theorem for closed symmetric monoidal categories, conjectured by Saunders Mac Lane in the 60’s and proved by Sergei Soloviev in the 90’s, says the following: Theorem A diagram in a free closed symmetric monoidal category is commutative if and only if all its instantiations in vector spaces are commutative.

Longo G. () Coherence and valid isomorphism in closed categories applications of proof theory to category theory in a computer sclentist perspective. In: Pitt D.H., Rydeheard D.E., Dybjer P., Pitts A.M., Poigné A. (eds) Category Theory and Computer by: 1.

(3) Free monoidal categories are directly connected with the coherence issues. The coherence theorem for monoidal categories is equivalent to the statement that the free monoidal category generated by an arbitrary set is a preorder. Moreover if Y-- V is a strict monoidal functor, it is clear that,a(carries I t I j.

to # t IV. t I C.M Kelly, SMactan, Coherence in closed categories 4. Central rnorphms ME) and in This section will use Theorem 3. 1 to handle, for a closed category Y, that part of the coherence Cited by: The trick is to construct an appropriate monoidal category It (B) It(B) in a way that the coherence in one variable for It (B) It(B) and one special object of It (B) It(B) is the general coherence for B B.

The coherence theorem in CWM (Theorem VII) is stated in a different way, but the main bulk of its proof is actually devoted to prove the above coherence theorem.

The coherence theorem for monoidal categories, like many coherence theorems, has several forms (or, alternatively, refers to several different theorems): Every “formal” diagram in a monoidal category made up of associators and unitors commutes.

Every diagram in a free monoidal category made up of associators and unitors commutes. This note covers the following topics: Monoidal categories, The pentagon axiom, Basic properties of unit objects in monoidal categories, monoidal categories, Monoidal functors, equivalence of monoidal categories, Morphisms of monoidal functors, MacLane's strictness theorem, The MacLane coherence theorem, Invertible objects, Exactness of the tensor product, Semisimplicity of the unit object.

At the heart of matters is the coherence theorem, of which this book provides a definitive treatment, as well as covering related results.

Along the way the author treats such material as the Gray tensor product and gives a construction of the fundamental 3-groupoid of a by: Introduction to Category Theory This note teaches the basics of category theory, in a way that is accessible and relevant to computer scientists.

The emphasis is on gaining a good understanding the basic definitions, examples, and techniques, so that students are equipped for further study on their own of more advanced topics if required. aged. Nearer to our topic, we have the coherence theorem for symmetric monoidal closed categories proved by Kelly and Mac Lane in [32] (see the end of §), where it also seems it was easier to conjecture the theorem than to prove it.

From the inception of proof nets. This book is an introduction to 2-categories and bicategories, assum- and the Coherence Theorem for bicategories. Grothendieck fibrations and the providing a symmetric monoidal closed structure on the category of 2-categories and 2-functors (Theorem ).

2-categoricalrestrictions. Coherence for compact closed categories For compact closed categories, since none of the axioms involves an equation between objects, it is easy to give the objects of Fa/: they are just what we called in 3 the objects of G.~l, which form the free ((3,I,()')-algebra on ob -l We next describe the arrows of a graph H.-,/with these by: This note explains the following topics related to Category Theory: Duality, Universal and couniversal properties, Limits and colimits, Biproducts in Vect and Rel, Functors, Natural transformations, Yoneda'a Lemma, Adjoint Functors, Cartesian Closed Categories, The Curry-Howard-Lambek Isomorphism, Induction and Coinduction, Stream programming examples and Monads.

The necessary and sufficient conditions of commutativity of all the diagrams of canonical maps in any closed category V are obtained. The main condition is that for every object A in V the first dual A ∗ is isomorphic to the third dual A ∗∗∗.It is also shown that isomorphism of A and A ∗∗ (without additional conditions) is sufficient for full by: Higher Dimensional Categories an illustrated guide book.

This work gives an explanatory introduction to various definitions of higher dimensional category. The emphasis is on ideas rather than formalities; the aim is to shed light on the formalities by emphasising the intuitions that lead there.

An introduction to Category Theory. The book is aimed primarily at the beginning graduate gives the de nition of this notion, goes through the various associated gadgetry such as functors, natural transformations, limits and colimits, and then explains adjunctions.

MacLane's strictness theorem, The MacLane coherence theorem. This paper presents a coherence theorem for star-autonomous categories exactly analogous to Kelly and Mac Lane’s coherence theorem for symmetric monoidal closed categories.

The proof of this. "Coherence is one of the most important books on business, business consulting, and leadership that has been published in the last decade. From beginning to end, it is richly detailed, deeply serious (but not ponderous), extremely illuminating, and altogether wise, and simply renders virtually all other books in this area woefully partial, fragmented, and by: 5.Instant download; Readable on all devices; Own it forever; Local sales tax included if applicable.Set is a coherent category.

More generally, any topos is coherent. Even more generally, any locally cartesian closed category with finite colimits is coherent, and thus any quasitopos is coherent. If C C is coherent, so is the functor category C D C^D for any D D. A slice category of a coherent category .