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Wednesday, May 20, 2020 | History

2 edition of Closed model categories and monoidal categories found in the catalog.

Closed model categories and monoidal categories

Donald Stanley

# Closed model categories and monoidal categories

## by Donald Stanley

Published in 1997 .
Written in English

The Physical Object
Paginationv, 174 leaves.
Number of Pages174
ID Numbers
Open LibraryOL20866378M

Example If C is a monoidal category, then so is its opposite tensor unit Iin Cop is the same as that in C, whereas the tensor product A Bin Cop is given by B Ain C, the associators in Cop are the inverses of those morphisms in C, and the left and right unitors of C swap roles in Cop. Monoidal categories have an important property called the interchange law, which governs the File Size: KB. Re: Quantum computation and symmetric monoidal categories. Concerning Frobenius algebras, one can actually drop the additional biproduct structure used in A categorical semantics of quantum protocols and encode classical data manipulations using dagger-compact Frobenius algebras, while, crucially, seemingly retaining all required expressive power. In Quantum measurements without sums Dusko.

Almost a year ago, Mike Stay and I proudly announced the completion of our Rosetta Stone paper, which explains how symmetric monoidal closed categories show up in physics, topology, logic and computation. But then Theo and Todd helped us spot a serious mistake in our description of a programming language that was supposed to be suitable for work inside any symmetric monoidal closed category. Keywords Model category, monoidal category, Dold-Kan equivalence, spectra 1 Introduction This paper is a sequel to [SS00] where we studied su cient conditions for extend-ing Quillen model category structures to the associated categories of monoids (rings), modules and algebras over a monoidal model category. Here we consider.

A closed monoidal category is a monoidal category where the functor ↦ ⊗ has a right adjoint, which is called the "internal Hom-functor" ↦ (,). Examples include cartesian closed categories such as Set, the category of sets, and compact closed categories such as . The book highlights in particular the distinguished role of equationally defined structures within the given lax-algebraic context and presents numerous new results ranging from topology and approach theory to domain theory. All the necessary pre-requisites in order and category theory are presented in the book.

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### Closed model categories and monoidal categories by Donald Stanley Download PDF EPUB FB2

Of a monoidal category, [13] for example. Monoidal objects in these categories also play an important role. For example monoidal ob j ects in topological spaces are t opological monoids and group-like topological monoids are essentially the same as loop spaces.

It is to these two topics, closed model categories and monoidal categories, and to the interactions between them that this thesis has. closed simplicial model categories [Q, II.2] are such compatibly enriched categories over the monoidal model category of simplicial sets.

We alsointroduce the monoid axiom which is the crucial ingredient for lifting the model category structure to monoids and modules.

Examples of monoidal model categories satisfying the monoid axiom are given. over the monoidal model category of simplicial sets, for example, is the same thing as a simplicial model category. Of course, the homotopy category of a monoidal model category is a closed monoidal category in a natural way, and similarly for modules and algebras.

The material in this chapter is all fairly straightforward,File Size: 1MB. Because the interest is in theorems obtainable by syntactic manipulations in sequential categories, the chapter is restricted to categories in which analogues of the export–import law hold.

The focus is on monoidal categories with additional structure. Such categories are denoted as closed categories. ARROW CATEGORIES OF MONOIDAL MODEL CATEGORIES 5 Theorem Suppose Mis a monoidal model category.

Then Ð→ M equipped with the projective model structure is a monoidal model category. Proof. We already know that Ð→ M is a symmetric monoidal closed category equipped with the projective model structure. We must show that it satis. According to G.M. Kelly's Examples of Non-monadic Structures on Categories (page 63), downloadable here, $\alpha$ is not necessarily an isomorphism if we only have a closed category with an adjoint to $[-,-]$.Unfortunately, as far I can Closed model categories and monoidal categories book, Kelly does not give a counterexample.

Thanks to Buschi Sergio in the comments for this question for pointing me to Day and LaPlaza's paper On Embedding. However, there is an obvious generalization of the Enrichment Axiom for monoidal model categories to the context of closed model categories enriched over a monoidal model category.

For the statement of this axiom, we need the following generalization of the map [ g, h ] from Section by: A monoidal model category is a model category which is also a closed monoidal category in a compatible way.

In particular, its homotopy category inherits a closed monoidal structure, as does the (infinity,1)-category that it presents. tionary, monoidal categories indeed correspond to monoids (which ex­ plains their name). Deﬁnition A monoidal subcategory of a monoidal category (C, ⊗, a, 1,ι) is a quintuple (D, ⊗, a, 1,ι), where D⊂C is a subcate­ gory closed under the tensor product of objects and morphisms and containing 1 and ι.

Deﬁnition opFile Size: KB. However, you will need more than just enriched monoidal categories in order to build an equivalence between the words "Monoidal category $\mathcal C$ enriched over $\mathcal V$", and "Monoidal category $\mathcal C$ with a braided monoidal functor $\mathcal V \to Z(\mathcal C)$".

On traced monoidal closed categories Article (PDF Available) in Mathematical Structures in Computer Science 19(2) April with 59 Reads How we measure 'reads'. Another important kind of monoidal category is a closed monoidal category.

A closed monoidal category is a monoidal category where the functor has a right adjoint (see Adjoint Functors and Monads) also known as the “internal Hom functor”, which is like a Hom functor that takes values in the category itself instead of in sets, and is denoted.

monoidal category, for example, a ring in the category of abelian groups under tensor product. To work with this symmetric monoidal product it must be compatible with the model category structure, which leads to the deﬁnition of a monoidal model category; see Deﬁnition To obtain a model category.

A closed category $\mathcal{V}$ is a symmetric monoidal category in which each functor ${-}\otimes b: \mathcal{V} \rightarrow \mathcal{V}$ has a specified right-adjoint $({-})^b: \mathcal{V} \rightarrow \mathcal{V}$.

Some examples of closed monoidal categories are. PDF | The category theory provides possibilities to model many important features of computer science. We used the symmetric monoidal closed category | Find, read and cite all the research you.

Traced monoidal categories Yank Balance Remark. Instead of ribbons it is permissible to draw strings with integer tso record the twists I.n fact th, e integers can be put anywher e on the string because thi iss known to be possible in balanced monoidal categories, and can be extende tdo traced monoidal categories becaus of slidinge.

over M discussed above passes to the homotopy categories. Theorem Let M be a monoidal model category, and let A be a monoid in M. If the category of left A-modules is a closed model category with ﬁbrations and weak equivalences created in M, then.

proofs) on traced monoidal categories, Int-construction, monoidal closed categories and monoidal adjunctions in Section 2 and 3, makingthe paper n 5 is devoted for applications in models of linear logic and xed-point computation; for.

Let $(\mathbf{M},\otimes,1)$ be a closed monoidal category and $(\mathbf{C},\oplus,0)$ an $\mathbf{M}$-enriched monoidal rmore, assume that we have a copowering $\odot:\mathbf{M}\times\mathbf{C}\to \mathbf{C}$.Is there a canonical morphism $$(A\odot X)\oplus (B\odot Y)\to (A\otimes B)\odot (X\oplus Y)$$ The question came to my mind because in order to spell.

A monoidal model category is a model category with a compatible closed monoidal structure. Such things abound in nature; simplicial sets and chain complexes of abelian groups are examples. Given a monoidal model category, one can consider monoids and modules over a given monoid.

This paper introduces the following new constructions on stable event structures: the tensor product, the linear function space, and the exponential. It results in a monoidal closed category Cited by: 3.point to one article, we provide our own comprehensive development of Prof, the compact closed monoidal weak 2-category of categories, profunctors, and natural transformations in Sectiondrawing on many sources.

The idea of using profunctors to enable the development of formal category theory inside a 2-category can be attributed to WoodFile Size: KB.cosmos V to be a model category in the next section, but we ignore model category theory for the moment.

When ⊗ is the cartesian product, we say that V is cartesian closed, but the same category V can admit other symmetric monoidal structures. Examples We give examples of cosmoi V. (i) The category Set of sets is closed cartesian monoidal.