Kan extensions in Enriched Category Theory

The small object argument is a transfinite construction which, starting from a set of maps in a category, generates a weak factorisation system on that category. As useful as it is, the small object argument has some problematic aspects: it possesses no universal property; it does not converge; and it does not seem to be related to other transfinite constructions occurring in categorical algebra. In this paper, we give an "algebraic" refinement of the small object argument, cast in terms of Grandis and Tholen's natural weak factorisation systems, which rectifies each of these three deficiencies.

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“…For G 3 , we have the following well-known result, which was first stated at this level of generality by Dubuc [7]; but see also [2].…”

Section: 5mentioning

confidence: 71%

Understanding the Small Object Argument

Garner

2008

Appl Categor Struct

226

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“…Let E 0 be the -projective category associated to B 0 with projection K 0 WD L as in the proof of Proposition 8. 26. Then E and K as defined in Remark 8.19 are a cofibrant replacement for B in the -model structure.…”

Section: Remark 831mentioning

confidence: 97%

Model structures on the category of small double categories

Fiore

Paoli

Pronk

2008

Algebr. Geom. Topol.

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In this paper we obtain several model structures on DblCat, the category of small double categories. Our model structures have three sources. We first transfer across a categorification-nerve adjunction. Secondly, we view double categories as internal categories in Cat and take as our weak equivalences various internal equivalences defined via Grothendieck topologies. Thirdly, DblCat inherits a model structure as a category of algebras over a 2-monad. Some of these model structures coincide and the different points of view give us further results about cofibrant replacements and cofibrant objects. As part of this program we give explicit descriptions for and discuss properties of free double categories, quotient double categories, colimits of double categories, horizontal nerve and horizontal categorification.18D05, 18G55; 55P99, 55U10

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“…by the existence of the functor in the upper line and the fact that C ‫ލ‬ has coequalisers (see [1,Section 4], [3,Appendix] and [6]). …”

mentioning

confidence: 99%

Comodules and Contramodules

Wisbauer

2010

Glasgow Math. J.

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Abstract. Algebras A and coalgebras C over a commutative ring R are defined by properties of the (endo)functors A ⊗ R − and C ⊗ R − on the category of R-modules R ‫.ލ‬ Generalising these notions, monads and comonads were introduced on arbitrary categories, and it turned out that some of their basic relations do not depend on the specific properties of the tensor product. In particular, the adjoint of any comonad is a monad (and vice versa), and hence, for any coalgebra C, Hom R (C, −), the right adjoint of C ⊗ R −, is a monad on R ‫.ލ‬ The modules for the monad Hom R (C, −) were called contramodules by Eilenberg-Moore and the purpose of this talk is to outline the related constructions and explain the relationship between C-comodules and Ccontramodules. The results presented grew out from cooperation with G. Böhm, T. Brzeziński and B. Mesablishvili.2010 Mathematics Subject Classification. 16T15, 18C20. Preliminaries.In this section we recall some formalisms for module categories and their transfer to arbitrary categories. Throughout R will denote a commutative ring with unit and R ‫ލ‬ the category of (left) R-modules. We recall (once again) the basic definitions explicitly since they provide the prototype of the constructions to follow.For any R-modules N, X, N ⊗ R X is again an R-module and thus we have a functor (endofunctor) x r r r r r r r r r r r A.

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Kan extensions in Enriched Category Theory

Cited by 190 publications

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Understanding the Small Object Argument

Understanding the Small Object Argument

Model structures on the category of small double categories

Comodules and Contramodules

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