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Monad interleaving: a construction of the operad for Leinster’s weak ωcategories, Preprint, 2003, available at http://arxiv.org/abs/math/0309336
"... We show how to “interleave ” the monad for operads and the monad for contractions on the category Coll of collections, to construct the monad for the operadswithcontraction of Leinster. We first decompose the adjunction for operads and the adjunction for contractions into a chain of adjunctions ea ..."
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We show how to “interleave ” the monad for operads and the monad for contractions on the category Coll of collections, to construct the monad for the operadswithcontraction of Leinster. We first decompose the adjunction for operads and the adjunction for contractions into a chain of adjunctions each of which acts on only one dimension of the underlying globular sets at a time. We then exhibit mutual stability conditions that enable us to alternate the dimensionbydimension free functors. Hence we give an explicit construction of a left adjoint
Multitensors and monads on categories of enriched graphs
"... Abstract. In this paper we unify the developments of [Batanin, 1998], [BataninWeber, 2011] and [Cheng, 2011] into a single framework in which the interplay between multitensors on a category V, and monads on the category GV of graphs enriched in V, is taken as fundamental. The material presented he ..."
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Abstract. In this paper we unify the developments of [Batanin, 1998], [BataninWeber, 2011] and [Cheng, 2011] into a single framework in which the interplay between multitensors on a category V, and monads on the category GV of graphs enriched in V, is taken as fundamental. The material presented here is the conceptual background for subsequent work: in [BataninCisinskiWeber, 2012] the Gray tensor product of 2categories and the Crans tensor product [Crans, 1999] of Gray categories are exhibited as existing within our framework, and in [Weber, 2013] the explicit construction of the funny tensor product of categories is generalised to a large class of Batanin operads. 1.
A QUILLEN APPROACH TO DERIVED CATEGORIES AND TENSOR PRODUCTS
, 2006
"... Abstract. We put a monoidal model category structure (in the sense of Quillen) on the category of chain complexes of quasicoherent sheaves over a quasicompact and semiseparated scheme X. The approach generalizes and simplifies the method used by the author in [Gil04] and [Gil06] to build monoidal ..."
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Abstract. We put a monoidal model category structure (in the sense of Quillen) on the category of chain complexes of quasicoherent sheaves over a quasicompact and semiseparated scheme X. The approach generalizes and simplifies the method used by the author in [Gil04] and [Gil06] to build monoidal model structures on the category of chain complexes of modules over a ring and chain complexes of sheaves over a ringed space. Indeed, much of the paper is dedicated to showing that in any Grothendieck category G, a nice enough class of objects F (which we call a Kaplansky class) induces a model structure on the category Ch(G) of chain complexes. We also find simple conditions to put on F which will guarantee that our model structure in monoidal. We see that the common model structures used in practice are all induced by such Kaplansky classes. 1.
Bibliography for Practical Foundations of Mathematics
, 1999
"... Algebra, pages 263297. Pergamon Press, 1970. [Knu74] Donald Knuth. Surreal Numbers. AddisonWesley, 1974. [Koc81] Anders Kock. Synthetic Differential Geometry. Number 51 in London Mathematical Society Lecture Notes. Cambridge University Press, 1981. [Koc95] Anders Kock. Monads for which structu ..."
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Algebra, pages 263297. Pergamon Press, 1970. [Knu74] Donald Knuth. Surreal Numbers. AddisonWesley, 1974. [Koc81] Anders Kock. Synthetic Differential Geometry. Number 51 in London Mathematical Society Lecture Notes. Cambridge University Press, 1981. [Koc95] Anders Kock. Monads for which structures are adjoint to units. Journal of Pure and Applied Algebra, 104:4159, 1995. [Kol25] Andrei Kolmogorov. On the principle of excluded middle. Matematiceskii Sbornik, 32:646667, 1925. In Russian; English translation in [vH67], pages 414437. [Koy82] C. P. J. Koymans. Models of the lambda calculus. Information and Control, 52:206 332, 1982. [Kre58] Georg Kreisel. Mathematical significance of consistency proofs. Journal of Symbolic Logic, 23:155182, 1958. [Kre67] Georg Kreisel. Informal rigour and completeness proofs. In Imre Lakatos, editor, Problems in the Philosophy of Mathematics. NorthHolland, 1967. [Kre68] Georg Kreisel. A survey of proof theory. Journal of Symbolic Logic,...