We prove that the marked triangulation functor from the category of marked cubical sets equipped with a model structure for (n-trivial, saturated) comical sets to the category of marked simplicial set equipped with a model structure for (n-trivial, saturated) complicial sets is a Quillen equivalence. Our proof is based on the theory of cones, previously developed by the first two authors together with Lindsey and Sattler.Proof. We will apply [Ols09, Theorem 2.2.5] (with L taken to be all monomorphisms). Our functorial cylinder is given byNote that (1), ( 2) and (4) imply that X ⊗ (∂ 0 , ∂ 1 ) is a monomorphism for each X since it can be written as i X ⊗(∂ 0 , ∂ 1 ) where i X ∶ 0 → X is the unique map. Similarly, since any map from a terminal object (and in particular ∂ 0 , ∂ 1 ) is a monomorphism, we can deduce that X ⊗ ∂ 0 and X ⊗ ∂ 1 are always
We construct a model structure on the category of cubical sets with connections whose cofibrations are the monomorphisms and whose fibrant objects are defined by the right lifting property with respect to inner open boxes, the cubical analogue of inner horns. We show that this model structure is Quillen equivalent to the Joyal model structure on simplicial sets via the triangulation functor. As an application, we show that cubical quasicategories admit an elegant and canonical notion of a mapping space between two objects.
This paper deals with computing the global dimension of endomorphism rings of maximal Cohen-Macaulay (=MCM) modules over commutative rings. Several examples are computed. In particular, we determine the global spectra, that is, the sets of all possible finite global dimensions of endomorphism rings of MCM-modules, of the curve singularities of type An for all n, Dn for n ≤ 13 and E 6,7,8 and compute the global dimensions of Leuschke's normalization chains for all ADE curves, as announced in [DFI15]. Moreover, we determine the centre of an endomorphism ring of a MCM-module over any curve singularity of finite MCM-type. In general, we describe a method for the computation of the global dimension of an endomorphism ring End R M , where R is a Henselian local ring, using add(M )-approximations. When M = 0 is a MCM-module over R and R is Henselian local of Krull dimension ≤ 2 with a canonical module and of finite MCM-type, we use Auslander-Reiten theory and Iyama's ladder method to explicitly construct these approximations.Contents 1 1 We only consider MCM-modules because there is a rich structure theory which also features representation theoretic methods, having their origin in representation theory of Artin algebras, see [LW12,Yos90].
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