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Register a new userLeft Bousfield localization without left properness
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Given a combinatorial (semi-)model category $M$ and a set of morphisms $C$, we establish the existence of a semi-model category $L_C M$ satisfying the universal property of the left Bousfield localization in the category of semi-model categories. Our main tool is a semi-model categorical version of a result of Jeff Smith, that appears to be of independent interest. Our main result allows for the localization of model categories that fail to be left proper. We give numerous examples and applications, related to the Baez-Dolan stabilization hypothesis, localizations of algebras over operads, chromatic homotopy theory, parameterized spectra, $C^*$-algebras, enriched categories, dg-categories, functor calculus, and Voevodskys work on radditive functors.
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For a model category, we prove that taking the category of coalgebras over a comonad commutes with left Bousfield localization in a suitable sense. Then we prove a general existence result for left-induced model structure on the category of coalgebras over a comonad in a left Bousfield localization. Next we provide several equivalent characterizations of when a left Bousfield localization preserves coalgebras over a comonad. These results are illustrated with many applications in chain complexes, (localized) spectra, and the stable module category.
We prove the equivalence of several hypotheses that have appeared recently in the literature for studying left Bousfield localization and algebras over a monad. We find conditions so that there is a model structure for local algebras, so that localization preserves algebras, and so that localization lifts to the level of algebras. We include examples coming from the theory of colored operads, and applications to spaces, spectra, and chain complexes.
We develop a minimal left-right symmetric model based on the gauge group $SU(3)_C otimes SU(2)_L otimes SU(2)_R otimes U(1)_{B-L}$ wherein the Higgs triplets conventionally employed for symmetry breaking are replaced by Higgs doublets. Majorana masses for the right-handed neutrinos $( u_R$) are induced via two-loop diagrams involving a charged scalar field $eta^+$. This setup is shown to provide excellent fits to neutrino oscillation data via the seesaw mechanism for the entire range of the $W_R^pm$ mass, from TeV to the GUT scale. When the $W_R^pm$ mass is at the TeV scale, the $ u_R$ masses turn out to be in the MeV range. We analyze constraints from low energy experiments, early universe cosmology and from supernova 1987a on such a scenario and show its consistency. We also study collider implications of a relatively light $eta^+$ scalar through its decay into multi-lepton final states and derive a lower limit of 390 GeV on its mass from the LHC, which can be improved to 555 GeV in its high luminosity run.
A Lie 2-group $G$ is a category internal to the category of Lie groups. Consequently it is a monoidal category and a Lie groupoid. The Lie groupoid structure on $G$ gives rise to the Lie 2-algebra $mathbb{X}(G)$ of multiplicative vector fields, see (Berwick-Evans -- Lerman). The monoidal structure on $G$ gives rise to a left action of the 2-group $G$ on the Lie groupoid $G$, hence to an action of $G$ on the Lie 2-algebra $mathbb{X}(G)$. As a result we get the Lie 2-algebra $mathbb{X}(G)^G$ of left-invariant multiplicative vector fields. On the other hand there is a well-known construction that associates a Lie 2-algebra $mathfrak{g}$ to a Lie 2-group $G$: apply the functor $mathsf{Lie}: mathsf{Lie Groups} to mathsf{Lie Algebras}$ to the structure maps of the category $G$. We show that the Lie 2-algebra $mathfrak{g}$ is isomorphic to the Lie 2-algebra $mathbb{X}(G)^G$ of left invariant multiplicative vector fields.
We study localization at a prime in homotopy type theory, using self maps of the circle. Our main result is that for a pointed, simply connected type $X$, the natural map $X to X_{(p)}$ induces algebraic localizations on all homotopy groups. In order to prove this, we further develop the theory of reflective subuniverses. In particular, we show that for any reflective subuniverse $L$, the subuniverse of $L$-separated types is again a reflective subuniverse, which we call $L$. Furthermore, we prove results establishing that $L$ is almost left exact. We next focus on localization with respect to a map, giving results on preservation of coproducts and connectivity. We also study how such localizations interact with other reflective subuniverses and orthogonal factorization systems. As key steps towards proving the main theorem, we show that localization at a prime commutes with taking loop spaces for a pointed, simply connected type, and explicitly describe the localization of an Eilenberg-Mac Lane space $K(G,n)$ with $G$ abelian. We also include a partial converse to the main theorem.
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