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Register a new user2-categorical opfibrations, Quillens Theorem B, and $S^{-1}S$
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In this paper we show that the strict and lax pullbacks of a 2-categorical opfibration along an arbitrary 2-functor are homotopy equivalent. We give two applications. First, we show that the strict fibers of an opfibration model the homotopy fibers. This is a version of Quillens Theorem B amenable to applications. Second, we compute the $E^2$ page of a homology spectral sequence associated to an opfibration and apply this machinery to a 2-categorical construction of $S^{-1}S$. We show that if $S$ is a symmetric monoidal 2-groupoid with faithful translations then $S^{-1}S$ models the group completion of $S$.
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We prove a bicategorical analogue of Quillens Theorem A. As an application, we deduce the well-known result that a pseudofunctor is a biequivalence if and only if it is essentially surjective on objects, essentially full on 1-cells, and fully faithful on 2-cells.
In this paper, we will be concerned with the explicit classification of closed, oriented, simply-connected spin manifolds in dimension eight with vanishing cohomology in the odd dimensions. The study of such manifolds was begun by Stefan Muller. In order to understand the structure of these manifolds, we will analyze their minimal handle presentations and describe explicitly to what extent these handle presentations are determined by the cohomology ring and the characteristic classes. It turns out that the cohomology ring and the characteristic classes do not suffice to reconstruct a manifold of the above type completely. In fact, the group ${rm Aut_0}bigl(#_{i=1}^b(S^2times S^5)bigr)/{rm Aut}_0bigl(#_{i=1}^b (S^2times D^6)bigr)$ of automorphisms of $#_{i=1}^b(S^2times S^5)$ which induce the identity on cohomology modulo those which extend to $#_{i=1}^b(S^2times D^6)$ acts on the set of oriented homeomorphy classes of manifolds with fixed cohomology ring and characteristic classes, and we will be also concerned with describing this group and some facts about the above action.
Bimonoidal categories are categorical analogues of rings without additive inverses. They have been actively studied in category theory, homotopy theory, and algebraic $K$-theory since around 1970. There is an abundance of new applications and questions of bimonoidal categories in mathematics and other sciences. This work provides a unified treatment of bimonoidal and higher ring-like categories, their connection with algebraic $K$-theory and homotopy theory, and applications to quantum groups and topological quantum computation. With ample background material, extensive coverage, detailed presentation of both well-known and new theorems, and a list of open questions, this work is a user friendly resource for beginners and experts alike.
We describe a category of undirected graphs which comes equipped with a faithful functor into the category of (colored) modular operads. The associated singular functor from modular operads to presheaves is fully faithful, and its essential image can be classified by a Segal condition. This theorem can be used to recover a related statement, due to Andre Joyal and Joachim Kock, concerning a larger category of undirected graphs whose functor to modular operads is not just faithful but also full.
In this work, we perform a systematical investigation about the possible hidden and doubly heavy molecular states with open and hidden strangeness from interactions of $D^{(*)}{bar{D}}^{(*)}_{s}$/$B^{(*)}{bar{B}}^{(*)}_{s}$, ${D}^{(*)}_{s}{bar{D}}^{(*)}_{s}$/${{B}}^{(*)}_{s}{bar{B}}^{(*)}_{s}$, ${D}^{(*)}D_{s}^{(*)}$/${B}^{(*)}B_{s}^{(*)}$, and $D_{s}^{(*)}D_{s}^{(*)}$/$B_{s}^{(*)}B_{s}^{(*)}$ in a quasipotential Bethe-Salpeter equation approach. The interactions of the systems considered are described within the one-boson-exchange model, which includes exchanges of light mesons and $J/psi/Upsilon$ meson. Possible molecular states are searched for as poles of scattering amplitudes of the interactions considered. The results suggest that recently observed $Z_{cs}(3985)$ can be assigned as a molecular state of $D^*bar{D}_s+Dbar{D}^*_s$, which is a partner of $Z_c(3900)$ state as a $Dbar{D}^*$ molecular state. The calculation also favors the existence of hidden heavy states $D_sbar{D}_s/B_sbar{B}_s$ with spin parity $J^P=0^+$, $D_sbar{D}^*_s/B_sbar{B}^*_s$ with $1^{+}$, and $D^*_sbar{D}^*_s/B^*_sbar{B}^*_s$ with $0^+$, $1^+$, and $2^+$. In the doubly heavy sector, the bound states can be found from the interactions $(D^*D_s+DD^*_s)/(B^*B_s+BB^*_s)$ with $1^+$, $D_sbar{D}_s^*/B_sbar{B}_s^*$ with $1^+$, $D^*D^*_s/B^*B^*_s$ with $1^+$ and $2^+$, and $D^*_sD^*_s/B^*_sB^*_s$ with $1^+$ and $2^+$. Some other interactions are also found attractive, but may be not strong enough to produce a bound state. The results in this work are helpful for understanding the $Z_{cs}(3985)$, and future experimental search for the new molecular states.
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