51 pages LaTeX with 9 encapsulated Postscript figures
发表信息：
In Higher Category Theory, eds. Ezra Getzler and Mikhail Kapranov, Contemp. Math. 230, American Mathematical Society, Providence, Rhode Island, 1998, pp. 1-36.
Categorification is the process of finding category-theoretic analogs of set-theoretic concepts by replacing sets with categories, functions with functors, and equations between functions by natural isomorphisms between functors, which in turn should satisfy certain equations of their own, called `coherence laws'. Iterating this process requires a theory of `n-categories', algebraic structures having objects, morphisms between objects, 2-morphisms between morphisms and so on up to n-morphisms. After a brief introduction to n-categories and their relation to homotopy theory, we discuss algebraic structures that can be seen as iterated categorifications of the natural numbers and integers. These include tangle n-categories, cobordism n-categories, and the homotopy n-types of the loop spaces Omega^k S^k. We conclude by describing a definition of weak n-categories based on the theory of operads.
We identify the Grothendieck group of certain direct sum of singular blocks of the highest weight category for sl(n) with the n-th tensor power of the fundamental (two-dimensional) sl(2)-module. The action of U(sl(2)) is given by projective functors and the commuting action of the Temperley-Lieb algebra by Zuckerman functors. Indecomposable projective functors correspond to Lusztig canonical basis in U(sl(2)). In the dual realization the n-th tensor power of the fundamental representation is identified with a direct sum of parabolic blocks of the highest weight category. Translation across the…
In this paper I define certain interesting 2-functors from the Khovanov-Lauda 2-category which categorifies quantum sl(k), for any k>1, to a 2-category of universal sl(3) foams with corners. For want of a better name I use the term "foamation" to indicate those 2-functors. I conjecture the existence of similar 2-functors to the 2-category of sl(n) foams with corners, for any n>1.
We construct 2-functors from a 2-category categorifying quantum sl(n) to 2-categories categorifying the irreducible representation of highest weight $ 2 ω_k. $
A modular tensor category provides the appropriate data for the construction of a three-dimensional topological field theory. We describe the following analogue for two-dimensional conformal field theories: a 2-category whose objects are symmetric special Frobenius algebras in a modular tensor category and whose morphisms are categories of bimodules. This 2-category provides sufficient ingredients for constructing all correlation functions of a two-dimensional rational conformal field theory. The bimodules have the physical interpretation of chiral data, boundary conditions, and topological de…
Motivated by generalizing Khovanov's categorification of the Jones polynomial, we study functors $F$ from thin posets $P$ to abelian categories $\mathcal{A}$. Such functors $F$ produce cohomology theories $H^*(P,\mathcal{A},F)$. We find that CW posets, that is, face posets of regular CW complexes, satisfy conditions making them particularly suitable for the construction of such cohomology theories. We consider a category of tuples $(P,\mathcal{A},F,c)$, where $c$ is a certain $\{1,-1\}$-coloring of the cover relations in $P$, and show the cohomology arising from a tuple $(P,\mathcal{A},F,c)$ i…
We define virtual braid groups of type B and construct a morphism from such a group to the group of isomorphism classes of some invertible complexes of bimodules up to homotopy.
In this paper, we study the diagrammatic categorification of the fermion algebra. We construct a graphical category corresponding to the one-dimensional fermion algebra, and we investigate the properties of this category. The categorical analogues of the Fock states are some kind of 1-morphisms in our category, and the dimension of the vector space of 2-morphisms is exactly the inner product of the corresponding Fock states. All the results in our categorical framework coincide exactly with those in normal quantum mechanics.
We make a first step towards categorification of the dendriform operad, using categories of modules over the Tamari lattices. This means that we describe some functors that correspond to part of the operad structure.
A diagrammatic presentation of functors and natural transformations and the virtues of biadjointness are discussed. We then review a graphical description of the category of Soergel bimodules and a diagrammatic categorification of positive halves of quantum groups. These notes are a write-up of Takagi lectures given by the author in Hokkaido University in June 2009.
We consider a possible framework to categorify the exponential map exp(-f) given the categorification of a generator f of $\frak{sl}_2$ by Lauda. In this setup the Taylor expansions of exp(-f) and exp(f) turn into complexes built out of categorified divided powers of f. Hom spaces between tensor powers of categorified f are given by diagrammatics combining nilHecke algebra relations with those for a additional "short strand" generator. The proposed framework is only an approximation to categorification of exponentiation, because the functors categorifying exp(f) and exp(-f) are not invertible.
We prove that the notion of Drinfeld center defines a functor from the category of indecomposable multi-tensor categories with morphisms given by bimodules to that of braided tensor categories with morphisms given by monoidal bimodules. Moreover, we apply some ideas from the physics of topological orders to prove that the center functor restricted to indecomposable multi-fusion categories (with additional conditions on the target category) is fully faithful. As byproducts, we provide new proofs to some important known results in fusion categories. In physics, this fully faithful functor gives …
In this note we consider different versions of coinduction functors between categories of comodules for corings induced by a morphism of corings. In particular we introduce a new version of the coinduction functor in the case of locally projective corings as a composition of suitable ``Trace'' and ``Hom'' functors and show how to derive it from a more general coinduction functor between categories of type $σ\lbrack M].$ In special cases (e.g. the corings morphism is part of a morphism of measuring $α$-pairings or the corings have the same base ring), a version of our functor is shown to be iso…
Given a fusion category $\mathcal{C}$ and an indecomposable $\mathcal{C}$-module category $\mathcal{M}$, the fusion category $\mathcal{C}^*_\mathcal{M}$ of $\mathcal{C}$-module endofunctors of $\mathcal{M}$ is called the (Morita) dual fusion category of $\mathcal{C}$ with respect to $\mathcal{M}$. We describe tensor functors between two arbitrary duals $\mathcal{C}^*_\mathcal{M}$ and $\mathcal{D}^*_\mathcal{N}$ in terms of data associated to $\mathcal{C}$ and $\mathcal{D}$. We apply the results to $G$-equivariantizations of fusion categories and group-theoretical fusion categories. We describe…
Universidade Federal de Santa Catarina（圣卡塔琳娜联邦大学）；Università di Napoli Federico II（那不勒斯费德里科二世大学）；INFN Sezione di Napoli（意大利国家核物理研究所那不勒斯分部）；Instytut Matematyczny, Polska Akademia Nauk（波兰科学院数学研究所）
Categorification is a process of lifting structures to a higher categorical level. The original structure can then be recovered by means of the so-called "decategorification" functor. Algebras are typically categorified to additive categories with additional structure and decategorification is usually given by the (split) Grothendieck group. In this expository article we study an alternative decategorification functor given by the trace or the zeroth Hochschild--Mitchell homology. We show that this form of decategorification endows any 2-representation of the categorified quantum sl(n) with an…
We study a 2-functor that assigns to a bimodule category over a finite k-linear tensor category a k-linear abelian category. This 2-functor can be regarded as a category-valued trace for 1-morphisms in the tricategory of finite tensor categories. It is defined by a universal property that is a categorification of Hochschild homology of bimodules over an algebra. We present several equivalent realizations of this 2-functor and show that it has a coherent cyclic invariance. Our results have applications to categories associated to circles in three-dimensional topological field theories with defe…
We construct knot invariants categorifying the quantum knot variants for all representations of quantum groups. We show that these invariants coincide with previous invariants defined by Khovanov for sl(2) and sl(3) and by Mazorchuk-Stroppel and Sussan for sl(n). Our technique uses categorifications of the tensor product representations of Kac-Moody algebras and quantum groups, constructed a prequel to this paper. These categories are based on the pictorial approach of Khovanov and Lauda. In this paper, we show that these categories are related by functors corresponding to the braiding and (co…
Certain quantization problems are equivalent to the construction of morphisms from "quantum" to "classical" props. Once such a morphism is constructed, Hensel's lemma shows that it is in fact an isomorphism. This gives a new, simple proof that any Etingof-Kazhdan quantization functor is an equivalence of categories between quantized universal enveloping (QUE) algebras and Lie bialgebras over a formal series ring (dequantization). We apply the same argument to construct dequantizations of formal solutions of the quantum Yang-Baxter equation and of quasitriangular QUE algebras. We also give stru…
We define the Hopf superalgebra U_T sl(1,1), which is a variant of the quantum supergroup U_q sl(1,1), and its tensor product representations V_1^{\otimes n} for n>0. We construct families of DG algebras A, B and R_n, and consider the DG categories DGP(A), DGP(B) and DGP(R_n), which are full DG subcategories of the categories of DG A-, B- and R_n-modules generated by certain distinguished projective modules. Their 0th homology categories HP(A), HP(B), and HP(R_n) are triangulated and give algebraic formulations of the contact categories of an annulus, a twice punctured disk, and an n times pun…
