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Lucian M. Ionescu 《Applied Categorical Structures》2002,10(1):35-47
We review several known categorification procedures, and introduce a functorial categorification of group extensions (Section 4.1) with applications to non-Abelian group cohomology (Section 4.2). The obstruction to the existence of group extensions (Section 4.2.4, Equation (9)) is interpreted as a coboundary condition (Proposition 4.5). 相似文献
14.
We construct the intertwining operator superalgebras and vertex tensor categories for the superconformal unitary minimal models and other related models.
15.
Clemens Berger 《Advances in Mathematics》2002,169(1):118-175
16.
E. Vasserot 《Compositio Mathematica》2002,131(1):51-60
It was proved by Ginzburg, Mirkovic and Vilonen that the G(O)-equivariant perverse sheaves on the affine Grassmannian of a connected reductive group G form a tensor category equivalent to the tensor category of finite dimensional representations of the dual group G
. In this paper we construct explicitly the action of G
on the global cohomology of a perverse sheaf. 相似文献
17.
Let T be a monad over a category A. Then a homotopy structure for A, defined by a cocylinder P : A A, or path-endofunctor, can be lifted to the category A
T
of Eilenberg–Moore algebras over T, provided that P is consistent with T in a natural sense, i.e. equipped with a natural transformation : T P P T satisfying some obvious axioms. In this way, homotopy can be lifted from well-known, basic situations to various categories of algebras for instance, from topological spaces to topological semigroups, or spaces over a fixed space (fibrewise homotopy), or actions of a fixed topological group (equivariant homotopy); from categories to strict monoidal categories; from chain complexes to associative chain algebras. The interest is given by the possibility of lifting the homotopy operations (as faces, degeneracy, connections, reversion, interchange, vertical composition, etc.) and their axioms from A to A
T
, just by verifying the consistency between these operations and : T P P T. When this holds, the structure we obtain on our category of algebras is sufficiently powerful to ensure the main general properties of homotopy. 相似文献
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We develop a notion of an n-fold monoidal category and show that it corresponds in a precise way to the notion of an n-fold loop space. Specifically, the group completion of the nerve of such a category is an n-fold loop space, and free n-fold monoidal categories give rise to a finite simplicial operad of the same homotopy type as the classical little cubes operad used to parametrize the higher H-space structure of an n-fold loop space. We also show directly that this operad has the same homotopy type as the n-th Smith filtration of the Barratt-Eccles operad and the n-th filtration of Berger's complete graph operad. Moreover, this operad contains an equivalent preoperad which gives rise to Milgram's small model for when n=2 and is very closely related to Milgram's model of for n>2. 相似文献
20.
On the Structure of Modular Categories 总被引:1,自引:0,他引:1
For a braided tensor category C and a subcategory K there isa notion of a centralizer CC K, which is a full tensor subcategoryof C. A pre-modular tensor category is known to be modular inthe sense of Turaev if and only if the center Z2C CCC (not tobe confused with the center Z1 of a tensor category, relatedto the quantum double) is trivial, that is, consists only ofmultiples of the tensor unit, and dimC 0. Here , the Xi being the simple objects. We prove several structural properties of modular categories.Our main technical tool is the following double centralizertheorem. Let C be a modular category and K a full tensor subcategoryclosed with respect to direct sums, subobjects and duals. ThenCCCCK = K and dim K·dim CCK = dim C. We give several applications. (1) If C is modular and K is a full modular subcategory,then L=CCK is also modular and C is equivalent as a ribbon categoryto the direct product: . Thus every modular category factorizes (non-uniquely, in general)into prime modular categories. We study the prime factorizationsof the categories D(G)-Mod, where G is a finite abelian group. (2) If C is a modular *-category and K is a full tensorsubcategory then dim C dim K · dim Z2K. We give exampleswhere the bound is attained and conjecture that every pre-modularK can be embedded fully into a modular category C with dim C=dimK·dim Z2K. (3) For every finite group G there is a braided tensor*-category C such that Z2CRep,G and the modular closure/modularization is non-trivial. 2000 MathematicsSubject Classification 18D10. 相似文献