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This series of lectures reviews the remarkable feature of quantum topology: There are unexpected direct relations among algebraic structures and the combinatorics of knots and manifolds. The 6j symbols, Hopf algebras, triangulations of 3-manifolds, Temperley-Lieb algebra, and braid groups are reviewed in the first three lectures. In the second lecture, we discuss parentheses structures and 2-categories of surfaces in 3-space in relation to the Temperley-Lieb algebras. In the fourth lecture, we give diagrammatics of 4 dimensional triangulations and their relations to the associahedron, a higher associativity condition. We prove that the 4-dimensional Pachner moves can be decomposed in terms of singular moves, and lower dimensional relations. In our last lecture, we give a combinatorial description of knotted surfaces in 4-space and their isotopies. MRCN: 57Q45 Key words: Reidemeister Moves, 2-categories, Movie Moves, Knotted Surfaces 1 1 Introduction In this series of tal...

The purpose of this paper is to develop some additional techniques for the weak n-categories defined by Tamsamani in [27] (which he calls n-nerves). The goal is to be able to define the internal Hom(A, B) for two n-nerves A and B, which should itself be an n-nerve. This in turn is for defining the n + 1-nerve nCAT of all n-nerves conjectured in

We develop the theory of n-stacks (or more generally Segal n-stacks which are ∞-stacks such that the morphisms are invertible above degree n). This is done by systematically using the theory of closed model categories (cmc). Our main results are: a definition of n-stacks in terms of limits, which should be perfectly general for stacks of any type of objects; several other characterizations of n-stacks in terms of “effectivity of descent data”; construction of the stack associated to an n-prestack; a strictification result saying that any “weak ” n-stack is equivalent to a (strict) n-stack; and a descent result saying that the (n + 1)-prestack of n-stacks (on a site) is an (n + 1)-stack. As for other examples, we start from a “left Quillen presheaf ” of cmc’s and introduce the associated Segal 1-prestack. For this situation, we prove a general descent result, giving sufficient conditions for this prestack to be a stack. This applies to the case of complexes, saying how complexes of sheaves of O-modules can be glued together via quasi-isomorphisms. This was the problem that originally motivated us. Résumé

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Abstract. We define Homotopy quantum field theories (HQFT) as Topological quantum field theories (TQFT) for manifolds endowed with extra structure in the form of a map into some background space X. We also build the category of homotopy cobordisms HCobord(n, X) such that an HQFT is a functor

this paper a new approach to the problem of constructing a quantum theory of gravity in the cosmological context is proposed. It is founded on results from four separate directions of investigation, which are: 1) A new point of view towards the interpretation problem in quantum cosmology[1, 2, 3, 4], which rejects the idea that a single quantum state, or a single Hilbert space, can provide a complete description of a closed system like the universe. Instead, the idea is to accept Bohr's original proposal that the quantum state requires for its interpretation a context in which we distinguish two subsystems of the universe-the quantum system and observer. However, we seek to relativize this split, so that the boundary between the part of the universe that is considered the system and that which might be considered the observer may be chosen arbitrarily. The idea is then that a quantum theory of cosmology is specified by giving an assignment of a Hilbert space and algebra of observables to every possible boundary that can be considered to split the universe into two such subsystems. A quantum state of the universe is then an assignment of a statistical state to every one of these Hilbert spaces, subject to certain conditions of consistency. Each of these states is interpreted to contain the information that an observer on one side of each boundary might have about the system of the other side. This formulation then accepts the idea that each observer can only have incomplete information about the universe, so that the most complete description possible of the universe is given by the whole collection of incomplete, but mutually compatible quantum state descriptions of all the possible observers. At the same time, the information of different observers is, to some extent, ...

In this paper we define a class of state-sum invariants of compact closed oriented piece-wise linear 4-manifolds using finite groups. The definition of these state-sums follows from the general abstract construction of 4-manifold invariants using spherical 2-categories, as we defined in [32], although it requires a slight generalization of that construction. We show that the state-sum invariants of Birmingham and Rakowski [11, 12, 13], who studied Dijkgraaf-Witten type invariants in dimension 4, are special examples of the general construction that we present in this paper. They showed that their invariants are nontrivial by some explicit computations, so our construction includes interesting examples already. Finally, we indicate how our construction is related to homotopy 3-types. This connection suggests that there are many more interesting examples of our construction to be found in the work on homotopy 3-types, such as [15], for example. 1 1

It has been difficult to see precisely the role played by strict n-categories in the nascent theory of n-categories, particularly as related to n-truncated homotopy types of spaces. We propose to show in a fairly general setting that one cannot obtain all 3-types by any reasonable realization functor 1 from strict 3-groupoids (i.e. groupoids in the sense of [20]). More precisely we show that one does not obtain the 3-type of S 2. The basic reason is that the Whitehead bracket is nonzero. This phenomenon is actually well-known, but in order to take into account the possibility of an arbitrary reasonable realization functor we have to write the argument in a particular way. We start by recalling the notion of strict n-category. Then we look at the notion of strict n-groupoid as defined by Kapranov and Voevodsky [20]. We show that their definition is equivalent to a couple of other natural-looking definitions (one of these equivalences was left as an exercise in [20]). At the end of these first sections, we have a picture of strict 3-groupoids having only one object and one 1-morphism, as being equivalent to abelian monoidal objects (G, +) in the category of groupoids, such that (π0(G), +) is a group. In the case in question, this group will be π2(S 2) = Z. Then comes the main