We introduce the notion of representable multicategory , which stands in the same relation to that of monoidal category as bration does to contravariant pseudofunctor (into Cat). We give an abstract reformulation of multicategories as monads in a suitable Kleisli bicategory of spans. We describe representability in elementary terms via universal arrows . We also give a doctrinal characterisation of representability based on a fundamental monadic adjunction between the 2-category of multicategories and that of strict monoidal categories. The first main result is the coherence theorem for representable multicategories, asserting their equivalence to strict ones, which we establish via a new technique based on the above doctrinal characterisation. The other main result is a 2-equivalence between the 2category of representable multicategories and that of monoidal categories and strong monoidal functors. This correspondence extends smoothly to one between bicategories and a se...

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To a bicategory B (in the sense of Benabou) we assign a simplicial set Ner(B), the (geometric) nerve of B, which completely encodes the structure of B as a bicategory. As a simplicial set Ner(B) is a subcomplex of its 2-Coskeleton and itself isomorphic to its 3-Coskeleton, what we call a 2-dimensional Postnikov complex. We then give, somewhat more delicately, a complete characterization of those simplicial sets which are the nerves of bicategories as certain 2-dimensional Postnikov complexes which satisfy certain restricted "exact horn-lifting" conditions whose satisfaction is controlled by (and here defines) subsets of (abstractly) invertible 2 and 1-simplices. Those complexes which have, at minimum, their degenerate 2-simplices always invertible and have an invertible 2-simplex # 1 2 (x 12 , x 01 ) present for each "composable pair" (x 12 , , x 01 ) # # 1 2 are exactly the nerves of bicategories. At the other extreme, where all 2 and 1-simplices are invertible, are those Kan complexes in which the Kan conditions are satisfied exactly in all dimensions > 2. These are exactly the nerves of bigroupoids -- all 2-cells are isomorphisms and all 1-cells are equivalences. Contents

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

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

Generalising a result of Brown and Loday, we give for n = 3 and 4, a decomposition of the group, dn NGn ; of boundaries of a simplicial group G as a product of commutator subgroups. Partial results are given for higher dimensions. Applications to 2-crossed modules and quadratic modules are discussed. A. M. S. Classication: 18G30, 55U10, 55P10. Introduction Simplicial groups occupy a place somewhere between homological group theory, homotopy theory, algebraic K-theory and algebraic geometry. In each sector they have played a signicant part in developments over quite a lengthy period of time and there is an extensive literature on their homotopy theory. In homotopy theory itself, they model all connected homotopy types and allow analysis of features of such homotopy types by a combination of group theoretic methods and tools from combinatorial homotopy theory. Simplicial groups have a natural structure of Kan complexes and so are potentially models for weak innity categories. They d...

There is an infinite series of notions, starting with symplectic manifolds (n = 0), Poisson manifolds (n = 1) and Courant algebroids (n = 2); there seems to be no name for higher n’s, so let us call the n’th term Σnmanifolds. Their overview is in Table 1 at the end of the paper. Except for the non-standard terminology, this table is well known (the connection with variational problems may be an exception); nevertheless, it seems interesting to write a short informal review. We’ll be mostly concerned with homotopy (or integration) of Σn-manifolds. The only non-trivial column is the one about quantization; for this reason, it won’t be mentioned anymore. The paper is based on a straightforward use of the basic idea of Sullivan’s Rational homotopy theory [1] in differential geometry. Its connection with symplectic geometry is from [2]. 1 Integration of Lie algebroids (after Dennis Sullivan) Let us begin with a simple construction of a groupoid Γ out of a Lie algebroid A → M. Intuitively, A consists of infinitesimal morphisms of Γ; to get all the morphisms, we have to compose them along curves. Thus, consider a

In [2] Baez and Dolan established their stabilization hypothesis as one of a list of the key properties that a good theory of higher categories should have. It is the analogue for n-categories of the well-known stabilization theorems in homotopy theory. To explain the statement, recall that Baez-Dolan introduce the notion of k-uply monoidal n-category which is an n + k-category having only one i-morphism for all i < k. This includes the notions previously defined and examined by many authors, of monoidal (resp. braided monoidal, symmetric monoidal) category (resp. 2-category) and so forth, as is explained in [2] [4]. See the bibliographies of those preprints as well as that of the the recent preprint [9] for many references concerning these types of objects. In the case where the n-category in question is an n-groupoid, this notion is—except for truncation at n—the same thing as the notion of k-fold iterated loop space, or “Ek-space ” which appears in Dunn [10] (see also some anterior references from there). The fully stabilized notion of k-uply monoidal n-categories for k ≫ n is what Grothendieck calls Picard n-categories in [12]. The stabilization hypothesis [2] states that for n + 2 ≤ k ≤ k ′ , the k-uply monoidal

. A finitary monad A on the category of globular sets provides basic algebraic operations from which more involved `pasting' operations can be derived. To makes this rigorous, we define A-computads and construct a monad on the category of A-computads whose algebras are A-algebras; an action of the new monad encapsulates the pasting operations. When A is the monad whose algebras are n-categories, an A-computad is an n-computad in the sense of R.Street. When A is associated to a higher operad (in the sense of the author) , we obtain pasting in weak n-categories. This is intended as a first step towards proving the equivalence of the various definitions of weak n-category now in the literature. Introduction This work arose as a reflection on the foundation of higher dimensional category theory. One of the main ingredients of any proposed definition of weak n-category is the shape of diagrams (pasting scheme) we accept to be composable. In a globular approach [3] each k-cell has a source ...

In this paper the 2-category Rep 2MatC (G) of (weak) representations of an arbitrary (weak) 2-group G on (some version of) Kapranov and Voevodsky’s 2-category of (complex) 2-vector spaces is studied. In particular, the set of equivalence classes of representations is computed in terms of the invariants π0(G), π1(G) and [α]∈H 3 (π0(G), π1(G)) classifying G. Also the categories of morphisms (up to equivalence) and the composition functors are determined explicitly. As a consequence, we obtain that the monoidal category