Nerves and classifying spaces for bicategories

We show that the classifying space functor $$B:\mathcal {M}on \rightarrow {\mathcal {T}\! op}^*$$ from the category of topological monoids to the category of based spaces is left adjoint to the Moore loop space functor $$\Omega ':{\mathcal {T}\! op}^*\rightarrow \mathcal {M}on$$ after we have localized $$\mathcal {M}on$$ with respect to all homomorphisms whose underlying maps are homotopy equivalences and $${\mathcal {T}\! op}^*$$ with respect to all based maps which are (not necessarily based) homotopy equivalences. It is well-known that this localization of $${\mathcal {T}\! op}^*$$ exists, and we show that the localization of $$\mathcal {M}on$$ is the category of monoids and homotopy classes of homotopy homomorphisms. To make this statement precise we have to modify the classical definition of a homotopy homomorphism, and we discuss the necessary changes. The adjunction is induced by an adjunction up to homotopy $$B:\mathcal {H}\mathcal {M}on^{w}\leftrightarrows {\mathcal {T}\! op}^w:\Omega '$$ between the category of well-pointed monoids and homotopy homomorphisms and the category of well-pointed spaces. This adjunction is shown to lift to diagrams. As a consequence, the well-known derived adjunction between the homotopy colimit and the constant diagram functor can also be seen to be induced by an adjunction up to homotopy before taking homotopy classes. As applications we among other things deduce a more algebraic version of the group completion theorem and show that the classifying space functor preserves homotopy colimits up to natural homotopy equivalences.

The Grothendieck construction for model categories

We relate the relative nerve N f (D) of a diagram of simplicial sets f : D sSet with the Grothendieck construction GrF of a simplicial functor F : D sCat in the case where f = NF .We further show that any strict monoidal simplicial category C gives rise to a functor C : op sCat, and that the relative nerve of NC is the operadic nerve N (C).Finally, we show that all the above constructions commute with appropriately defined opposite functors.

Categorical Foundations for K-theory

We build on the correspondence between Petri nets and free symmetric strict monoidal categories already investigated in the literature, and present a categorical semantics for Petri nets with guards. This comes in two flavors: Deterministic and with side-effects. Using the Grothendieck construction, we show how the guard semantics can be internalized in the net itself.

Classifying spaces for braided monoidal categories and lax diagrams of bicategories

Classical algebraic K-theory: The functors K0, K1, K2

The mathematical aspects of the notion of tensor categories are reviewed in connection with quantum symmetries of operator algebras. Let us begin with brief historical backgrounds for sources of tensor categories: • Abstract charcaterizations of Tannaka duals (1970–) – Algebraic Geometry (Grothendieck-Saavedra Rivano, Deligne-Milne) – Superselection Sectors (Doplicher-Haag-Roberts) • Low-Dimensional Physics (Braid statistics–Braided Tensor Categories) In all these, classical and quantum symmetries appear as tensor cateogries. Let me recall now classical Tannaka duality for compact groups. 1. Tannaka Duality Let G be a compact group and R = R(G) be the category of finite-dimensional representations of G: For G-modules V and W , Hom(V,W ) = the space of intertwiners and we have the operations of taking tensor products V,W ⇝ V ⊗W and taking conjugations V ⇝ V ∗. The group G is then recovered from the category R by g = {gV }, gV : V → V is a unitary satisfying gV⊗W = gV ⊗ gW , gV ∗ = tg−1 V , V gV −−−→ V f y yf W −−−→ gW W The fact that the category R is furnished with the operation of tensor products is abstracted into the notion of tensor cateogry. 2. Tensor Categories A tensor category is a category C such that • Hom(X,Y ) is a complex vector space, • a bivariant functor X,Y ⇝ X ⊗ Y is given, • a special object I (unit object) is given and these satisfy (X ⊗ Y )⊗ Z = X ⊗ (Y ⊗ Z), I ⊗X = X = X ⊗ I. Remark • X is just a symbol and may not be a vector space. • Commutativity X ⊗ Y = Y ⊗X is not required. In most physical applications, it is natural to require the Positivity/Unitarity: We shall work with C*-categories in which the so-called *-operation Hom(X, Y ) ∋ f 7→ f ∗ ∈ Hom(Y,X) is furnished so that f ∗f ≥ 0 in certain sense. Example 1 (i) Let N be a *-algebra (more precisely a von Neumann algebra) and consider a Hilbert space X on which N acts in a bimodule fashion. Then the totality {NXN} forms a C*-tensor category: Hom(X, Y ) = {f : X → Y ; f(aξb) = af(ξ)b, a, b ∈ N, ξ ∈ X}, X ⊗ Y = N(X ⊗N Y )N . (ii) In the theory of superselection sectors, there appear C*-tensor categories of endomorphisms: Let N be as above and consider unital *-endomorphisms of N , say ρ. Then the totality {ρ ∈ End(N)} forms a C*-tensor category by Hom(ρ, σ) = {a ∈ N ; ρ(x)a = aσ(x),∀x ∈ N}, ρ⊗ σ = ρ ◦ σ (the composite endomorphism). Remark The second example is a special case of the first one as each ρ ∈ End(N) gives rise to a bimodule NL (N)ρN , where L (N) denotes the regular representation (Hilbert space) with the left action of N by left multiplication, wheareas the right action of N is the combination of right multiplication and the endomorphism ρ. More interesting examples are provided by positive energy representations of loop groups (a geometric realization of WZW models): Let G be a compact Lie group, say SU(N), and π be a positive energy (projective) representation on a Hilbert space X of level l ≥ 1. Recall that irreducible positive energy representations of level l are parametrized by Young diagrams l = (l1, . . . , lN) satisfying l1 − lN ≤ l in such a way that the representation of G = SU(N) on the lowest energy subspace X(0) is the irreducible representation associated to l. Let N = π0(C (Sleft, G)) ′′ be the von Neumann algebra in the vacuum representation π0 of level l with Sleft denoting the left semicircle. Then X is an N -N bimodule in an obvious way and the totality {NXN} turns out to form a C*-tensor category, an operator-algebraic version of WZW-models (JonesWassermann).

Homotopy colimits of algebras over [formula omitted]-operads and iterated loop spaces

Traces in monoidal derivators, and homotopy colimits

Thomason’s Homotopy Colimit Theorem has been extended to bicategories and this extension can be adapted, through the delooping principle, to a corresponding theorem for diagrams of monoidal categories. In this version, we show that the homotopy type of the diagram can also be represented by a genuine simplicial set nerve associated with it. This suggests the study of a homotopy colimit theorem, for diagrams B of braided monoidal categories, by means of a simplicial set nerve of the diagram. We prove that it is weak homotopy equivalent to the homotopy colimit of the diagram, of simplicial sets, obtained from composing B with the geometric nerve functor of braided monoidal categories.

In [TVa], Bertrand Toën and Michel Vaquié defined a scheme theory for a closed monoidal category ( ⊗1). In this article, we define a notion of smoothness in this relative (and not necessarily additive) context which generalizes the notion of smoothness in the category of rings. This generalisation consists in replacing homological finiteness conditions by homotopical ones, using the Dold-Kan correspondence. To do this, we provide the category s of simplicial objects in a monoidal category and all the categories sA-mod, sA-alg (A ∈ sComm()) with compatible model structures using the work of Rezk [R]. We then give a general notion of smoothness in sComm(). We prove that this notion is a generalisation of the notion of smooth morphism in the category of rings and is stable under composition and homotopy pushouts. Finally we provide some examples of smooth morphisms, in particular in ℕ-alg and Comm(Set).

Given a symmetric monoidal category ${\mathcal C}$ with product $\sqcup $ , where the neutral element for the product is an initial object, we consider the poset of $\sqcup $ -complemented subobjects of a given object X. When this poset has finite height, we define decompositions and partial decompositions of X which are coherent with $\sqcup $ , and order them by refinement. From these posets, we define complexes of frames and partial bases, augmented Bergman complexes and related ordered versions. We propose a unified approach to the study of their combinatorics and homotopy type, establishing various properties and relations between them. Via explicit homotopy formulas, we will be able to transfer structural properties, such as Cohen-Macaulayness. In well-studied scenarios, the poset of $\sqcup $ -complemented subobjects specializes to the poset of free factors of a free group, the subspace poset of a vector space, the poset of nondegenerate subspaces of a vector space with a nondegenerate form, and the lattice of flats of a matroid. The decomposition and partial decomposition posets, the complex of frames and partial bases together with the ordered versions, either coincide with well-known structures, generalize them, or yield new interesting objects. In these particular cases, we provide new results along with open questions and conjectures.

Cohomological results in monoid and category theory via classifying spaces

Rudolf Bock Werner Wittkowski Histologie – Interaktives Lernprogramm. 1. Auflage 2004 Urban & Fischer bei Elsevier GmbH. München Jena 3-437-43060-2 (CD-Rom, Audio CD für Windows 95/98/ME/NT/2000/XP, € 24,95, sFr 40.00)