We define a new monoidal category on collections (shuffle composition). Monoids in this category (shuffle operads) turn out to bring a new insight in the theory of symmetric operads. For this category, we develop the machinery of Gröbner bases for operads, and present operadic versions of Bergman’s Diamond Lemma and Buchberger’s algorithm. This machinery can be applied to study symmetric operad...

The category of all binary relations between arbitrary sets turns out to be a certain symmetric monoidal category Rel with an additional structure characterized by a family d = (dA : A → A⊗A | A ∈ |Rel|) of diagonal morphisms, a family t = (tA : A → I | A ∈ |Rel|) of terminal morphisms, and a family∇ = (∇A : A⊗A → A | A ∈ |Rel|) of diagonal inversions having certain properties. Using this prope...

For an arbitrary symmetric monoidal∞-category V, we define the factorization homology of V-enriched∞-categories over (possibly stratified) 1-manifolds and study its basic properties. In the case that V is cartesian symmetric monoidal, by considering the circle and its self-covering maps we obtain a notion of unstable topological cyclic homology, which we endow with an unstable cyclotomic trace ...

A symmetric monoidal category is a category equipped with an associative and commutative (binary) product and an object which is the unit for the product. In fact, those properties only hold up to natural isomorphisms which satisfy some coherence conditions. The coherence theorem asserts the commutativity of all linear diagrams involving the left and right unitors, the associator and the braidi...

By definition, a bialgebra H in a braided monoidal category (C, τ) is an algebra and coalgebra whose multiplication and comultiplication (and unit and counit) are compatible; the compatibility condition involves the braiding τ . The present paper is based upon the following simple observation: If H is a Hopf algebra, that is, if an antipode exists, then the compatibility condition of a bialgebr...

Let there be given, on a category C, a pair (*, *) of adjoint monoidal closed-category-valued pseudofunctors. Thus, to each object X ∈ C is associated a closed category DX , with unit object OX ; and to each C-map ψ : X → Y , adjoint monoidal functors DX ψ∗ ←−− −→ ψ∗ DY . There are also, as before, compatibilities—expressed by commutative diagrams—among adjunction, pseudofunctoriality, and mono...

In [6] Higson showed that the formal properties of the Kasparov KK -theory groups are best understood if one regards KK (A, B) for separable C∗-algebras A, B as the morphism set of a category KK . In category language the composition and exterior KK product give KK the structure of a symmetric monoidal category which is enriched over abelian groups. We show that the enrichment of KK can be lift...

Isomorphic objects in symmetric monoidal closed categories† K O S T A D O Š E N ‡ and Z O R A N P E T R I Ć § ‡ University of Toulouse III, Institut de Recherche en Informatique de Toulouse, 118 route de Narbonne, 31062 Toulouse cedex, France and Mathematical Institute, Knez Mihailova 35, P.O. Box 367, 11001 Belgrade, Yugoslavia § University of Belgrade, Faculty of Mining and Geology, Djušina 7...

The notion of commutative monad was denned by the author in [4]. The content of the present paper may briefly be stated: The category of algebras for a commutative monad can in a canonical way be made into a closed category, the two adjoint functors connecting the category of algebras with the base category are in a canonical way closed functors, and the frontand end-adjunctions are closed tran...

In the categorical approach to the foundations of quantum theory, one begins with a symmetric monoidal category, the objects of which represent physical systems, and the morphisms of which represent physical processes. Usually, this category is taken to be at least compact closed, and more often, dagger compact, enforcing a certain self-duality, whereby preparation processes (roughly, states) a...