THH and Traces of Enriched Categories

We prove that topological Hochschild homology (THH) arises from a presheaf of circles on a certain combinatorial category, which gives a universal construction of THH for any enriched $\infty $-category. Our results rely crucially on an elementary, model-independent framework for enriched higher-category theory, which may be of independent interest. For those interested only in enriched category theory, read Sections 1.3 and 2.

Hamiltonian Analysis for the Scalar Electrodynamics as 3BF Theory

The higher category theory can be employed to generalize the B F action to the so-called 3 B F action, by passing from the notion of a gauge group to the notion of a gauge 3-group. The theory of scalar electrodynamics coupled to Einstein–Cartan gravity can be formulated as a constrained 3 B F theory for a specific choice of the gauge 3-group. The complete Hamiltonian analysis of the 3 B F action for the choice of a Lie 3-group corresponding to scalar electrodynamics is performed. This analysis is the first step towards a canonical quantization of a 3 B F theory, an important stepping stone for the quantization of the complete scalar electrodynamics coupled to Einstein–Cartan gravity formulated as a 3 B F action with suitable simplicity constraints. It is shown that the resulting dynamic constraints eliminate all propagating degrees of freedom, i.e., the 3 B F theory for this choice of a 3-group is a topological field theory, as expected.

Homotopical patch theory

Homotopy type theory is an extension of Martin-Löf type theory, based on a correspondence with homotopy theory and higher category theory. In homotopy type theory, the propositional equality type is proof-relevant, and corresponds to paths in a space. This allows for a new class of datatypes, called higher inductive types, which are specified by constructors not only for points but also for paths. In this paper, we consider a programming application of higher inductive types. Version control systems such as Darcs are based on the notion of patches—syntactic representations of edits to a repository. We show how patch theory can be developed in homotopy type theory. Our formulation separates formal theories of patches from their interpretation as edits to repositories. A patch theory is presented as a higher inductive type. Models of a patch theory are given by maps out of that type, which, being functors, automatically preserve the structure of patches. Several standard tools of homotopy theory come into play, demonstrating the use of these methods in a practical programming context.

Cofree compositions of coalgebras

International audience We develop the notion of the composition of two coalgebras, which arises naturally in higher category theory and the theory of species. We prove that the composition of two cofree coalgebras is cofree and give conditions which imply that the composition is a one-sided Hopf algebra. These conditions hold when one coalgebra is a graded Hopf operad $\mathcal{D}$ and the other is a connected graded coalgebra with coalgebra map to $\mathcal{D}$. We conclude with examples of these structures, where the factor coalgebras have bases indexed by the vertices of multiplihedra, composihedra, and hypercubes. Nous développons la notion de composition de coalgèbres, qui apparaît naturellement dans la théorie des catégories d'ordre supérieur et dans la théorie des espèces. Nous montrons que la composée de deux coalgèbres colibres est colibre et nous donnons des conditions qui impliquent que la composée est une algèbre de Hopf unilatérale. Ces conditions sont valables quand l'une des coalgèbres est une opérade de Hopf graduée $\mathcal{D}$ et l'autre est une coalgèbre graduée connexe avec un morphisme vers $\mathcal{D}$. Nous concluons avec des exemples de ces structures, où les coalgèbres composées ont des bases indexées par les sommets de multiplièdres, de composièdres, et d'hypercubes.

Elementary remarks on units in monoidal categories

We explore an alternative definition of unit in a monoidal category originally due to Saavedra: a Saavedra unit is a cancellable idempotent (in a 1-categorical sense). This notion is more economical than the usual notion in terms of left-right constraints, and is motivated by higher category theory. To start, we describe the semi-monoidal category of all possible unit structures on a given semi-monoidal category and observe that it is contractible (if non-empty). Then we prove that the two notions of units are equivalent in a strong functorial sense. Next, it is shown that the unit compatibility condition for a (strong) monoidal functor is precisely the condition for the functor to lift to the categories of units, and it is explained how the notion of Saavedra unit naturally leads to the equivalent non-algebraic notion of fair monoidal category, where the contractible multitude of units is considered as a whole instead of choosing one unit. To finish, the lax version of the unit comparison is considered. The paper is self-contained. All arguments are elementary, some of them of a certain beauty.