We generalize Dress and Müller’s main result in [5]. We observe that their result can be seen as a characterization of free algebras for certain monad on the category of species. This perspective allows to formulate a general exponential principle in a symmetric monoidal category. We show that for any groupoid G, the category !̂G of presheaves on the symmetric monoidal completion !G of G satisfi...

So-called ∗-autonomous, or “Frobenius”, category structures occur widely in mathematical quantum theory. This trend was observed in [3], mainly in relation to Hopf algebroids, and continued in [8] with a general account of Frobenius monoids. Below we list some of the ∗-autonomous partially ordered sets A = (A , p, j, S) that appear in the literature, an abstract definition of ∗-autonomous promo...

extends to concurrent systems with state changes the body of theory developed within the algebraic semantics approach. It is both a foundational tool and the kernel language of several implementation e orts (Cafe, ELAN, Maude). extends (unconditional) rewriting logic since it takes into account state changes with side e ects and synchronization. It is especially useful for de ning compositional...

The category of Hilbert modules may be interpreted as a naive quantum field theory over a base space. Open subsets of the base space are recovered as idempotent subunits, which form a meet-semilattice in any firm braided monoidal category. There is an operation of restriction to an idempotent subunit: it is a graded monad on the category, and has the universal property of algebraic localisation...

Over a monoidal model category, under some mild assumptions, we equip the categories of colored PROPs and their algebras with projective model category structures. A BoardmanVogt style homotopy invariance result about algebras over cofibrant colored PROPs is proved. As an example, we define homotopy topological conformal field theories and observe that such structures are homotopy invariant.

We show that if (M,⊗, I) is a monoidal model category then REnd M (I) is a (weak) 2-monoid in sSet. This applies in particular when M is the category of A-bimodules over a simplicial monoid A: the derived endomorphisms of A then form its Hochschild cohomology, which therefore becomes a simplicial 2-monoid.

We introduce the notion of symplectic microfolds and symplectic micromorphisms between them. They form a monoidal category, which is a version of the “category” of symplectic manifolds and canonical relations obtained by localizing them around lagrangian submanifolds in the spirit of Milnor’s microbundles.

Monoidal computer is a categorical model of intensional computation, where many different programs correspond to the same input-output behavior. The upshot of yet another model of computation is that a categorical formalism should provide a high-level language for theory of computation, flexible enough to allow abstracting away the low level implementation details when they are irrelevant, or t...

This talk covers most of section 4 in the Mathew-Naumann-Noel paper [MNN15]. We first discuss nilpotence in an arbitrary symmetric monoidal stable ∞-category. We then discuss the historical origins of nilpotence in the stable homotopy category, namely the Ravenel conjectures and the Nilpotence theorem proved by Devinatz-Hopkins-Smith.