Tensor products of finitely presented functors

We provide explicit constructions for various ingredients of right exact monoidal structures on the category of finitely presented functors. As our main tool, we prove a multilinear version of the universal property of so-called Freyd categories, which in turn is used in the proof of correctness of our constructions. Furthermore, we compare our construction with the Day convolution of arbitrary additive functors. Day convolution always yields a closed monoidal structure on the category of all additive functors. In contrast, right exact monoidal structures for finitely presented functor categories are not necessarily closed. We provide a necessary criterion for being closed that relies on the underlying category having weak kernels and a so-called finitely presented prointernal hom structure. Our results are stated in a constructive way and thus serve as a unified approach for the implementation of tensor products in various contexts.

Network Models from Petri Nets with Catalysts

Petri networks and network models are two frameworks for the compositional design of systems of interacting entities. Here we show how to combine them using the concept of a `catalyst': an entity that is neither destroyed nor created by any process it engages in. In a Petri net, a place is a catalyst if its in-degree equals its out-degree for every transition. We show how a Petri net with a chosen set of catalysts gives a network model. This network model maps any list of catalysts from the chosen set to the category whose morphisms are all the processes enabled by this list of catalysts. Applying the Grothendieck construction, we obtain a category fibered over the category whose objects are lists of catalysts. This category has as morphisms all processes enabled by some list of catalysts. While this category has a symmetric monoidal structure that describes doing processes in parallel, its fibers also have premonoidal structures that describe doing one process and then another while reusing the catalysts.

The Gray Monoidal Product of Double Categories

The category of double categories and double functors is equipped with a symmetric closed monoidal structure. For any double category $${\mathbb {A}}$$A, the corresponding internal hom functor "Equation missing" sends a double category $${\mathbb {B}}$$B to the double category whose 0-cells are the double functors $${\mathbb {A}} \rightarrow {\mathbb {B}}$$A→B, whose horizontal and vertical 1-cells are the horizontal and vertical pseudo transformations, respectively, and whose 2-cells are the modifications. Some well-known functors of practical significance are checked to be compatible with this monoidal structure.

The Braided Monoidal Structure on the Category of Comodules of Bimonads

We investigate how the category of comodules of bimonads can be made into a monoidal category. It suffices that the monad and comonad in question are bimonads, with some extra compatibility relation. On a monoidal category of comodules of bimonads, we construct a braiding and get the necessary and sufficient conditions making it a braided monoidal category. As an application, we consider the category of comodules of corings and the category of entwined modules.

Arrow categories of monoidal model categories

We prove that the arrow category of a monoidal model category, equipped with the pushout product monoidal structure and the projective model structure, is a monoidal model category. This answers a question posed by Mark Hovey, in the course of his work on Smith ideals. As a corollary, we prove that the projective model structure in cubical homotopy theory is a monoidal model structure. As illustrations we include numerous examples of non-cofibrantly generated monoidal model categories, including chain complexes, small categories, pro-categories, and topological spaces.

Hybrid linear logic, revisited

HyLL (Hybrid Linear Logic) is an extension of intuitionistic linear logic (ILL) that has been used as a framework for specifying systems that exhibit certain modalities. In HyLL, truth judgements are labelled by worlds (having a monoidal structure) and hybrid connectives (at and ↓) relate worlds with formulas. We start this work by showing that HyLL's axioms and rules can be adequately encoded in linear logic (LL), so that one focused step in LL will correspond to a step of derivation in HyLL. This shows that any proof in HyLL can be exactly mimicked by a LL focused derivation. Another extension of LL that has extensively been used for specifying systems with modalities is Subexponential Linear Logic (SELL). In SELL, the LL exponentials (!, ?) are decorated with labels representing locations, and a pre-order on such labels defines the provability relation. We propose an encoding of HyLL into SELL⋒ (SELL plus quantification over locations) that gives better insights about the meaning of worlds in HyLL. More precisely, we identify worlds as locations, and show that a flat subexponential structure is sufficient for representing any world structure in HyLL. This shows that HyLL's monoidal structure is not reflected in LL derivations, hence not increasing the expressiveness of LL, from a proof theoretical point of view. We conclude by proposing the notion of fixed points in multiplicative additive HyLL (μHyMALL), which can be encoded into multiplicative additive linear logic with fixed points (μMALL). As an application, we propose encodings of Computational Tree Logic (CTL) into both μMALL and μHyMALL. In the former, states are represented as atoms in the linear context, hence reflecting a more operational view of CTL connectives. In the latter, worlds represent states of the transition system, thus exhibiting a pleasant similarity with the semantics of CTL.

A categorical reconstruction of crystals and quantum groups at q = 0

The quantum co-ordinate algebra Aq(g) associated to a Kac–Moody Lie algebra g forms a Hopf algebra whose comodules are direct sums of finite-dimensional irreducible Uq(g) modules. In this paper, we investigate whether an analogous result is true when q=0. We classify crystal bases as coalgebras over a comonadic functor on the category of pointed sets and encode the monoidal structure of crystals into a bicomonadic structure. In doing this, we prove that there is no coalgebra in the category of pointed sets whose comodules are equivalent to crystal bases. We then construct a bialgebra over Z whose based comodules are equivalent to crystals, which we conjecture is linked to Lusztig’s quantum group at v=∞.

Antipodes of monoidal decomposition spaces

We introduce a notion of antipode for monoidal (complete) decomposition spaces, inducing a notion of weak antipode for their incidence bialgebras. In the connected case, this recovers the usual notion of antipode in Hopf algebras. In the non-connected case, it expresses an inversion principle of more limited scope, but still sufficient to compute the Möbius function as [Formula: see text], just as in Hopf algebras. At the level of decomposition spaces, the weak antipode takes the form of a formal difference of linear endofunctors [Formula: see text], and it is a refinement of the general Möbius inversion construction of Gálvez–Kock–Tonks, but exploiting the monoidal structure.