Kan extensions and cartesian monoidal categories

One fundamental aspect of Lawvere's categorical semantics is that every algebraic theory (eg. of monoid, of Lie algebra) induces a free construction (eg. of free monoid, of free Lie algebra) computed as a Kan extension. Unfortunately, the principle fails when one shifts to linear variants of algebraic theories, like Adams and Mac Lane's PROPs, and similar PROs and PROBs. Here, we introduce the notion of T -algebraic theory for a pseudomonad T — a mild generalization of equational doctrine — in order to describe these various kinds of "algebraic theories". Then, we formulate two conditions (the first one combinatorial, the second one algebraic) which ensure that the free model of aT -algebraic theory exists and is computed as an Kan extension. The proof is based on Bénabou's theory of distributors, and of an axiomatization of the colimit computation in Wood's proarrow equipments.

2003

The definition and calculus of extraordinary natural transformations is extended to a context internal to any autonomous monoidal bicategory. The original calculus is recaptured from the geometry of the monoidal bicategory V-Mod whose objects are categories enriched in a cocomplete symmetric monoidal category V and whose morphisms are modules.

Journal of Topology, 2016

arXiv (Cornell University), 2020

We give a new syntax independent definition of the notion of a finitely presented generalized algebraic theory as an initial object in a category of categories with families (cwfs) with extra structure. To this end we define inductively how to build a valid signature Σ for a generalized algebraic theory and the associated category CwFΣ of cwfs with a Σ-structure and cwf-morphisms that preserve Σ-structure on the nose. Our definition refers to the purely semantic notions of uniform family of contexts, types, and terms. Furthermore, we show how to syntactically construct initial cwfs with Σ-structures. This result can be viewed as a generalization of Birkhoff's completeness theorem for equational logic. It is obtained by extending Castellan, Clairambault, and Dybjer's construction of an initial cwf. We provide examples of generalized algebraic theories for monoids, categories, categories with families, and categories with families with extra structure for some type formers of dependent type theory. The models of these are internal monoids, internal categories, and internal categories with families (with extra structure) in a category with families. Finally, we show how to extend our definition to some generalized algebraic theories that are not finitely presented, such as the theory of contextual categories with families.

Applied Categorical Structures, 2022

One way of interpreting a left Kan extension is as taking a kind of “partial colimit”, whereby one replaces parts of a diagram by their colimits. We make this intuition precise by means of the partial evaluations sitting in the so-called bar construction of monads. The (pseudo)monads of interest for forming colimits are the monad of diagrams and the monad of small presheaves, both on the (huge) category CAT of locally small categories. Throughout, particular care is taken to handle size issues, which are notoriously delicate in the context of free cocompletion. We spell out, with all 2-dimensional details, the structure maps of these pseudomonads. Then, based on a detailed general proof of how the restriction-of-scalars construction of monads extends to the case of pseudoalgebras over pseudomonads, we consider a morphism of monads between them, which we call image. This morphism allows in particular to generalize the idea of confinal functors, i.e. of functors which leave colimits i...

1980

This paper is a slight revision of a paper I wrote in 1980 and never submitted for publication. I would greatly appreciate any information about more recent work on this topic.

2011

In this paper, we state and prove precise theorems on the classification of the category of (braided) categorical groups and their (braided) monoidal functors, and some applications obtained from the basic studies on monoidal functors between categorical groups.

Theory and Applications of Categories, 2008

In a recent paper, Daisuke Tambara defined two-sided actions on an endomodule (= endodistributor) of a monoidal V-category A. When A is autonomous (= rigid = compact), he showed that the V-category (that we call Tamb(A)) of soequipped endomodules (that we call Tambara modules) is equivalent to the monoidal centre Z[A , V ] of the convolution monoidal V-category [A , V ]. Our paper extends these ideas somewhat. For general A , we construct a promonoidal V-category DA (which we suggest should be called the double of A) with an equivalence [DA , V ] Tamb(A). When A is closed, we define strong (respectively, left strong) Tambara modules and show that these constitute a V-category Tamb s (A) (respectively, Tamb ls (A)) which is equivalent to the centre (respectively, lax centre) of [A , V ]. We construct localizations D s A and D ls A of DA such that there are equivalences Tamb s (A) [D s A , V ] and Tamb ls (A) [D ls A , V ]. When A is autonomous, every Tambara module is strong; this implies an equivalence Z[A , V ] [DA , V ].

Mathematical Structures in Computer Science

We give a syntax independent formulation of finitely presented generalized algebraic theories as initial objects in categories of categories with families (cwfs) with extra structure. To this end, we simultaneously define the notion of a presentation Σ of a generalized algebraic theory and the associated category CwFΣ of small cwfs with a Σ-structure and cwf-morphisms that preserve Σ-structure on the nose. Our definition refers to the purely semantic notion of uniform family of contexts, types, and terms in CwFΣ. Furthermore, we show how to syntactically construct an initial cwf with a Σ-structure. This result can be viewed as a generalization of Birkhoff’s completeness theorem for equational logic. It is obtained by extending Castellan, Clairambault, and Dybjer’s construction of an initial cwf. We provide examples of generalized algebraic theories for monoids, categories, categories with families, and categories with families with extra structure for some type formers of Martin-Löf...

Journal of Pure and Applied Algebra, 1987