Theories of analytic monads

We characterize the equational theories and Lawvere theories that correspond to the categories of analytic and polynomial monads on Set, and hence also the categories of the symmetric and rigid operads in Set. We show that the category of analytic monads is equivalent to the category of regular-linear theories. The category of polynomial monads is equivalent to the category of rigid theories, i.e. regular-linear theories satisfying an additional global condition. This solves a problem A. Carboni and P. T. Johnstone. The Lawvere theories corresponding to these monads are identified via some factorization systems.Comment: 29 pages. v2: minor correction

Weights for Objects of Monoids

The main objective of the paper is to define the construction of the object of monoids, over a monoidal category object in any 2-category with finite products, as a weighted limit. To simplify the definition of the weight, we use matrices of symmetric (possibly colored) operads that define some auxiliary categories and 2-categories. Systematic use of these matrices of operads allows us to define several similar objects as weighted limits. We show, among others, that the constructions of the object of bi-monoids over a symmetric monoidal category object or the object of actions of monoids along an action of a monoidal category object can be also described as weighted limits.Comment: 19 page

Rigidity is undecidable

We show that the problem `whether a finite set of regular-linear axioms defines a rigid theory' is undecidable.Comment: 8 page

Monads of regular theories

We characterize the category of monads on $Set$ and the category of Lawvere theories that are equivalent to the category of regular equational theories.Comment: 36 page

A Model Structure for Enriched Coloured Operads

We prove that, under certain conditions, the model structure on a monoidal model category $\mathcal{V}$ can be transferred to a model structure on the category of $\mathcal{V}$-enriched coloured (symmetric) operads. As a particular case we recover the known model structure on simplicial operads.Comment: 44 pages, Preliminary version, comments are welcom

Operads as polynomial 2-monads

In this article we give a construction of a polynomial 2-monad from an operad and describe the algebras of the 2-monads which then arise. This construction is different from the standard construction of a monad from an operad in that the algebras of our associated 2-monad are the categorified algebras of the original operad. Moreover it enables us to characterise operads as categorical polynomial monads in a canonical way. This point of view reveals categorical polynomial monads as a unifying environment for operads, Cat-operads and clubs. We recover the standard construction of a monad from an operad in a 2-categorical way from our associated 2-monad as a coidentifier of 2-monads, and understand the algebras of both as weak morphisms of operads into a Cat-operad of categories. Algebras of operads within general symmetric monoidal categories arise from our new associated 2-monad in a canonical way. When the operad is sigma-free, we establish a Quillen equivalence, with respect to the model structures on algebras of 2-monads found by Lack, between the strict algebras of our associated 2-monad, and those of the standard one.Comment: 54 pages. References updated and paper restructured for clarity thanks to the advice of a diligent referee. To appear in Theory and Applications of Categorie

Partial Evaluations and the Compositional Structure of the Bar Construction

An algebraic expression like $3 + 2 + 6$ can be evaluated to $11$, but it can also be \emph{partially evaluated} to $5 + 6$. In categorical algebra, such partial evaluations can be defined in terms of the $1$-skeleton of the bar construction for algebras of a monad. We show that this partial evaluation relation can be seen as the relation internal to the category of algebras generated by relating a formal expression to its result. The relation is transitive for many monads which describe commonly encountered algebraic structures, and more generally for BC monads on \Set, defined by the underlying functor and multiplication being weakly cartesian. We find that this is not true for all monads: we describe a finitary monad on \Set for which the partial evaluation relation on the terminal algebra is not transitive. With the perspective of higher algebraic rewriting in mind, we then investigate the compositional structure of the bar construction in all dimensions. We show that for algebras of BC monads, the bar construction has fillers for all \emph{directed acyclic configurations} in $\Delta^n$, but generally not all inner horns. We introduce several additional \emph{completeness} and \emph{exactness} conditions on simplicial sets which correspond via the bar construction to composition and invertibility properties of partial evaluations, including those arising from \emph{weakly cartesian} monads. We characterize and produce factorizations of pushouts and certain commutative squares in the simplex category in order to provide simplified presentations of these conditions and relate them to more familiar properties of simplicial sets.Comment: 90 Pages. This work arose out of the 2019 Applied Category Theory Adjoint School. The fourth author recently gave a talk on this project at the MIT Categories Seminar, recording available at https://www.youtube.com/watch?v=kMqUj3Kq1p8&list=PLhgq-BqyZ7i6Vh4nxlyhKDAMhlv1oWl5n&index=2&t=0