1 Monoidal ∞-Categories 4 1.1 Monoidal Structures and Algebra Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Cartesian Monoidal Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3 Subcategories of Monoidal ∞-Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.4 Free Algebras . . . . . . . . . . . . . . . . . . ....

Models of iterated computation, such as (completely) iterative monads, often depend on a notion of guardedness, which guarantees unique solvability of recursive equations and requires roughly that recursive calls happen only under certain guarding operations. On the other hand, many models of iteration do admit unguarded iteration. Solutions are then no longer unique, and in general not even de...

Distributive laws between monads (triples) were defined by Jon Beck in the 1960s; see [1]. They were generalized to monads in 2-categories and noticed to be monads in a 2-category of monads; see [2]. Mixed distributive laws are comonads in the 2-category of monads [3]; if the comonad has a right adjoint monad, the mate of a mixed distributive law is an ordinary distributive law. Particular case...

Operads were originally defined as V-operads, that is, enriched in a symmetric or braided monoidal category V. The symmetry or braiding in V is required in order to describe the associativity axiom the operads must obey, as well as the associativity that must be a property of the action of an operad on any of its algebras. A sequence of categorical types that filter the category of monoidal cat...

Models of iterated computation, such as (completely) iterative monads, often depend on a notion of guardedness, which guarantees unique solvability of recursive equations and requires roughly that recursive calls happen only under certain guarding operations. On the other hand, many models of iteration do admit unguarded iteration. Solutions are then no longer unique, and in general not even de...

The structure theorem of Joyal, Street and Verity says that every traced monoidal category C arises as a monoidal full subcategory of the tortile monoidal category IntC. In this paper we focus on a simple observation that a traced monoidal category C is closed if and only if the canonical inclusion from C into IntC has a right adjoint. Thus, every traced monoidal closed category arises as a mon...

The notion of monads originates from the category theory. It became popular in the programming languages community after Moggi proposed a way to use monads to structure denotational semantics. Wadler and others showed how this can be eeectively used as a methodology for building interpreters. Monads are capable of capturing individual language features in a modular way. This paper evaluates two...

The most familiar example of higher, or vertically iterated enrichment is that in the definition of strict n-category. We begin with strict n-categories based on a general symmetric monoidal category V. Motivation is offered through a comparison of the classical and extended versions of topological quantum field theory. A sequence of categorical types that filter the category of monoidal catego...

In this paper we propose a new categorical formulation of attribute grammars in traced symmetric monoidal categories. The new formulation, called monoidal attribute grammars, concisely captures the essence of the classical attribute grammars. We study monoidal attribute grammars in two categories: Rel and ωCPPO. It turns out that in Rel monoidal attribute grammars correspond to the graphtheoret...

In this paper we construct extensions of Set-monads – and, more generally, of lax Rel-monads – into lax monads of the bicategory Mat(V) of generalized V-matrices, whenever V is a well-behaved lattice equipped with a tensor product. We add some guiding examples.