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 . . . . . . . . . . . . . . . . . . ....

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...

Let SMC denote the 2-category with objects small symmetric monoidal categories, 1cells symmetric monoidal functors and 2-cells monoidal natural transformations. It is shown that the category quotient of SMC by the congruence generated by its 2-cells is symmetric monoidal closed. 1 Summary of results Thomason’s famous result claims that symmetric monoidal categories model all connective spectra ...

The goal of this paper is to prove coherence results with respect to relational graphs for monoidal endofunctors, i.e. endofunctors of a monoidal category that preserve the monoidal structure up to a natural transformation that need not be an isomorphism. These results are proved first in the absence of symmetry in the monoidal structure, and then with this symmetry. In the later parts of the p...

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 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...

Some research works have mentioned the similarity of autopoiesis with (M,R) systems proposed by Rosen, from the perspective of closedness of the systems. However, there are some difference between the aspects of closedness required for autopoiesis and (M,R) systems. This paper aims at clarifying these differences to investigate the possibility of algebraic description of living systems, based o...

In this paper, we obtain several new generalized KKM-type theorems under a new coercivity condition and the condition of intersectionally closedness which improves condition of transfer closedness. As applications, we obtain new versions of equilibrium problem, minimax inequality, coincidence theorem, fixed point theorem and an existence theorem for an 1-person game.

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...