Suppose $A$ and $C$ two symmetric monoidal categories. Let's say that $A$ is small and $C$ is locally presentable, and let's assume also that the tensor product on $C$ preserves co …

the category of right comodule of coalgebra is a monoidal category according the following the associativity constraint is defined as a_{U,V,W}((u\otimes v)\otimes w)=u_0\otimes …

I am novice in the algebraic K- theory and don' t know if this is the right place for the following questions. So some people might consider them as basic questions. Consider an e …

It is well-known that any symmetric monoidal category is equivalent to a strict symmetric monoidal category. The construction of this strict monoidal category is rather technical a …

In Higher Operads, Higher Categories, Leinster nicely characterizes operads among monoidal categories (as PROs). Roughly speaking, a monoidal category comes from an operad if its s …

Toen-Vaquié construct a category of schemes relative to some complete cocomplete closed symmetric monoidal category $C$. Affine schemes correspond by definition 1:1 to commutative …

Let $C$ be a monoidal category, not assumed to be symmetric. Assume that the underlying category of $C$ is nice enough, for example cocomplete, perhaps even presentable. A semigrou …

Consider the $2$-categories $\mathsf{MonCat}$ of monoidal categories, with strong monoidal functors and monoidal transformations, $\mathsf{SymMonCat}$ of symmetric monoidal categ …

It is well-known that the (augmented) simplex category is the universal monoidal category with a monoid object. What about a commutative analogue? Consider the category $\mathsf{Fi …

I would like to know if the following definition is already known or even well-known. Is there a reference in the literature? Do you know prominent examples? Definition. Let $\mat …

Background By a cocomplete symmetric monoidal category $C$ I mean a symmetric monoidal category whose underlying category is cocomplete and such that $- \otimes X : C \to C$ is co …

As a special case of a general construction I have constructed "accidentally" a tensor product of quivers aka directed multigraphs (aka directed graphs for category theorists). Pro …

Is the notion of a commutative monoidal category, where every product is isomorphic to its opposite, but not necessarily functorially, known not to be useful? I have not been able …

Let $\mathcal{T}$ be a commutative algebraic theory (for example sets, abelian groups, commutative monoids, but not groups etc.). References include the nlab and Borceux' Handbook …

Take a Cartesian (or monoidal) closed category; define Reader monad for a given object $E$ as $X \mapsto X^E$; and take a strong monad $M$ (strong means preserves product or tensor …