Related papers: On monoidal equivalences and Ann-equivalences

We give an example of a morphism of simplicial sets which is a monomorphism, bijective on 0-simplices, and a weak categorical equivalence, but which is not inner anodyne. This answers an open question of Joyal. Furthermore, we use this…

We construct an equivalence between the 2-categories VMonCat of rigid V-monoidal categories for a braided monoidal category V and VModTens of oplax braided functors from V into the Drinfeld centers of ordinary rigid monoidal categories. The…

Given a small category C, a C-module M is a functor from C to the category of finite-dimensional vector spaces over a field k. Associated to M is its local structure, given as a functor from C to the category of bi-closed multi-flags over…

We show that an $\infty$-category $\mathcal{M}$ with a closed left action of a monoidal $\infty$-category $\mathcal{V}$ is completely determined by the $\mathcal{V}$-valued graph of morphism objects equipped with the structure of a…

We lift the standard equivalence between fibrations and indexed categories to an equivalence between monoidal fibrations and monoidal indexed categories, namely weak monoidal pseudofunctors to the 2-category of categories. In doing so, we…

We introduce and investigate the category of factorization of a multiplicative, commutative, cancellative, pre-ordered monoid $A$, which we denote $\mathcal{F}(A)$. The objects of $\mathcal{F}(A)$ are factorizations of elements of $A$, and…

We give a direct proof that the category of strict $\omega$-categories is monadic over the category of polygraphs.

Constructions of spectra from symmetric monoidal categories are typically functorial with respect to strict structure-preserving maps, but often the maps of interest are merely lax monoidal. We describe conditions under which one can…

Let $\mathcal{M}$ be a small $n$-abelian category. We show that the category of absolutely pure group valued functors over $\mathcal{M}$, denote by $\mathcal{L}_2(\mathcal{M},\mathcal{G})$, is an abelian category and $\mathcal{M}$ is…

We offer two proofs that categories weakly enriched over symmetric monoidal categories can be strictified to categories enriched in permutative categories. This is a "many 0-cells" version of the strictification of bimonoidal categories to…

In this paper, we classify finite categories with two objects such that one of the endomorphism monoids is a group. We prove that having a group on one side affects the structure of the other endomorphism monoid, and we prove that it is…

We present a Rocq library for monoidal categories, which includes a decision procedure for proving equality of morphisms as well as notations that make it possible to reason as if they were strict, inferring MacLane isomorphims…

We introduce categorical models of $N_\infty$ spaces, which we call normed symmetric monoidal categories (NSMCs). These are ordinary symmetric monoidal categories equipped with compatible families of norm maps, and when specialized to a…

A braided monoidal category may be considered a $3$-category with one object and one $1$-morphism. In this paper, we show that, more generally, $3$-categories with one object and $1$-morphisms given by elements of a group $G$ correspond to…

The category of strict polynomial functors inherits an internal tensor product from the category of divided powers. To investigate this monoidal structure, we consider the category of representations of the symmetric group which admits a…

We present a new method, involving monads and comonads from category theory, to help establish a certain type of equivalence of subcategories. As a case study we consider the category of topological gradings of $C^*$-algebras over a fixed…

This is a major update of the previous version. The methods of the paper are now fully constructive and the style is "formalization ready" with the emphasis on the possibility of formalization both in type theory and in constructive set…

It is proved that MacLane's coherence results for monoidal and symmetric monoidal categories can be extended to some other categories with multiplication; namely, to relevant, affine and cartesian categories. All results are formulated in…

We introduce categories $\M$ and $\S$ internal in the tricategory $\Bicat_3$ of bicategories, pseudofunctors, pseudonatural transformations and modifications, for matrices and spans in a 1-strict tricategory $V$. Their horizontal…

For every functor $\mathcal{F} : \mathcal{K} \to \mathbf{C}$, where $\mathcal{K}$ is a small category and $\mathbf{C}$ is a model category which satisfies some mild hypotheses, we define a model category $\mathbf{C}^m$ of…