14 problems

Uniqueness conjecture. There are no other braids underlying the composition of a product of enriched categories besides braid (1) and its inverse that fulfill both obligations.

Let $\vV$-theories be the enriched theories considered in the paper. A $\vV$-theory is cartesian when it is equipped with natural copy and discard structure. Comparison conjecture.…

Let $({\mathcal{T}},{\mathcal{V}})$ be the context of $({\mathcal{T}},{\mathcal{V}})$-categories, let $(X,a)$ and $(Y,b)$ be such categories, and let $f\colon (X,a)\to(Y,b)$ be a…

Let $(\mathscr{V},\otimes,I)$ be a presentably symmetric monoidal $(\infty,1)$-category. Let $\mathscr{M}\mathrm{od}_{\mathscr{V}}$ be the $(\infty,2)$-category whose objects are…

Let $\mathbb U$ be the ultrafilter monad on $\bf Set$. An ultracategory is a small ultracategory in the sense of Clementino and Tholen. Ultracategory conjecture. The $\mathbb U$-ca…

Let $\mathscr{M}$ be a monoidal model category, and write $\operatorname{Ho}(\mathscr{M})$ for its homotopy category and $\operatorname{Ho}(\mathbf{sSet})$ for the homotopy categor…

Let $\mathbb{C}$ be a category with the structures and notation used above, let $C$ be an object of $\mathbb{C}$, and let $\bar{A}=(A,f_A)$ and $\bar{B}=(B,f_B)$ be algebras in…

Let $\mathcal{L}$ be a Lawvere theory admitting a symmetric monoidal structure compatible with finite products. Consider the $\infty$-categories of…

Let $\mathcal{A}$ be a bounded $s\ell$-category, let $B\in Alg_{s\ell}$, and let … be a functor. The coend $L=End^\wedge(T)$ and its lifting $\widetilde{T}$ exist. General Tannaka…

Let $\mathcal{A}$ be a bounded complete $s\ell$-category, let $B$ be a locale, and let … be an open and faithful $s\ell$-functor. Recognition conjecture. The associated Tannakian l…

General recognition conjecture. Under these hypotheses, if $F$ is an $s \ell$-enriched open and faithful functor, then $\widetilde{F}$ is an equivalence.

Enriched icon conjecture. For every symmetric monoidal $(n+1)$-category ${\mathcal {E}}$, there is a symmetric monoidal $(n+1)$-category ${\mathcal {E}}\text{-Icon}$ of categories…

Enriched model comparison conjecture. The category of categories enriched in $(n-1)\mathcal{QC}at$ is Quillen equivalent to

Let $C$ be a small category, let $… and suppose that$M$is a cartesian model category. For an object$X$of$M\operatorname{-Cat}$, let$hX…