6 problems

Let $(\operatorname{\mathcal{L}},\eta)$ be a lasso on a category $\mathsf{C}$. For monomorphisms $f_1\colon X_1\hookrightarrow Y$ and $f_2\colon X_2\to Y_{/f}$ arranged as in the s…

A schema is a small category, and for a schema $\mathsf{T}$, $\mathsf{Set}^{\mathsf{T}}$ denotes its associated copresheaf category. Let $\mathsf{Lasso}(\mathsf{Set}^{\mathsf{T}})$…

Let $\mathrm{OrbSpc}$ be the category of orbispaces, let $\overline{\mathrm{OrbSpc}}$ be its enlargement obtained by gluing cells under arbitrary maps, and let $\mathrm{GloSpc}$ be…

Let $h$ be an associative presheaf, and let $\mathcal{A}(h)$ denote its pattern Hopf algebra. Freeness conjecture. The pattern Hopf algebra $\mathcal{A}(h)$ is free. This would est…

Gaunt-colimit characterization. The restricted Yoneda functor

Let $\mathscr{T}$ be a triangulated category, let $\mathscr{B}$ be a small full subcategory closed under the shift functor $\Sigma$, and let $\mathscr{C}$ be the thick full subcate…