Stability conditions on morphisms in a category

Let $\mathrm{h}\mathscr{C}$ be the homotopy category of a stable infinity category $\mathscr{C}$. Then the homotopy category $\mathrm{h}\mathscr{C}^{\Delta^{1}}$ of morphisms in the stable infinity category $\mathscr{C}$ is also triangulated. Hence the space $\mathsf{Stab}\,{ \mathrm{h}\mathscr{C}^{\Delta^{1}}}$ of stability conditions on $\mathrm{h}\mathscr{C}^{\Delta^{1}}$ is well-defined though the non-emptiness of $\mathsf{Stab}\,{ \mathrm{h}\mathscr{C}^{\Delta^{1}}}$ is not obvious. Our basic motivation is a comparison of the homotopy type of $\mathsf{Stab}{\mathrm{h}\mathscr{C}}$ and that of $\mathsf{Stab}{\mathrm{h}\mathscr{C}^{\Delta^{1}}}$. Under the motivation we show that functors $d_{0}$ and $d_{1} \colon \mathscr{C}^{\Delta^{1}} \rightrightarrows \mathscr{C}$ induce continuous maps from $\mathsf{Stab} {\mathrm{h}\mathscr{C}}$ to $\mathsf{Stab}{\mathrm{h}\mathscr{C}^{\Delta^{1}}}$ contravariantly where $d_{0}$ (resp. $d_{1}$) takes a morphism to the target (resp. source) of the morphism. As a consequence, if $\mathsf{Stab}{\mathrm{h}\mathscr{C}}$ is nonempty then so is $\mathsf{Stab}{\mathrm{h}\mathscr{C}^{\Delta^{1}}}$. Assuming $\mathscr{C}$ is the derived infinity category of the projective line over a field, we further study basic properties of $d_{0}^{*}$ and $d_{1}^{*}$. In addition, we give an example of a derived category which does not have any stability condition.Comment: 26 pages, comments are welcome. For v2, added subsection 2.2 which gives a description of a Serre functor of the category of morphisms in $\mathbf D$. For v3, the proof of Proposition 3.3 has been updated. For v5, Section 6 was added. For v6, modified the proof of Proposition 6.1. For v7, minor revision for Proposition 6.1, final versio

Iterated wreath product of the simplex category and iterated loop spaces

Generalising Segal's approach to 1-fold loop spaces, the homotopy theory of $n$-fold loop spaces is shown to be equivalent to the homotopy theory of reduced $\Theta_n$-spaces, where $\Theta_n$ is an iterated wreath product of the simplex category $\Delta$. A sequence of functors from $\Theta_n$ to $\Gamma$ allows for an alternative description of the Segal-spectrum associated to a $\Gamma$-space. In particular, each Eilenberg-MacLane space $K(\pi,n)$ has a canonical reduced $\Theta_n$-set model

Module categories for AnA_n web categories from A~n−1\tilde{A}_{n-1}-buildings

We equip the category of vector bundles over the vertices of a locally finite $\tilde{A}_{n-1}$ building $\Delta$ with the structure of a module category over a category of type $A_{n}$ webs in positive characteristic. This module category is a $q$-analogue of the $Rep(SL_{n})$ action on vector bundles over the $sl_n$ weight lattice. We show our module categories are equivariant with respect to symmetries of the building, and when a group $G$ acts simply transitively on the vertices of $\Delta$ this recovers the fiber functors constructed by Jones.Comment: 29 pages. Author contact: [email protected]

Tilting modules and highest weight theory for reduced enveloping algebras

Let $G$ be a reductive algebraic group over an algebraically closed field of characteristic $p>0$, and let ${\mathfrak g}$ be its Lie algebra. Given $\chi\in{\mathfrak g}^{*}$ in standard Levi form, we study a category ${\mathscr C}_\chi$ of graded representations of the reduced enveloping algebra $U_\chi({\mathfrak g})$. Specifically, we study the effect of translation functors and wall-crossing functors on various highest-weight-theoretic objects in ${\mathscr C}_\chi$, including tilting modules. We also develop the theory of canonical $\Delta$-flags and $\overline{\nabla}$-sections of $\Delta$-flags, in analogy with similar concepts for algebraic groups studied by Riche and Williamson.Comment: 45 page