arXiv subjects
Publications and source records attributed to Takeshi Torii.
In this paper we study multiplicative structures on comodules over bialgebras in the setting of $\infty$-categories. We show that the $\infty$-category of comodules over an $(\mathcal{O},\mathbf{Ass})$-bialgebra in a mixed $(\mathcal{O},\mathbf{Ass})$-duoidal $\infty$-category has the structure of an $\mathcal{O}$-monoidal $\infty$-category for any $\infty$-operad $\mathcal{O}$.
A duoidal category is a category equipped with two monoidal structures in which one is (op)lax monoidal with respect to the other. In this paper we introduce duoidal $\infty$-categories which are counterparts of duoidal categories in the setting of $\infty$-categories. There are three kinds of functors between duoidal $\infty$-categories, which are called bilax, double lax, and double oplax monoidal functors. We make three formulations of $\infty$-categories of duoidal $\infty$-categories according to which functors we take. Furthermore, corresponding to the three kinds of functors, we define bimonoids, double monoids, and double comonoids in duoidal $\infty$-categories.
In this paper we give an example of duoidal $\infty$-categories. We introduce map $\mathcal{O}$-monoidales in an $\mathcal{O}$-monoidal $(\infty,2)$-category for an $\infty$-operad $\mathcal{O}^{\otimes}$. We show that the endomorphism mapping $\infty$-category of a map $\mathcal{O}$-monoidale is a coCartesian $(Δ^{\rm op},\mathcal{O})$-duoidal $\infty$-category. After that, we introduce a convolution product on the mapping $\infty$-category from an $\mathcal{O}$-comonoidale to an $\mathcal{O}$-monoidale. We show that the $\mathcal{O}$-monoidal structure on the duoidal endomorphism mapping $\infty$-category of a map $\mathcal{O}$-monoidale is equivalent to the convolution product on the mapping $\infty$-category from the dual $\mathcal{O}$-comonoidale to the map $\mathcal{O}$-monoidale.
Twisted arrow $\infty$-categories of $(\infty,1)$-categories were introduced by Lurie, and they have various applications in higher category theory. Abell\'{a}n Garc\'{i}a and Stern gave a generalization to twisted arrow $\infty$-categories of $(\infty,2)$-categories. In this paper we introduce another simple model for twisted arrow $\infty$-categories of $(\infty,2)$-categories.
Let ${\rm N}_m(R) = \{ (a_{ij}) \in {\rm M}_m(R) \mid a_{11} = a_{22} = \cdots = a_{mm} \mbox{ and } a_{ij} = 0 \mbox{ for any } i > j \}$ for a commutative ring $R$. Then ${\rm N}_m(R)$ is a quadratic monomial algebra over $R$. We calculate ${\rm HH}^{\ast}({\rm N}_m(R), {\rm M}_m(R)/{\rm N}_m(R))$ as $R$-modules. We also determine the $R$-algebra structure of the Hochschild cohomology ring ${\rm HH}^{\ast}({\rm N}_m(R), {\rm N}_m(R))$. For $m \ge 3$, ${\rm HH}^{\ast}({\rm N}_m(R), {\rm N}_m(R))$ is an infinitely generated algebra over $R$ and has no Batalin-Vilkovisky algebra structure giving the Gerstenhaber bracket.
There are two dual equivalences between the $\infty$-category of $\mathcal{O}$-monoidal $\infty$-categories with right adjoint lax $\mathcal{O}$-monoidal functors and that with left adjoint oplax $\mathcal{O}$-monoidal functors, where $\mathcal{O}$ is an $\infty$-operad. We study the space of equivalences between these two $\infty$-categories, and show that the two equivalences equipped with compatible $\mathcal{O}$-monoidal presheaf functors are canonically equivalent.
We give another proof of the fact that there is a dual equivalence between the $\infty$-category of monoidal $\infty$-categories with left adjoint oplax monoidal functors and that with right adjoint lax monoidal functors by constructing a perfect pairing between them.
There is a natural action of a kind of Hecke algebra $\mathcal{H}_n$ on the $n$th Morava $E$-theory of spaces. We construct Hecke operators in an amalgamated cohomology theory of the $n$th and the $(n+1)$st Morava $E$-theories. These operations are natural extensions of the Hecke operators in the $(n+1)$st Morava $E$-theory, and they induce an action of the Hecke algebra $\mathcal{H}_{n+1}$ on the $n$th Morava $E$-theory of spaces. We study a relationship between the actions of the Hecke algebras $\mathcal{H}_n$ and $\mathcal{H}_{n+1}$ on the $n$th Morava $E$-theory, and show that the $\mathcal{H}_{n+1}$-module structure is obtained from the $\mathcal{H}_n$-module structure by the restriction along an algebra homomorphism from $\mathcal{H}_{n+1}$ to $\mathcal{H}_n$.
In this paper we study duoidal structures on $\infty$-categories of operadic modules. Let $\mathcal{O}^{\otimes}$ be a small coherent $\infty$-operad and let $\mathcal{P}^{\otimes}$ be an $\infty$-operad. If a $\mathcal{P}\otimes\mathcal{O}$-monoidal $\infty$-category $\mathcal{C}^{\otimes}$ has a sufficient supply of colimits, then we show that the $\infty$-category ${\rm Mod}_A^{\mathcal{O}}(\mathcal{C})$ of $\mathcal{O}$-$A$-modules in $\mathcal{C}^{\otimes}$ has a structure of $(\mathcal{P},\mathcal{O})$-duoidal $\infty$-category for any $\mathcal{P}\otimes\mathcal{O}$-algebra object $A$.
In this paper we introduce a notion of $\mathbf{O}$-monoidal $\infty$-categories for a finite sequence $\mathbf{O}^{\otimes}$ of $\infty$-operads, which is a generalization of the notion of higher monoidal categories in the setting of $\infty$-categories. We show that the $\infty$-category of coCartesian $\mathbf{O}$-monoidal $\infty$-categories and right adjoint lax $\mathbf{O}$-monoidal functors is equivalent to the opposite of the $\infty$-category of Cartesian $\mathbf{O}_{\rm rev}$-monoidal $\infty$-categories and left adjoint oplax $\mathbf{O}_{\rm rev}$-monoidal functors, where $\mathbf{O}^{\otimes}_{\rm rev}$ is a sequence obtained by reversing the order of $\mathbf{O}^{\otimes}$.
We introduce graded $\mathbb{E}_{\infty}$-rings and graded modules over them, and study their properties. We construct projective schemes associated to connective $\mathbb{N}$-graded $\mathbb{E}_{\infty}$-rings in spectral algebraic geometry. Under some finiteness conditions, we show that the $\infty$-category of almost perfect quasi-coherent sheaves over a spectral projective scheme $\mathrm{Proj}\,(A)$ associated to a connective $\mathbb{N}$-graded $\mathbb{E}_{\infty}$-ring $A$ can be described in terms of $\mathbb{Z}$-graded $A$-modules.
Let ${\rm Mold}_{n, d}$ be the moduli of rank $d$ subalgebras of ${\rm M}_n$ over ${\Bbb Z}$. For $x \in {\rm Mold}_{n, d}$, let ${\mathcal A}(x) \subseteq {\rm M}_n(k(x))$ be the subalgebra of ${\rm M}_n$ corresponding to $x$, where $k(x)$ is the residue field of $x$. In this article, we apply Hochschild cohomology to ${\rm Mold}_{n, d}$. The dimension of the tangent space $T_{{\rm Mold}_{n, d}/{\Bbb Z}, x}$ of ${\rm Mold}_{n, d}$ over ${\Bbb Z}$ at $x$ can be calculated by the Hochschild cohomology $H^{1}({\mathcal A}(x), {\rm M}_n(k(x))/{\mathcal A}(x))$. We show that $H^{2}({\mathcal A}(x), {\rm M}_n(k(x))/{\mathcal A}(x)) = 0$ is a sufficient condition for the canonical morphism ${\rm Mold}_{n, d} \to {\Bbb Z}$ being smooth at $x$. We also calculate $H^{i}(A, {\rm M}_n(k)/A)$ for several $R$-subalgebras $A$ of ${\rm M}_n(R)$ over a commutative ring $R$. In particular, we summarize the results on $H^{i}(A, {\rm M}_n(k)/A)$ for all $k$-subalgebras $A$ of ${\rm M}_n(k)$ over an algebraically closed field $k$ in the case $n=2, 3$.
In this paper we study quasi-categories of comodules over coalgebras in a stable homotopy theory. We show that the quasi-category of comodules over the coalgebra associated to a Landweber exact S-algebra depends only on the height of the associated formal group. We also show that the quasi-category of E(n)-local spectra is equivalent to the quasi-category of comodules over the coalgebra A\otimes A for any Landweber exact S_(p)-algebra A of height n at a prime p. Furthermore, we show that the category of module objects over a discrete model of the Morava E-theory spectrum in the K(n)-local discrete symmetric G_n-spectra is a model of the K(n)-local category, where G_n is the extended Morava stabilizer group.
In this paper we study the category of discrete G-spectra for a profinite group G. We consider an embedding of module objects in spectra into a category of module objects in discrete G-spectra, and study the relationship between the embedding and the homotopy fixed points functor. We also consider an embedding of module objects in terms of quasi-categories, and show that the two formulations of embeddings are equivalent in some circumstances.
In this paper we study the topology of the moduli spaces of representations of degree $2$ for free monoids. We calculate the virtual Hodge polynomials of the character varieties for several types of $2$-dimensional representations. Furthermore, we count the number of isomorphism classes for each type of $2$-dimensional representations over any finite field ${\Bbb F}_q$, and show that the number coincides with the virtual Hodge polynomial evaluated at $q$.
Let n \geq 1 and let p be any prime. Also, let E_n be the Lubin-Tate spectrum, G_n the extended Morava stabilizer group, and K(n) the nth Morava K-theory spectrum. Then work of Devinatz and Hopkins and some results due to Behrens and the first author of this note, show that if X is a finite spectrum, then the localization L_{K(n)}(X) is equivalent to the homotopy fixed point spectrum (L_{K(n)}(E_n \wedge X))^{hG_n}, which is formed with respect to the continuous action of G_n on L_{K(n)}(E_n \wedge X). In this note, we show that this equivalence holds for any (S-cofibrant) spectrum X. Also, we show that for all such X, the strongly convergent Adams-type spectral sequence abutting to π_\ast(L_{K(n)}(X)) is isomorphic to the descent spectral sequence that abuts to π_\ast((L_{K(n)}(E_n \wedge X))^{hG_n}).
In this note some generalization of the Chern character is discussed from the chromatic point of view. We construct a multiplicative G_{n+1}-equivariant natural transformation Θfrom some height (n+1) cohomology theory E^*(-) to the height n cohomology theory K^*(-)\hat{\otimes}_F L, where K^*(-) is essentially the n-th Morava K-theory. As a corollary, it is shown that the G_n-module K^*(X) can be recovered from the G_{n+1}-module E^*(X). We also construct a lift of Θto a natural transformation between characteristic zero cohomology theories.
We have shown that the n-th Morava K-theory K^*(X) for a CW-spectrum X with action of Morava stabilizer group G_n can be recovered from the system of some height-(n+1) cohomology groups E^*(Z) with G_{n+1}-action indexed by finite subspectra Z. In this note we reformulate and extend the above result. We construct a symmetric monoidal functor F from the category of E^{vee}_*(E)-precomodules to the category of K_{*}(K)-comodules. Then we show that K^*(X) is naturally isomorphic to the inverse limit of F(E^*(Z)) as a K_{*}(K)-comodule.