Exponential Functors Research Articles

Exponential functors, R–matrices and twists

In this paper we show that each polynomial exponential functor on complex finite-dimensional inner product spaces is defined up to equivalence of monoidal functors by an involutive solution to the Yang-Baxter equation (an involutive $R$-matrix), which determines an extremal character on $S_{\infty}$. These characters are classified by Thoma parameters, and Thoma parameters resulting from polynomial exponential functors are of a special kind. Moreover, we show that each $R$-matrix with Thoma parameters of this kind yield a corresponding polynomial exponential functor. In the second part of the paper we use these functors to construct a higher twist over $SU(n)$ for a localisation of $K$-theory that generalises the one given by the basic gerbe. We compute the indecomposable part of the rational characteristic classes of these twists in terms of the Thoma parameters of their $R$-matrices.

On the Structure of Graded Commutative Exponential functors

Abstract We investigate the structure of graded commutative exponential functors. We give applications of these structure results, including computations of the homology of the symmetric groups and of extensions in the category of strict polynomial functors.

Some extension groups between exponential functors

Let F be the category of functors that send a finite-dimensional vector space over F2 to a vector space over F2. In this note, we describe the first extension groups between some exponential functors such as ExtF1(S⁎,Λ⁎), ExtF1(S4⁎,Λ⁎), and ExtF1(S4⁎,S4⁎), where S⁎, Λ⁎, S4⁎ are the symmetric power, the exterior power, and the truncated symmetric power at the power 4, respectively. Three main techniques are used: tri-graded Hopf algebra structure of the extension groups between two exponential functors, the polynomial filtration of these functors, and the hypercohomology spectral sequences.

Bar complexes and extensions of classical exponential functors

Nous calculons les groupes d’Ext entre foncteurs exponentiels classiques (i.e. puissances symétriques, extérieures ou divisées), et leur précomposition par le foncteur de torsion de Frobenius. Notre méthode repose sur les constructions bar, et relie ces calculs d’Ext avec l’homologie des espaces d’Eilenberg et Mac Lane.

Koszul duality and extensions of exponential functors

We study Koszul duality in the category of strict polynomial functors. We compute Koszul duals of various functors and apply these results to the problem of calculating Ext-groups between exponential functors. The main application is a full description of the Ext-groups between twisted exterior and divided powers and between twisted symmetric and divided powers which completes the program started in [V. Franjou, E. Friedlander, A. Scorichenko, A. Suslin, General linear and functor cohomology over finite fields, Ann. of Math. 150 (2) (1999) 663–728].

Deriving calculus with cotriples

We construct a Taylor tower for functors from pointed categories to abelian categories via cotriples associated to cross effect functors. The tower was inspired by Goodwillie’s Taylor tower for functors of spaces, and is related to Dold and Puppe’s stable derived functors and Mac Lane’s Q Q -construction. We study the layers, D n F = fiber ⁡ ( P n F → P n − 1 F ) D_{n}F = \operatorname {fiber}(P_{n}F\rightarrow P_{n-1}F) , and the limit of the tower. For the latter we determine a condition on the cross effects that guarantees convergence. We define differentials for functors, and establish chain and product rules for them. We conclude by studying exponential functors in this setting and describing their Taylor towers.

Corrections to “on right adjoints to exponential functors”

Corrections to “on right adjoints to exponential functors”