By R. Schrader, R. Seiler, D.A. Uhlenbrock
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Extra resources for Mathematical Problems in Theoretical Physics
Example text
By the lemma, this is an I-adic topological A ⊗ A-algebra, where I := lim I I n /I n+1 = SymA P . ←− n , and It remains to define on G := Spf E the structure of a formal groupoid on Y . , the DG pull-back of the (Spec C)a+1 -scheme Y a+1 by the diagonal embedding Spec C → (Spec C)a+1 ). Thus G0 = Y , G1 = G. Now the G· form a formal simplicial scheme in the obvious way, and each standard projection Ga → G × . . × G (a times) Y Y is an isomorphism. , the “classifying space” BG of G). The details are left to the reader.
Is the tensor category dual to Mo! ). If we are in the k-linear situation, then we call M∗! abelian if M is an abelian k-category, the PI∗ are left exact functors, and ⊗! is right exact. A compound tensor functor τ ∗! : N∗! → M∗! between the compound tensor categories is a compound pseudo-tensor functor such that all the canonical morphisms νI : τ (⊗! , τ ! : N! → M! is a tensor I I functor. Such τ ∗! amounts to a pair (τ ∗ , τ ! ) where τ ∗ : N∗ → M∗ , τ ! : N! → M! are, respectively, the pseudo-tensor and tensor extensions of the same functor τ : N → M which commute with ⊗IS,T maps.
L[1]}, M ) → ∗ Pn−1 ({L, . . , L}, M ) which is minus the sum of the canonical “convolution” maps for each of the n arguments. The above constructions are functorial in the obvious manner. Remarks. (i) If for every n the inner P object P∗n ({L, . . 1), then one has the inner Chevalley complex C(L, M ) defined in the obvious way. It carries a canonical ∗ action of L† . 7). 2). Then for every n ≥ 0 and every N ∈ M one has N ⊗ L◦⊗n = P∗n ({L, . . 7). 1) ∼ h(C(L, M )) −→ C(L, M ). 16), so L is a DG Lie∗ super algebra, M a DG super L-module.