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**On the algebraic \(L\)-theory of \(\Delta \)-sets.**
*(English)*
Zbl 1256.19004

We start by quoting from the authors abstract: “The algebraic \(L\)-groups \(L_{\ast} (\mathcal A, X)\) are defined for an additive category \(\mathcal A\) with chain duality and a \(\Delta\)-set X, and identified with the generalized homology groups \(H_\ast(X; \mathcal L_\bullet(\mathcal A))\) of \(X\) with coefficients in the algebraic \(L\)-spectrum \(\mathcal L_\bullet(\mathcal A)\). Previously such groups had only been defined for simplicial complexes.”

There were two main observations and motivations which led to this generalization. The first being that the homotopy theory \(\Delta\)-sets introduced by C. P. Rourke and B. J. Sanderson [Q. J. Math., Oxf. II. Ser. 22, 321–338 (1971; Zbl 0226.55019)] is the same as that of simplicial complexes, and in addition \(\Delta\)-sets are smaller and the quotient of a \(\Delta\)-set under group action is also a \(\Delta\)-set. The second being that it may not be possible to extend an involution \(T: \mathcal A\to\mathcal A\) on an additive category \(\mathcal A\) to the functor category \(\mathcal A_\ast(X)\) for an arbitrary category \(X\), as defined in the paper under review. The authors prove in Proposition 5.7 that this is possible if \(X\) is a \(\Delta\)-set. Having proven this they go on to extend the result 13.7 of T. Fimmel [Math. Nachr. 190, 51–122 (1998; Zbl 0903.55008)], proved for simplicial complexes, to the Proposition 5.10 for \(\Delta\)-sets.

There were two main observations and motivations which led to this generalization. The first being that the homotopy theory \(\Delta\)-sets introduced by C. P. Rourke and B. J. Sanderson [Q. J. Math., Oxf. II. Ser. 22, 321–338 (1971; Zbl 0226.55019)] is the same as that of simplicial complexes, and in addition \(\Delta\)-sets are smaller and the quotient of a \(\Delta\)-set under group action is also a \(\Delta\)-set. The second being that it may not be possible to extend an involution \(T: \mathcal A\to\mathcal A\) on an additive category \(\mathcal A\) to the functor category \(\mathcal A_\ast(X)\) for an arbitrary category \(X\), as defined in the paper under review. The authors prove in Proposition 5.7 that this is possible if \(X\) is a \(\Delta\)-set. Having proven this they go on to extend the result 13.7 of T. Fimmel [Math. Nachr. 190, 51–122 (1998; Zbl 0903.55008)], proved for simplicial complexes, to the Proposition 5.10 for \(\Delta\)-sets.

Reviewer: Himadri Kumar Mukerjee (Shillong)