Permutative categories, multicategories and algebraic<i>K</i>–theory

Elmendorf, Anthony; Mandell, Michael A.

doi:10.2140/agt.2009.9.2391

Cited by 37 publications

(66 citation statements)

References 11 publications

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“…Using earlier work of Elmendorf and Mandell (see [EM06] and [EM09]) refining Segal's Γ-space construction, in Theorem 3.6 we show that the K-theory of Lawvere theories admits the structure of a lax symmetric monoidal functor with respect to the Kronecker product, which is a generalization of the tensor products of rings.…”

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confidence: 86%

Generalizations of Loday's assembly maps for Lawvere's algebraic theories

Bohmann¹,

Szymik²

2021

Preprint

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Loday's assembly maps approximate the K-theory of group rings by the K-theory of the coefficient ring and the corresponding homology of the group. We present a generalization that places both ingredients on the same footing. Building on Elmendorf-Mandell's multiplicativity results and our earlier work, we show that the K-theory of Lawvere theories is lax monoidal. This result makes it possible to present our theory in a user-friendly way without using higher categorical language. It also allows us to extend the idea to new contexts and set up a non-abelian interpolation scheme, raising novel questions. Numerous examples illustrate the scope of our extension.

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confidence: 86%

Generalizations of Loday's assembly maps for Lawvere's algebraic theories

Bohmann¹,

Szymik²

2021

Preprint

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“…The symmetric monoidal category Alg K has an associated Set-valued colored operad (see, e.g., [21]) that we denote by the same symbol Alg K . Concretely, the objects are the objects of Alg K and the sets of operations are given by…”

Section: Definition 25mentioning

confidence: 99%

Categorification of algebraic quantum field theories

et al. 2021

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This paper develops a concept of 2-categorical algebraic quantum field theories (2AQFTs) that assign locally presentable linear categories to spacetimes. It is proven that ordinary AQFTs embed as a coreflective full 2-subcategory into the 2-category of 2AQFTs. Examples of 2AQFTs that do not come from ordinary AQFTs via this embedding are constructed by a local gauging construction for finite groups, which admits a physical interpretation in terms of orbifold theories. A categorification of Fredenhagen’s universal algebra is developed and also computed for simple examples of 2AQFTs.

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“…The category Cat * has a coproduct ∨ and smash product ∧ defined analogously to the wedge product and smash product of based spaces. See [EM09,Construction 4.19] for the construction of smash products in the general setting of based objects in a symmetric monoidal category V.…”

Section: Preliminaries On Symmetric Spectramentioning

confidence: 99%

A multiplicative comparison of Segal and Waldhausen K-Theory

Bohmann

Osorno

2019

Math. Z.

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In this paper, we establish a multiplicative equivalence between two multiplicative algebraic K-theory constructions, Elmendorf and Mandell's version of Segal's K-theory and Blumberg and Mandell's version of Waldhausen's S• construction. This equivalence implies that the ring spectra, algebra spectra, and module spectra constructed via these two classical algebraic K-theory functors are equivalent as ring, algebra or module spectra, respectively. It also allows for comparisions of spectrally enriched categories constructed via these definitions of Ktheory. As both the Elmendorf-Mandell and Blumberg-Mandell multiplicative versions of K-theory encode their multiplicativity in the language of multicategories, our main theorem is that there is multinatural transformation relating these two symmetric multifunctors that lifts the classical functor from Segal's to Waldhausen's construction. Along the way, we provide a slight generalization of the Elmendorf-Mandell construction to symmetric monoidal categories.Second, we focus on K-theory as a construction on Waldhausen categories, rather than any of the classes of infinity categories. Since the classical Waldhausen and Segal constructions apply only to more specific types of input, their uniqueness is not addressed by any of the uniqueness or universality results above. The present paper provides this kind of multiplicative comparison between the classical versions of algebraic K-theory.

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Permutative categories, multicategories and algebraicK–theory

Cited by 37 publications

References 11 publications

Generalizations of Loday's assembly maps for Lawvere's algebraic theories

Generalizations of Loday's assembly maps for Lawvere's algebraic theories

Categorification of algebraic quantum field theories

A multiplicative comparison of Segal and Waldhausen K-Theory

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