Universality of multiplicative infinite loop space machines

Gepner, David; Groth, Moritz; Nikolaus, Thomas

doi:10.2140/agt.2015.15.3107

Cited by 77 publications

(100 citation statements)

References 21 publications

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“…We have just asserted that SymMon 8 is an example of a presentable 8category with a closed symmetric monoidal tensor product, but this is certainly not well-known. We believe it was first observed by Gepner, Groth, and Nikolaus [11], as an example of the following lemma:…”

Section: Semiring and Module Categoriesmentioning

confidence: 72%

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On the commutative algebra of categories

Berman

2018

Algebr. Geom. Topol.

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We discuss what it means for a symmetric monoidal category to be a module over a commutative semiring category. Each of the categories of (1) cartesian monoidal categories, (2) semiadditive categories, and (3) connective spectra can be recovered in this way as categories of modules over a commutative semiring category (or 8-category in the last case). This language provides a simultaneous generalization of the formalism of algebraic theories (operads, PROPs, Lawvere theories) and stable homotopy theory, with essentially a variant of algebraic K-theory bridging between the two. Principle 1.1. Set and Ab classify the properties of symmetric monoidal categories being cocartesian monoidal (respectively additive), just as Zr 1 2 s and Z{2 characterize properties of abelian groups.What we have said is not literally true. The problem is that Set and Ab are not finitely generated as semiring categories, and will therefore have poor algebraic properties. However, there are two ways to resolve this problem, and either way the principle becomes true.

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Section: Semiring and Module Categoriesmentioning

confidence: 72%

“…Although we are not aware of a specific work which includes all of these conditions under the name 'solid', none of the conditions are new. So the reader who objects that this 'definition' requires proof (that all the conditions are equivalent) may consult [11] for all but the last condition.…”

Section: Solid Rings Ring Spectra and Semiring Categoriesmentioning

confidence: 99%

On the commutative algebra of categories

Berman

2018

Algebr. Geom. Topol.

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“…As promised in [5], we prove: Proof. As in [9], the product map Sp ě0 b Sp ě0 Ñ Sp ě0 is an equivalence. Identifying Sp ě0 with Mdl Burn and noting that Mdl is symmetric monoidal, we find that Burn b Burn Ñ Burn is also an equivalence.…”

Section: Applications Of Lawvere Theoriesmentioning

confidence: 94%

Higher Lawvere theories

Berman

2020

Journal of Pure and Applied Algebra

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We survey Lawvere theories at the level of 8-categories, as an alternative framework for higher algebra (rather than 8-operads). From a pedagogical perspective, they make many key definitions and constructions less technical. They are also better-suited than operads for equivariant homotopy theory and its relatives.Our main result establishes a universal property for the 8-category of Lawvere theories, which completely characterizes the relationship between a Lawvere theory and its 8-category of models. Many familiar properties of Lawvere theories follow directly.As a consequence, we prove that the Burnside category is a classifying object for additive categories, as promised in an earlier paper, and as part of a more general correspondence between enriched Lawvere theories and module Lawvere theories.

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“…We will need later a stronger version of that result where we can drop the assumption that we are already given a natural transformation. To prove this stronger result we have to rely on results of [GGN15] which are obtained in the setting of ∞-categories. Thus we will also state the result in the setting of ∞-categories.…”

Section: Preliminaries On Dendroidal Setsmentioning

confidence: 99%

Homology of dendroidal sets

Bašić

Nikolaus

2017

Homology, Homotopy and Applications

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We define for every dendroidal set X a chain complex and show that this assignment determines a left Quillen functor. Then we define the homology groups H n (X) as the homology groups of this chain complex. This generalizes the homology of simplicial sets. Our main result is that the homology of X is isomorphic to the homology of the associated spectrum K(X) as discussed in [BN14] and [Nik14]. Since these homology groups are sometimes computable we can identify some spectra K(X) which we could not identify before.

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Universality of multiplicative infinite loop space machines

Cited by 77 publications

References 21 publications

On the commutative algebra of categories

On the commutative algebra of categories

Higher Lawvere theories

Homology of dendroidal sets

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