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Free Commutative Monoids in Homotopy Type Theory [article]

Vikraman Choudhury, Marcelo Fiore
2021 arXiv   pre-print
We develop a constructive theory of finite multisets, defining them as free commutative monoids in Homotopy Type Theory.  ...  These presentations correspond to equational theories including a commutation axiom.  ...  Commutative Monoids We start by giving the definition of commutative monoids in type theory; they are monoids with an additional commutation axiom. Definition 2.1 (Commutative monoid).  ... 
arXiv:2110.05412v1 fatcat:fdak3n72tnfkvgqnccwys5ll4i

E∞-monoids with coherent homotopy inverses are Abelian groups

R. Schwänzl, R.M. Vogt
1989 Topology  
If X in addition is coherently homotopy commutative it is called an E, monoid. Interest in such homotopy monoid structures arose from the following well-known fact. PROPOSITION 3.2.  ...  LOOSELY speaking, an A, monoid is an H-space X whose multiplication is homotopy associative with a homotopy unit in a coherent way, i.e. the homotopies fit together up to higher homotopies which in turn  ...  The morphisms in its image are called the set operations of 0. (ii) %A--the theory of commutative monoids. %?A([n], [l]) is the free commutative monoid on n generators xi, . . . , x., %?.  ... 
doi:10.1016/0040-9383(89)90006-2 fatcat:ixzetvog4zcohoz46r6pnvhwx4

On the homotopy theory of monoids

Carol M. Hurwitz
1989 Journal of the Australian Mathematical Society  
This immediately gives a monoid whose classifying space is of the same homotopy type as that of the small category.  ...  In other words, is it possible to construct a group G as above whose classifying space has the homotopy type of a finite CW-complex. Such a group is said to be geometrically finite.  ...  Thus, for any connected category in Cat/, we can find a monoid with the same homotopy type.  ... 
doi:10.1017/s1446788700031621 fatcat:4e4qjjx5mbcklos3pbcmhi4i5q

Logarithmic structures on topological

Steffen Sagave
2014 Geometry and Topology  
In our setup, we obtain canonical logarithmic structures on connective K-theory spectra which approximate the respective periodic spectra.  ...  The inclusion of the p-complete Adams summand into the p-complete connective complex K-theory spectrum is compatible with these logarithmic structures.  ...  In order to prove the theorem, we begin with giving a more explicit description of the homotopy type of the commutative J -space monoid D(x).  ... 
doi:10.2140/gt.2014.18.447 fatcat:k7ecwm5xabg43d3txvfcbsiq44

E_n ring spectra and Dyer-Lashof operations [article]

Tyler Lawson
2020 arXiv   pre-print
This is a preliminary version of a chapter written for the Handbook of Homotopy Theory.  ...  In particular, we discuss Dyer-Lashof operations and their evolving role in the study of iterated loop spaces, E_n-algebras, and E_n-ring spectra.  ...  For example, there is essentially no workable obstruction theory for the construction of commutative rings of any type in equivariant stable theory.  ... 
arXiv:2002.03889v1 fatcat:knxpwymiqjftzchm63ofdbwheq

Localization sequences for logarithmic topological Hochschild homology

John Rognes, Steffen Sagave, Christian Schlichtkrull
2015 Mathematische Annalen  
Our results apply, for example, to connective covers of periodic ring spectra like real and complex topological K-theory.  ...  We study the logarithmic topological Hochschild homology of ring spectra with logarithmic structures and establish localization sequences for this theory.  ...  In order to prepare for subsequent computations of log THH, we describe the homotopy type and the homology of B cy (N ) and B rep (N ) (or rather of their geometric realizations) for a free commutative  ... 
doi:10.1007/s00208-015-1202-3 fatcat:jdw3avaxtjefzptofugt73mkxu

Homotopy composition of cospans

Joachim Kock, David I. Spivak
2017 Communications in Contemporary Mathematics  
In this note we observe that the same construction yields also general commutative Frobenius algebras, if just the pushouts are changed to homotopy pushouts.  ...  It is well known that the category of finite sets and cospans, composed by pushout, contains the universal special commutative Frobenius algebra.  ...  The symmetric monoidal category Cospan, whose objects are finite sets and whose morphisms are homotopy classes of cospans, is the free symmetric monoidal category on a commutative Frobenius object.  ... 
doi:10.1142/s0219199716500474 fatcat:ldhjtuy2fjaixfzjrhkybmsko4

Page 915 of Mathematical Reviews Vol. , Issue 85c [page]

1985 Mathematical Reviews  
This requires that E,. be interpreted in terms of Cech type rather than singular type stable homotopy theory. Suitable foundations have been developed by R. W.  ...  A monoid is said to be partially free if it is the free product of a free semigroup and a free group. B. Mitchell {J.  ... 

Commutative Γ-rings do not model all commutative ring spectra

Tyler Lawson
2009 Homology, Homotopy and Applications  
We show that the free E ∞ -algebra on a zero-cell cannot be modeled by a commutative Γ-ring.  ...  The free E ∞ -algebra on S 0 cannot be realized by a commutative Γ-ring. Proof. The free E ∞ -algebra on a spectrum X has the homotopy type P(X) = k 0 (X ∧k ) hΣ k .  ...  Γ-rings (S[M ], R). ( 1 ) In particular, if M is the free commutative topological monoid N S on a set S, then we get an adjunction Map(S, R(S 0 )) ∼ = Map comm. Γ-rings S[N S ], R .  ... 
doi:10.4310/hha.2009.v11.n2.a9 fatcat:6znmzff3ubaa7f6wqkp2b4uecu

Categorified algebra and equivariant homotopy theory [article]

John D. Berman
2018 arXiv   pre-print
This sets up a parallel between equivariant homotopy theory and motivic homotopy theory, where Burnside constructions are analogous to Morita theory.  ...  Finally, we provide evidence for a formal duality between naive and genuine equivariant homotopy theory, in the form of a group-theoretic Eilenberg-Watts Theorem.  ...  What are called commutative algebra objects in the higher category theory literature are called commutative monoid objects in the classical category theory literature.  ... 
arXiv:1805.08745v1 fatcat:neegkwfatrhplaedzscrmdhexy

Higher‐dimensional algebra and topological quantum field theory

John C. Baez, James Dolan
1995 Journal of Mathematical Physics  
We give evidence for this hypothesis and describe its relation to stable homotopy theory.  ...  We review progress towards a definition of n-category suited for this purpose, and outline a program in which n-dimensional TQFTs are to be described as n-category representations.  ...  For example, a monoid object in Vect is an algebra. It turns out that 2Cob is the 'free rigid symmetric monoidal category on one commutative monoid object with nondegenerate trace'.  ... 
doi:10.1063/1.531236 fatcat:at2hzee7ajcsxd7d4d7bmlzq5i

Realizing homotopy group actions [article]

David Blanc, Debasis Sen
2014 arXiv   pre-print
If successful, we obtain a G-space X' realizing the given homotopy information, determined up to Bredon G-homotopy type.  ...  For any finite group G, we define the notion of a Bredon homotopy action of G, modelled on the diagram of fixed point sets (X_H)_H≤ G for a G-space X, together with a pointed homotopy action of the group  ...  In the model category of strictly associative monoids, we can replace h by another weak equivalence of monoids making A commute on the nose (cf.  ... 
arXiv:1210.2574v2 fatcat:qvttlnecmjbkvoxaedzb7bgzpe

The Bott cofiber sequence in deformation K-theory and simultaneous similarity in U( n)

TYLER LAWSON
2008 Mathematical proceedings of the Cambridge Philosophical Society (Print)  
We show that there is a homotopy cofiber sequence of spectra relating Carlsson's deformation K-theory of a group G to its "deformation representation ring," analogous to the Bott periodicity sequence relating  ...  connective K-theory to ordinary homology.  ...  We briefly sketch an identification of the homotopy type of this spectrum; a more general decomposition of the homotopy type for free products can be found in [10] .  ... 
doi:10.1017/s0305004108001928 fatcat:j6lbkdvzdvcshjm6zf6ae4wmqy

Strict algebraic models for rational parametrised spectra I [article]

Vincent Braunack-Mayer
2020 arXiv   pre-print
Building on Quillen's rational homotopy theory, we obtain algebraic models for the rational homotopy theory of parametrised spectra.  ...  In this article, we prove that the rational homotopy type of an X-parametrised spectrum is completely encoded by a Λ_X-representation and also by a C_X-comodule.  ...  But the rational homotopy theory of spectra is encoded by rational chain complexes and Quillen's rational homotopy theory provides a dg Lie algebra Λ X encoding the rational homotopy type of X.  ... 
arXiv:1910.14608v2 fatcat:vxfd4lx2bza5vcorz4hmjircpm

Stable homotopy of algebraic theories

Stefan Schwede
2001 Topology  
The simplicial objects in an algebraic category admit an abstract homotopy theory via a Quillen model category structure.  ...  For the theory of commutative algebras we obtain a ring spectrum which is related to AndreH }Quillen homology via certain spectral sequences.  ...  Monoids and groups. The theories of sets, monoids and groups have equivalent stable homotopy theories.  ... 
doi:10.1016/s0040-9383(99)00046-4 fatcat:fvifpxxnwzhtxolq62g4ivwhya
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