Lifting Coalgebra Modalities and $\mathsf{MELL}$ Model Structure to Eilenberg-Moore Categories

A categorical model of the multiplicative and exponential fragments of intuitionistic linear logic ($\mathsf{MELL}$), known as a \emph{linear category}, is a symmetric monoidal closed category with a monoidal coalgebra modality (also known as a linear exponential comonad). Inspired by Blute and Scot...

Full description

Saved in:
Bibliographic Details
Published in:Logical methods in computer science Vol. 15, Issue 4
Main Author: Lemay, Jean-Simon Pacaud
Format: Journal Article
Language:English
Published: Logical Methods in Computer Science e.V 07.11.2019
Subjects:
computer science - logic in computer science
ISSN:1860-5974, 1860-5974
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Abstract A categorical model of the multiplicative and exponential fragments of intuitionistic linear logic ($\mathsf{MELL}$), known as a \emph{linear category}, is a symmetric monoidal closed category with a monoidal coalgebra modality (also known as a linear exponential comonad). Inspired by Blute and Scott's work on categories of modules of Hopf algebras as models of linear logic, we study categories of algebras of monads (also known as Eilenberg-Moore categories) as models of $\mathsf{MELL}$. We define a $\mathsf{MELL}$ lifting monad on a linear category as a Hopf monad -- in the Brugui{\`e}res, Lack, and Virelizier sense -- with a special kind of mixed distributive law over the monoidal coalgebra modality. As our main result, we show that the linear category structure lifts to the category of algebras of $\mathsf{MELL}$ lifting monads. We explain how groups in the category of coalgebras of the monoidal coalgebra modality induce $\mathsf{MELL}$ lifting monads and provide a source for such groups from enrichment over abelian groups. Along the way we also define mixed distributive laws of symmetric comonoidal monads over symmetric monoidal comonads and lifting differential category structure. Comment: An extend abstract version of this paper appears in the conference proceedings of the 3rd International Conference on Formal Structures for Computation and Deduction (FSCD 2018), under the title "Lifting Coalgebra Modalities and $\mathsf{MELL}$ Model Structure to Eilenberg-Moore Categories"
AbstractList A categorical model of the multiplicative and exponential fragments of intuitionistic linear logic ($\mathsf{MELL}$), known as a \emph{linear category}, is a symmetric monoidal closed category with a monoidal coalgebra modality (also known as a linear exponential comonad). Inspired by Blute and Scott's work on categories of modules of Hopf algebras as models of linear logic, we study categories of algebras of monads (also known as Eilenberg-Moore categories) as models of $\mathsf{MELL}$. We define a $\mathsf{MELL}$ lifting monad on a linear category as a Hopf monad -- in the Brugui{\`e}res, Lack, and Virelizier sense -- with a special kind of mixed distributive law over the monoidal coalgebra modality. As our main result, we show that the linear category structure lifts to the category of algebras of $\mathsf{MELL}$ lifting monads. We explain how groups in the category of coalgebras of the monoidal coalgebra modality induce $\mathsf{MELL}$ lifting monads and provide a source for such groups from enrichment over abelian groups. Along the way we also define mixed distributive laws of symmetric comonoidal monads over symmetric monoidal comonads and lifting differential category structure. Comment: An extend abstract version of this paper appears in the conference proceedings of the 3rd International Conference on Formal Structures for Computation and Deduction (FSCD 2018), under the title "Lifting Coalgebra Modalities and $\mathsf{MELL}$ Model Structure to Eilenberg-Moore Categories"
A categorical model of the multiplicative and exponential fragments of intuitionistic linear logic ($\mathsf{MELL}$), known as a \emph{linear category}, is a symmetric monoidal closed category with a monoidal coalgebra modality (also known as a linear exponential comonad). Inspired by Blute and Scott's work on categories of modules of Hopf algebras as models of linear logic, we study categories of algebras of monads (also known as Eilenberg-Moore categories) as models of $\mathsf{MELL}$. We define a $\mathsf{MELL}$ lifting monad on a linear category as a Hopf monad -- in the Brugui{\`e}res, Lack, and Virelizier sense -- with a special kind of mixed distributive law over the monoidal coalgebra modality. As our main result, we show that the linear category structure lifts to the category of algebras of $\mathsf{MELL}$ lifting monads. We explain how groups in the category of coalgebras of the monoidal coalgebra modality induce $\mathsf{MELL}$ lifting monads and provide a source for such groups from enrichment over abelian groups. Along the way we also define mixed distributive laws of symmetric comonoidal monads over symmetric monoidal comonads and lifting differential category structure.
Author Lemay, Jean-Simon Pacaud
Author_xml – sequence: 1
  givenname: Jean-Simon Pacaud
  orcidid: 0000-0003-4124-3722
  surname: Lemay
  fullname: Lemay, Jean-Simon Pacaud
BookMark eNpNkE1Lw0AQhhdRUGt_gLccetBDdGe_mniTUD8gxUP1Ji6z2d2YErOySQ8i_nfTKuJcZnhfeAaeY7Lfhc4Rcgr0gnHFs8tyWaxSkGfiKjtnFPI9cgSZoqnM52L_331Ipn2_puNwDhlTR-SlbPzQdHVSBGxrZyImy2CxbYbG9Ql2Npk9v-Hw2vvP5aIsv2bb2rXJaoibathElwwhWTSt64yLdboMYYwKHFwd4kg4IQce295Nf_eEPN0sHou7tHy4vS-uy7QCCXkKCNaAQo_Sce9RSQDMKJMsR2ElMwIMN0o6Rg3zxnoxZ5IL8FAZZpXhE3L_w7UB1_o9Nm8YP3TARu-CEGuNcWiq1ulMKD7nmHubWyHGp4KqfE5ZVUlwzlYjC35YVQx9H53_4wHVO99661uD1EJneuubfwNgtHUD
ContentType Journal Article
DBID AAYXX
CITATION
DOA
DOI 10.23638/LMCS-15(4:8)2019
DatabaseName CrossRef
DOAJ Directory of Open Access Journals
DatabaseTitle CrossRef
DatabaseTitleList CrossRef
Database_xml – sequence: 1
  dbid: DOA
  name: Directory of Open Access Journals
  url: https://www.doaj.org/
  sourceTypes: Open Website
DeliveryMethod fulltext_linktorsrc
Discipline Computer Science
EISSN 1860-5974
ExternalDocumentID oai_doaj_org_article_846373a9fd9d4416a4069702cc51eedc
10_23638_LMCS_15_4_8_2019
GroupedDBID .4S
.DC
29L
2WC
5GY
5VS
AAFWJ
AAYXX
ADBBV
ADMLS
ADQAK
AENEX
AFPKN
ALMA_UNASSIGNED_HOLDINGS
ARCSS
BCNDV
CITATION
EBS
EJD
FRP
GROUPED_DOAJ
J9A
KQ8
MK~
ML~
M~E
OK1
OVT
P2P
TR2
TUS
XSB
ID FETCH-LOGICAL-c1519-1a1db16afa5e3ffa6511a802529a4d52b41b3b65e20b2fbdf4725341f1cb2d6b3
IEDL.DBID DOA
ISSN 1860-5974
IngestDate Fri Oct 03 12:51:22 EDT 2025
Sat Nov 29 08:04:21 EST 2025
IsDoiOpenAccess true
IsOpenAccess true
IsPeerReviewed true
IsScholarly true
Language English
License https://creativecommons.org/licenses/by/4.0
LinkModel DirectLink
MergedId FETCHMERGED-LOGICAL-c1519-1a1db16afa5e3ffa6511a802529a4d52b41b3b65e20b2fbdf4725341f1cb2d6b3
ORCID 0000-0003-4124-3722
OpenAccessLink https://doaj.org/article/846373a9fd9d4416a4069702cc51eedc
ParticipantIDs doaj_primary_oai_doaj_org_article_846373a9fd9d4416a4069702cc51eedc
crossref_primary_10_23638_LMCS_15_4_8_2019
PublicationCentury 2000
PublicationDate 2019-11-07
PublicationDateYYYYMMDD 2019-11-07
PublicationDate_xml – month: 11
  year: 2019
  text: 2019-11-07
  day: 07
PublicationDecade 2010
PublicationTitle Logical methods in computer science
PublicationYear 2019
Publisher Logical Methods in Computer Science e.V
Publisher_xml – name: Logical Methods in Computer Science e.V
SSID ssj0000331826
Score 2.1378253
Snippet A categorical model of the multiplicative and exponential fragments of intuitionistic linear logic ($\mathsf{MELL}$), known as a \emph{linear category}, is a...
SourceID doaj
crossref
SourceType Open Website
Index Database
SubjectTerms computer science - logic in computer science
Title Lifting Coalgebra Modalities and $\mathsf{MELL}$ Model Structure to Eilenberg-Moore Categories
URI https://doaj.org/article/846373a9fd9d4416a4069702cc51eedc
Volume 15, Issue 4
hasFullText 1
inHoldings 1
isFullTextHit
isPrint
journalDatabaseRights – providerCode: PRVAON
  databaseName: Directory of Open Access Journals
  customDbUrl:
  eissn: 1860-5974
  dateEnd: 99991231
  omitProxy: false
  ssIdentifier: ssj0000331826
  issn: 1860-5974
  databaseCode: DOA
  dateStart: 20040101
  isFulltext: true
  titleUrlDefault: https://www.doaj.org/
  providerName: Directory of Open Access Journals
– providerCode: PRVHPJ
  databaseName: ROAD: Directory of Open Access Scholarly Resources
  customDbUrl:
  eissn: 1860-5974
  dateEnd: 99991231
  omitProxy: false
  ssIdentifier: ssj0000331826
  issn: 1860-5974
  databaseCode: M~E
  dateStart: 20040101
  isFulltext: true
  titleUrlDefault: https://road.issn.org
  providerName: ISSN International Centre
link http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwrV1LSwMxEA4iHrz4FuuLHHpQIdgkm6TrTUvFQ1eEKngQQ55YkK7Ux0X0tzvJttKbFy97yC5L-GZ25ht28g1CbcOtV2BZQrk0pIghkKR-Shw4jCmcDK4T87AJdX3dvb8vb-ZGfaWesEYeuAHuFPIjV9yU0ZceUrc06aim6jDnBIX47lL0BdYzV0zlGMx5Is7Nb0zGwclOB1VvSKg4Ks66xywr68wlojm9_pxYLtfQypQR4vNmJ-toIYw30Ops2gKefnyb6HEwiqlHGffqNJsDqlxc1T7RaCh2sRl73H4A-vn0Gj-r_mDw1U63wzMeZoXY90nAbzXuQxDILV2kqmtY6iWhiDqVy1vo7rJ_27si0-kIxEGWLgk11FvAIxoReIxGAnUyXaAwrDSFF8wW1HIrRWAdy6L1sVBMQM6K1FnmpeXbaHFcj8MOwsJTy2QwQSlaMEVNqbyRnSCFV8JZ30InM6j0SyOCoaF4yLjqhKumQhe6qxOuLXSRwPx9MOlX5wWwqp5aVf9l1d3_eMkeWk4byicH1T5aBLjDAVpyH2-j18lhdhi4Vt_9H6z5xdw
linkProvider Directory of Open Access Journals
openUrl ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Lifting+Coalgebra+Modalities+and+%24%5Cmathsf%7BMELL%7D%24+Model+Structure+to+Eilenberg-Moore+Categories&rft.jtitle=Logical+methods+in+computer+science&rft.au=Jean-Simon+Pacaud+Lemay&rft.date=2019-11-07&rft.pub=Logical+Methods+in+Computer+Science+e.V&rft.eissn=1860-5974&rft.volume=15%2C+Issue+4&rft_id=info:doi/10.23638%2FLMCS-15%284%3A8%292019&rft.externalDBID=DOA&rft.externalDocID=oai_doaj_org_article_846373a9fd9d4416a4069702cc51eedc
thumbnail_l http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=1860-5974&client=summon
thumbnail_m http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=1860-5974&client=summon
thumbnail_s http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=1860-5974&client=summon