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Journal articles on the topic 'Monoidal structure'

Author: Grafiati

Published: 7 June 2025

Last updated: 2 August 2025

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1

Ma, Zizhu. "Generalized Enrichments of Categories for Operads." Algebra Colloquium 14, no. 01 (2007): 61–78. http://dx.doi.org/10.1142/s1005386707000077.

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Most enriched categories also have an ordinary category structure which is compatible with the enrichment on them. In this paper, enrichments in a monoidal category are generalized to arbitrary categories. These specialize to the classical enrichments when sets are regraded as discrete categories. We also generalize the definitions of PROs and PROPs as some generalized enrichments of categories. Then an operad in some monoidal category corresponds to a generalized PROP. Algebras of operads induce some special kind of monoidal functors. In the category of small categories, we construct several

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2

DOŠEN, KOSTA, and ZORAN PETRIĆ. "Coherence for monoidal endofunctors." Mathematical Structures in Computer Science 20, no. 4 (2010): 523–43. http://dx.doi.org/10.1017/s0960129510000022.

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The goal of this paper is to prove coherence results with respect to relational graphs for monoidal endofunctors, that is, endofunctors of a monoidal category that preserve the monoidal structure up to a natural transformation that need not be an isomorphism. These results are proved first in the absence of symmetry in the monoidal structure, and then with this symmetry. In the later parts of the paper, the coherence results are extended to monoidal endofunctors in monoidal categories that have diagonal or codiagonal natural transformations, or where the monoidal structure is given by finite p

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3

Groth, Moritz, Kate Ponto, and Michael Shulman. "The additivity of traces in monoidal derivators." Journal of K-theory 14, no. 3 (2014): 422–94. http://dx.doi.org/10.1017/is014005011jkt262.

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AbstractMotivated by traces of matrices and Euler characteristics of topological spaces, we expect abstract traces in a symmetric monoidal category to be “additive”. When the category is “stable” in some sense, additivity along cofiber sequences is a question about the interaction of stability and the monoidal structure.May proved such an additivity theorem when the stable structure is a triangulation, based on new axioms for monoidal triangulated categories. in this paper we use stable derivators instead, which are a different model for “stable homotopy theories”. We define and study monoidal

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4

White, David, and Donald Yau. "Arrow categories of monoidal model categories." MATHEMATICA SCANDINAVICA 125, no. 2 (2019): 185–98. http://dx.doi.org/10.7146/math.scand.a-114968.

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We prove that the arrow category of a monoidal model category, equipped with the pushout product monoidal structure and the projective model structure, is a monoidal model category. This answers a question posed by Mark Hovey, in the course of his work on Smith ideals. As a corollary, we prove that the projective model structure in cubical homotopy theory is a monoidal model structure. As illustrations we include numerous examples of non-cofibrantly generated monoidal model categories, including chain complexes, small categories, pro-categories, and topological spaces.

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DOŠEN, KOSTA, and ZORAN PETRIĆ. "Coherence for monoidal monads and comonads." Mathematical Structures in Computer Science 20, no. 4 (2010): 545–61. http://dx.doi.org/10.1017/s0960129510000034.

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The goal of this paper is to prove coherence results with respect to relational graphs for monoidal monads and comonads, that is, monads and comonads in a monoidal category such that the endofunctor of the monad or comonad is a monoidal functor (this means that it preserves the monoidal structure up to a natural transformation that need not be an isomorphism). These results are proved first in the absence of symmetry in the monoidal structure, and then with this symmetry. The monoidal structure is also allowed to be given with finite products or finite coproducts. Monoidal comonads with finite

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6

Logar, Alessandro, and Fabio Rossi. "Monoidal closed structures on categories with constant maps." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 38, no. 2 (1985): 175–85. http://dx.doi.org/10.1017/s144678870002303x.

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AbstractThe purpose of this paper is to study the so-called canonical monoidal closed structures on concrete categories with constant maps. First of all we give an example of a category of this kind where there exists a non canonical monoidal closed structure. Later, we give a technique to construct a class of suitable full subcategories of the category of T0-spaces, such that all monoidal closed structures on them are canonical. Finally we show that “almost all” useful categories of topological compact spaces admit no monoidal closed structures whatsoever.

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7

Femic, Bojana. "Enrichment and internalization in tricategories, the case of tensor categories and alternative notion to intercategories." Filomat 38, no. 8 (2024): 2601–60. https://doi.org/10.2298/fil2408601f.

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This paper emerged as a result of tackling the following three issues. Firstly, we would like the well known embedding of bicategories into pseudo double categories to be monoidal, which it is not if one uses the usual notion of a monoidal pseudo double category. Secondly, in [3] the question was raised: which would be an alternative notion to intercategories of Grandis and Par?, so that monoids in B?hm?s monoidal category (Dbl,?) of strict double categories and strict double functors with a Gray type monoidal product be an example of it? We obtain and prove that precisely the monoidal structu

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8

BARNES, DAVID. "A monoidal algebraic model for rational SO(2)-spectra." Mathematical Proceedings of the Cambridge Philosophical Society 161, no. 1 (2016): 167–92. http://dx.doi.org/10.1017/s0305004116000219.

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AbstractThe category of rational SO(2)–equivariant spectra admits an algebraic model. That is, there is an abelian category ${\mathcal A}$(SO(2)) whose derived category is equivalent to the homotopy category of rational SO(2)–equivariant spectra. An important question is: does this algebraic model capture the smash product of spectra?The category ${\mathcal A}$(SO(2)) is known as Greenlees' standard model, it is an abelian category that has no projective objects and is constructed from modules over a non–Noetherian ring. As a consequence, the standard techniques for constructing a monoidal mod

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9

HASEGAWA, MASAHITO. "On traced monoidal closed categories." Mathematical Structures in Computer Science 19, no. 2 (2009): 217–44. http://dx.doi.org/10.1017/s0960129508007184.

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The structure theorem of Joyal, Street and Verity says that every traced monoidal category arises as a monoidal full subcategory of the tortile monoidal category Int. In this paper we focus on a simple observation that a traced monoidal category is closed if and only if the canonical inclusion from into Int has a right adjoint. Thus, every traced monoidal closed category arises as a monoidal co-reflexive full subcategory of a tortile monoidal category. From this, we derive a series of facts for traced models of linear logic, and some for models of fixed-point computation. To make the paper mor

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10

Joyal, André, Ross Street, and Dominic Verity. "Traced monoidal categories." Mathematical Proceedings of the Cambridge Philosophical Society 119, no. 3 (1996): 447–68. http://dx.doi.org/10.1017/s0305004100074338.

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11

Shen, Bingliang, and Xiaoguang Zou. "The Braided Monoidal Structure on the Category of Comodules of Bimonads." Algebra Colloquium 26, no. 04 (2019): 565–78. http://dx.doi.org/10.1142/s1005386719000427.

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We investigate how the category of comodules of bimonads can be made into a monoidal category. It suffices that the monad and comonad in question are bimonads, with some extra compatibility relation. On a monoidal category of comodules of bimonads, we construct a braiding and get the necessary and sufficient conditions making it a braided monoidal category. As an application, we consider the category of comodules of corings and the category of entwined modules.

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12

Dong, Lihong, and Shengxiang Wang. "Hom-ideal structure of monoidal Hom-algebras in a category." Journal of Algebra and Its Applications 14, no. 05 (2015): 1550072. http://dx.doi.org/10.1142/s0219498815500723.

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In this paper, firstly we study the central invariant ZH(L)0 of generalized Hom-Lie algebras L, which are monoidal Hom-Lie algebras in the category [Formula: see text] of H-modules for a triangular Hopf algebra (H, R). If V is an H-Hom-Lie ideal of [L, L], under certain conditions we obtain V0 ⊆ ZH(L)0, where V0 is the monoidal Hom-subalgebra of H-invariant of V. Secondly, we describe the H-Hom-Lie ideal structure of monoidal Hom-algebras in a module category.

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BUCKLEY, MITCHELL, RICHARD GARNER, STEPHEN LACK, and ROSS STREET. "The Catalan simplicial set." Mathematical Proceedings of the Cambridge Philosophical Society 158, no. 2 (2014): 211–22. http://dx.doi.org/10.1017/s0305004114000498.

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AbstractThe Catalan numbers are well known to be the answer to many different counting problems, and so there are many different families of sets whose cardinalities are the Catalan numbers. We show how such a family can be given the structure of a simplicial set. We show how the low-dimensional parts of this simplicial set classify, in a precise sense, the structures of monoid and of monoidal category. This involves aspects of combinatorics, algebraic topology, quantum groups, logic, and category theory.

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14

DONG, LIHONG, RUIFANG HUANG, and SHENGXIANG WANG. "GENERALIZED HOM-LIE STRUCTURE OF MONOIDAL HOM-ALGEBRAS IN A CATEGORY." Journal of Algebra and Its Applications 13, no. 05 (2014): 1350149. http://dx.doi.org/10.1142/s0219498813501491.

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In this paper, we study the structure of monoidal Hom-Lie algebras in the category Hℳ of H-modules for a triangular Hopf algebra (H, R) and in particular the H-Lie structure of a monoidal Hom-algebra in Hℳ by analogy with that of generalized Lie algebras.

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15

Laugwitz, Robert. "The relative monoidal center and tensor products of monoidal categories." Communications in Contemporary Mathematics 22, no. 08 (2019): 1950068. http://dx.doi.org/10.1142/s0219199719500688.

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This paper develops a theory of monoidal categories relative to a braided monoidal category, called augmented monoidal categories. For such categories, balanced bimodules are defined using the formalism of balanced functors. It is shown that there exists a monoidal structure on the relative tensor product of two augmented monoidal categories which is Morita dual to a relative version of the monoidal center. In examples, a category of locally finite weight modules over a quantized enveloping algebra is equivalent to the relative monoidal center of modules over its Borel part. A similar result h

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16

Kelly, G. M., and F. Rossi. "Topological categories with many symmetric monoidal closed structures." Bulletin of the Australian Mathematical Society 31, no. 1 (1985): 41–59. http://dx.doi.org/10.1017/s0004972700002264.

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It would seem from results of Foltz, Lair, and Kelly that symmetric monoidal closed structures, and even monoidal biclosed ones, are quite rare on one-sorted algebraic or essentially-algebraic categories. They showed many such categories to admit no such structures at all, and others to admit only one or two; no such category is known to admit an infinite set of such structures.Among concrete categories, topological ones are in some sense at the other extreme from essentially-algebraic ones; and one is led to ask whether a topological category may admit many such structures. On the category of

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17

BARTHA, MIKLÓS. "The monoidal structure of Turing machines." Mathematical Structures in Computer Science 23, no. 02 (2013): 204–46. http://dx.doi.org/10.1017/s0960129512000096.

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18

Calvo, M., A. M. Cegarra, and B. A. Heredia. "Structure and classification of monoidal groupoids." Semigroup Forum 87, no. 1 (2013): 35–79. http://dx.doi.org/10.1007/s00233-013-9470-2.

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19

Greenough, Justin. "Monoidal 2-structure of bimodule categories." Journal of Algebra 324, no. 8 (2010): 1818–59. http://dx.doi.org/10.1016/j.jalgebra.2010.06.018.

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20

Stout, Lawrence Neff. "When does a category built on a lattice with a monoidal structure have a monoidal structure?" Fuzzy Sets and Systems 161, no. 9 (2010): 1162–74. http://dx.doi.org/10.1016/j.fss.2009.12.018.

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21

LODAY, JEAN-LOUIS, and TODOR POPOV. "PARASTATISTICS ALGEBRA, YOUNG TABLEAUX AND THE SUPER PLACTIC MONOID." International Journal of Geometric Methods in Modern Physics 05, no. 08 (2008): 1295–314. http://dx.doi.org/10.1142/s0219887808003351.

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The parastatistics algebra is a superalgebra with (even) parafermi and (odd) parabose creation and annihilation operators. The states in the parastatistics Fock-like space are shown to be in one-to-one correspondence with the Super Semistandard Young Tableaux (SSYT) subject to further constraints. The deformation of the parastatistics algebra gives rise to a monoidal structure on the SSYT which is a super-counterpart of the plactic monoid.

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22

IONESCU, LUCIAN M. "A NOTE ON DUALITY, FROBENIUS ALGEBRAS AND TQFTS." Journal of Knot Theory and Its Ramifications 13, no. 08 (2004): 999–1006. http://dx.doi.org/10.1142/s0218216504003676.

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Topological quantum field theories (TQFTs) represent the structure present in cobordism categories. As an example, we review the correspondence between Frobenius algebras and (1+1)TQFTs. It is a corollary of the self-duality of the cobordism category, which is a rigid monoidal category generated by a Frobenius object (the circle). A self-dual definition of a Frobenius object without the use of a prefered dual is considered. The issue of duality as part of the definition of a TQFT is addressed. Note that duality is preserved by monoidal functors. Hermitian structures are modeled as a conjugatio

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HASUO, ICHIRO, and BART JACOBS. "Traces for coalgebraic components." Mathematical Structures in Computer Science 21, no. 2 (2011): 267–320. http://dx.doi.org/10.1017/s0960129510000551.

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This paper contributes a feedback operator, in the form of a monoidal trace, to the theory of coalgebraic, state-based modelling of components. The feedback operator on components is shown to satisfy the trace axioms of Joyal, Street and Verity. We employ McCurdy's tube diagrams, which are an extension of standard string diagrams for monoidal categories, to represent and manipulate component diagrams. The microcosm principle then yields a canonical ‘inner’ traced monoidal structure on the category of resumptions (elements of final coalgebras/components). This generalises an observation by Abra

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BULACU, DANIEL, and STEFAAN CAENEPEEL. "A MONOIDAL STRUCTURE ON THE CATEGORY OF RELATIVE HOPF MODULES." Journal of Algebra and Its Applications 11, no. 02 (2012): 1250026. http://dx.doi.org/10.1142/s0219498811005506.

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Let B be a bialgebra, and A be a left B-comodule algebra in a braided monoidal category [Formula: see text], and assume that A is also a coalgebra, with a not-necessarily associative or unital left B-action. Then we can define a right A-action on the tensor product of two relative Hopf modules, and this defines a monoidal structure on the category of relative Hopf modules if and only if A is a bialgebra in the category of left Yetter–Drinfeld modules over B. Some examples are given.

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25

Hasegawa, Masahito, and Jean-Simon Pacaud Lemay. "Traced Monads and Hopf Monads." Compositionality 5 (October 30, 2023): 10. http://dx.doi.org/10.32408/compositionality-5-10.

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A traced monad is a monad on a traced symmetric monoidal category that lifts the traced symmetric monoidal structure to its Eilenberg-Moore category. A long-standing question has been to provide a characterization of traced monads without explicitly mentioning the Eilenberg-Moore category. On the other hand, a symmetric Hopf monad is a symmetric bimonad whose fusion operators are invertible. For compact closed categories, symmetric Hopf monads are precisely the kind of monads that lift the compact closed structure to their Eilenberg-Moore categories. Since compact closed categories and traced

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26

Holstein, Julian, and Andrey Lazarev. "Enriched Koszul duality for dg categories." Documenta Mathematica 30, no. 4 (2025): 755–86. https://doi.org/10.4171/dm/1002.

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It is well known that the category of small dg categories \mathsf{dgCat} , though it is monoidal, does not form a monoidal model category. In this paper we construct a monoidal model structure on the category of pointed curved coalgebras \mathsf{ptdCoa}^{*} over a field \mathbf{k} and show that the Quillen equivalence relating it to \mathsf{dgCat} is monoidal. We also show that \mathsf{dgCat} is a \mathsf{ptdCoa}^{*} -enriched model category. As a consequence, the homotopy category of \mathsf{dgCat} is closed monoidal and is equivalent as a closed monoidal category to the homotopy category of

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ESTRADA, SERGIO, JAMES GILLESPIE, and SINEM ODABAŞI. "Pure exact structures and the pure derived category of a scheme." Mathematical Proceedings of the Cambridge Philosophical Society 163, no. 2 (2016): 251–64. http://dx.doi.org/10.1017/s0305004116000980.

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AbstractLet$\mathcal{C}$be closed symmetric monoidal Grothendieck category. We define the pure derived category with respect to the monoidal structure via a relative injective model category structure on the categoryC($\mathcal{C}$) of unbounded chain complexes in$\mathcal{C}$. We use λ-Purity techniques to get this. As application we define the stalkwise pure derived category of the category of quasi–coherent sheaves on a quasi-separated scheme. We also give a different approach by using the category of flat quasi–coherent sheaves.

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Cisinski, Denis-Charles, and Gonçalo Tabuada. "Symmetric monoidal structure on non-commutative motives." Journal of K-Theory 9, no. 2 (2011): 201–68. http://dx.doi.org/10.1017/is011011005jkt169.

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AbstractIn this article we further the study of non-commutative motives, initiated in [12, 43]. Our main result is the construction of a symmetric monoidal structure on the localizing motivator Motlocdg of dg categories. As an application, we obtain : (1) a computation of the spectra of morphisms in Motlocdg in terms of non-connective algebraic K-theory; (2) a fully-faithful embedding of Kontsevich's category KMMk of non-commutative mixed motives into the base category Motlocdg(e) of the localizing motivator; (3) a simple construction of the Chern character maps from non-connective algebraic K

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Aquilino, Cosima, and Rebecca Reischuk. "The monoidal structure on strict polynomial functors." Journal of Algebra 485 (September 2017): 213–29. http://dx.doi.org/10.1016/j.jalgebra.2017.05.009.

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Chatterjee, Saikat, Amitabha Lahiri, and Ambar N. Sengupta. "A Morphism Double Category and Monoidal Structure." Algebra 2013 (March 25, 2013): 1–8. http://dx.doi.org/10.1155/2013/460582.

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We provide a recipe for “fattening” a category that leads to the construction of a double category. Motivated by an example where the underlying category has vector spaces as objects, we show how a monoidal category leads to a law of composition, satisfying certain coherence properties, on the object set of the fattened category.

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COECKE, BOB, DUSKO PAVLOVIC, and JAMIE VICARY. "A new description of orthogonal bases." Mathematical Structures in Computer Science 23, no. 3 (2012): 555–67. http://dx.doi.org/10.1017/s0960129512000047.

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We show that an orthogonal basis for a finite-dimensional Hilbert space can be equivalently characterised as a commutative †-Frobenius monoid in the category FdHilb, which has finite-dimensional Hilbert spaces as objects and continuous linear maps as morphisms, and tensor product for the monoidal structure. The basis is normalised exactly when the corresponding commutative †-Frobenius monoid is special. Hence, both orthogonal and orthonormal bases are characterised without mentioning vectors, but just in terms of the categorical structure: composition of operations, tensor product and the †-fu

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32

Böhm, Gabriella. "The Gray Monoidal Product of Double Categories." Applied Categorical Structures 28, no. 3 (2019): 477–515. http://dx.doi.org/10.1007/s10485-019-09587-5.

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AbstractThe category of double categories and double functors is equipped with a symmetric closed monoidal structure. For any double category $${\mathbb {A}}$$A, the corresponding internal hom functor "Equation missing" sends a double category $${\mathbb {B}}$$B to the double category whose 0-cells are the double functors $${\mathbb {A}} \rightarrow {\mathbb {B}}$$A→B, whose horizontal and vertical 1-cells are the horizontal and vertical pseudo transformations, respectively, and whose 2-cells are the modifications. Some well-known functors of practical significance are checked to be compatible

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33

Stammeier, Nicolai. "The nature of generalized scales." International Journal of Algebra and Computation 29, no. 06 (2019): 1035–62. http://dx.doi.org/10.1142/s0218196719500401.

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The notion of a generalized scale emerged in recent joint work with Afsar–Brownlowe–Larsen on equilibrium states on [Formula: see text]-algebras of right Least Common Multiple (LCM) monoids, where it features as the key datum for the dynamics under investigation. This work provides the structure theory for such monoidal homomorphisms. We establish the uniqueness of the generalized scale and characterize its existence in terms of a simplicial graph arising from a new notion of irreducibility inside right LCM monoids. In addition, the method yields an explicit construction of the generalized sca

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GUIRAUD, YVES, and PHILIPPE MALBOS. "Coherence in monoidal track categories." Mathematical Structures in Computer Science 22, no. 6 (2012): 931–69. http://dx.doi.org/10.1017/s096012951100065x.

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We introduce homotopical methods based on rewriting on higher-dimensional categories to prove coherence results in categories with an algebraic structure. We express the coherence problem for (symmetric) monoidal categories as an asphericity problem for a track category and use rewriting methods on polygraphs to solve it. The setting is extended to more general coherence problems, viewed as 3-dimensional word problems in a track category, including the case of braided monoidal categories.

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Srivastava, Arun K., and S. P. Tiwari. "On Categories of Fuzzy Petri Nets." Advances in Fuzzy Systems 2011 (2011): 1–5. http://dx.doi.org/10.1155/2011/812040.

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We introduce the concepts of fuzzy Petri nets and marked fuzzy Petri nets along with their appropriate morphisms, which leads to two categories of such Petri nets. Some aspects of the internal structures of these categories are then explored, for example, their reflectiveness/coreflectiveness and symmetrical monoidal closed structure.

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36

Carqueville, Nils, and Ingo Runkel. "On the monoidal structure of matrix bi-factorizations." Journal of Physics A: Mathematical and Theoretical 43, no. 27 (2010): 275401. http://dx.doi.org/10.1088/1751-8113/43/27/275401.

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IM, Geun Bin, and G. M. Kelly. "A universal property of the convolution monoidal structure." Journal of Pure and Applied Algebra 43, no. 1 (1986): 75–88. http://dx.doi.org/10.1016/0022-4049(86)90005-8.

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38

Carlier, Louis, and Joachim Kock. "Antipodes of monoidal decomposition spaces." Communications in Contemporary Mathematics 22, no. 02 (2018): 1850081. http://dx.doi.org/10.1142/s0219199718500815.

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We introduce a notion of antipode for monoidal (complete) decomposition spaces, inducing a notion of weak antipode for their incidence bialgebras. In the connected case, this recovers the usual notion of antipode in Hopf algebras. In the non-connected case, it expresses an inversion principle of more limited scope, but still sufficient to compute the Möbius function as [Formula: see text], just as in Hopf algebras. At the level of decomposition spaces, the weak antipode takes the form of a formal difference of linear endofunctors [Formula: see text], and it is a refinement of the general Möbiu

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39

Blute, Richard F. "Hopf algebras and linear logic." Mathematical Structures in Computer Science 6, no. 2 (1996): 189–212. http://dx.doi.org/10.1017/s0960129500000943.

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It has recently become evident that categories of representations of Hopf algebras provide fundamental examples of monoidal categories. In this expository paper, we examine such categories as models of (multiplicative) linear logic. By varying the Hopf algebra, it is possible to model several variants of linear logic. We present models of the original commutative logic, the noncommutative logic of Lambek and Abrusci, the braided variant due to the author, and the cyclic logic of Yetter. Hopf algebras provide a unifying framework for the analysis of these variants. While these categories are mo

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40

Bahiraei, P., and J. Nazaripour. "⊗-Pure model structure on the category of N -complexes." Extracta Mathematicae 39, no. 1 (2024): 119–34. http://dx.doi.org/10.17398/2605-5686.39.1.119.

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Let G be a closed symmetric monoidal concrete Grothendieck category. In this paper, we introduce a model structure on (CN (G), P⊗dw ) the exact category of N -complexes with the degree-wise ⊗-pure exact structure. Our result is based on the Gillespie’s Theorem by introducing two compatible cotorsion pairs on this category.

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Riba, Colin. "Monoidal-closed categories of tree automata." Mathematical Structures in Computer Science 30, no. 1 (2020): 62–117. http://dx.doi.org/10.1017/s0960129519000173.

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AbstractThis paper surveys a new perspective on tree automata and Monadic second-order logic (MSO) on infinite trees. We show that the operations on tree automata used in the translations of MSO-formulae to automata underlying Rabin’s Tree Theorem (the decidability of MSO) correspond to the connectives of Intuitionistic Multiplicative Exponential Linear Logic (IMELL). Namely, we equip a variant of usual alternating tree automata (that we call uniform tree automata) with a fibered monoidal-closed structure which in particular handles a linear complementation of alternating automata. Moreover, t

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Gogioso, Stefano, Dan Marsden, and Bob Coecke. "Symmetric Monoidal Structure with Local Character is a Property." Electronic Proceedings in Theoretical Computer Science 287 (January 31, 2019): 179–90. http://dx.doi.org/10.4204/eptcs.287.10.

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43

Gunnlaugsdóttir, Elísabet. "Monoidal structure of the category of u+q-modules." Linear Algebra and its Applications 365 (May 2003): 183–99. http://dx.doi.org/10.1016/s0024-3795(02)00484-6.

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COCKETT, J. R. B., and J. S. LEMAY. "Integral categories and calculus categories." Mathematical Structures in Computer Science 29, no. 2 (2018): 243–308. http://dx.doi.org/10.1017/s0960129518000014.

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Differential categories are now an established abstract setting for differentiation. However, not much attention has been given to the process which is inverse to differentiation: integration. This paper presents the parallel development for integration by axiomatizing an integral transformation, sA: !A → !A ⊗ A, in a symmetric monoidal category with a coalgebra modality. When integration is combined with differentiation, the two fundamental theorems of calculus are expected to hold (in a suitable sense): a differential category with integration which satisfies these two theorems is called a c

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Bonchi, Filippo, Fabio Gadducci, Aleks Kissinger, Pawel Sobocinski, and Fabio Zanasi. "String Diagram Rewrite Theory I: Rewriting with Frobenius Structure." Journal of the ACM 69, no. 2 (2022): 1–58. http://dx.doi.org/10.1145/3502719.

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String diagrams are a powerful and intuitive graphical syntax, originating in theoretical physics and later formalised in the context of symmetric monoidal categories. In recent years, they have found application in the modelling of various computational structures, in fields as diverse as Computer Science, Physics, Control Theory, Linguistics, and Biology. In several of these proposals, transformations of systems are modelled as rewrite rules of diagrams. These developments require a mathematical foundation for string diagram rewriting: whereas rewrite theory for terms is well-understood, the

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Elmendorf, A. D. "Function spectra." Mathematical Proceedings of the Cambridge Philosophical Society 108, no. 1 (1990): 31–34. http://dx.doi.org/10.1017/s0305004100068924.

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Boardman's stable category (see [5]) is a closed category ([4], VII·7), and in the best of all possible worlds, the category of spectra underlying the stable category would be closed as well; this would make life considerably easier for those doing calculations in stable homotopy theory. Unfortunately none of the categories of spectra introduced to date are closed; only S, the category introduced in [2], is even symmetric monoidal. The problem with making S closed is that it comes equipped with an augmentation to I, the category of universes and linear isometries (called Un in [2]), which pres

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Bannwart, Julie. "When equivariant homotopy theory meets combinatorics." Pittsburgh Interdisciplinary Mathematics Review 3 (July 10, 2025): 1–27. https://doi.org/10.5195/pimr.2025.56.

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In many different ways, mathematics often amounts to finding and studying suitable \emph{algebraic structures} on various collections of objects. In any first class in algebra, one gets to know the bestiary of monoids, (abelian) groups, rings, modules, algebras, etc. Algebraic structures are omnipresent and can also be viewed in a broader sense. For instance, certain categories can be endowed with a commutative multiplicative structure, making them into a \emph{symmetric monoidal category}. The latter is a triple \((C,\otimes,\mathbb{1}_C)\), where \(C\) is a category, \(\otimes\) is a functor

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Diaconescu, Denisa, and George Georgescu. "On the Forcing Semantics for Monoidal t-norm Based Logic." JUCS - Journal of Universal Computer Science 13, no. (11) (2007): 1550–72. https://doi.org/10.3217/jucs-013-11-1550.

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MTL-algebras are algebraic structures for the Esteva-Godo monoidal t-norm based logic (MTL), a many-valued propositional calculus that formalizes the structure of the real interval [0, 1], induced by a left-continuous t-norm. Given a complete MTL-algebra Χ, we define the weak forcing value |φ|χ and the forcing value [φ]χ, for any formula φ of MTL in Χ. We establish some arithmetical properties of|.|χ and [.]χ, and prove the equality [φ]χ=||φ||χ, where ||φ||χ is the truth value of φ in Χ.

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Xu, Fei. "Tensor structure on kC-mod and cohomology." Proceedings of the Edinburgh Mathematical Society 56, no. 1 (2012): 349–70. http://dx.doi.org/10.1017/s0013091512000107.

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AbstractLet $\mathcal{C}$ be a finite category and let k be a field. We consider the category algebra $k\mathcal{C}$ and show that $k\mathcal{C}$-mod is closed symmetric monoidal. Through comparing $k\mathcal{C}$ with a co-commutative bialgebra, we exhibit the similarities and differences between them in terms of homological properties. In particular, we give a module-theoretic approach to the multiplicative structure of the cohomology rings of small categories. As an application, we prove that the Hochschild cohomology rings of a certain type of finite category algebras are finitely generated

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Volkov, Y., and S. Witherspoon. "Graded Lie structure on cohomology of some exact monoidal categories." Homology, Homotopy and Applications 26, no. 2 (2024): 79–98. http://dx.doi.org/10.4310/hha.2024.v26.n2.a4.

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