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Journal articles on the topic 'Adjunctions'

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Published: 3 June 2025

Last updated: 31 July 2025

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1

Ellerman, David. "Where Do Adjunctions Come From? Chimera Morphisms and Adjoint Functors in Category Theory." Foundations 5, no. 1 (2025): 10. https://doi.org/10.3390/foundations5010010.

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Category theory has foundational importance because it provides conceptual lenses to characterize what is important and universal in mathematics—with adjunction seeming to be the primary lens. Our topic is a theory showing “where adjoints come from”. The theory is based on object-to-object “chimera morphisms”, “heteromorphisms”, or “hets” between the objects of different categories (e.g., the insertion of generators as a set-to-group map). After showing that heteromorphisms can be treated rigorously using the machinery of category theory (bifunctors), we show that all adjunctions between two c

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2

KENDZIORRA, ANDREAS, and STEFAN E. SCHMIDT. "THE APPLICATION OF A CHARACTERIZATION OF ADJUNCTIONS." Journal of Algebra and Its Applications 12, no. 02 (2012): 1250155. http://dx.doi.org/10.1142/s0219498812501551.

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3

David, Ellerman. "BRAIN Journal - Brain Functors: A mathematical model of intentional perception and action." BRAIN - Broad Research in Artificial Intelligence and Neuroscience 7, no. 1 (2016): 5–17. https://doi.org/10.5281/zenodo.1044260.

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ABSTRACT Category theory has foundational importance because it provides conceptual lenses to characterize what is important and universal in mathematics - with adjunctions being the primary lens. If adjunctions are so important in mathematics, then perhaps they will isolate concepts of some importance in the empirical sciences. But the applications of adjunctions have been hampered by an overly restrictive formulation that avoids heteromorphisms or hets. By reformulating an adjunction using hets, it is split into two parts, a left and a right semiadjunction. Semiadjunctions (essentially a for

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4

Cheng, Eugenia, Nick Gurski, and Emily Riehl. "Cyclic multicategories, multivariable adjunctions and mates." Journal of K-Theory 13, no. 2 (2014): 337–96. http://dx.doi.org/10.1017/is013012007jkt250.

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AbstractA multivariable adjunction is the generalisation of the notion of a 2-variable adjunction, the classical example being the hom/tensor/cotensor trio of functors, ton+ 1 functors ofnvariables. In the presence of multivariable adjunctions, natural transformations between certain composites built from multivariable functors have “dual” forms. We refer to corresponding natural transformations as multivariable or parametrised mates, generalising the mates correspondence for ordinary adjunctions, which enables one to pass between natural transformations involving left adjoints to those involv

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5

Jay, C. Barry. "Local adjunctions." Journal of Pure and Applied Algebra 53, no. 3 (1988): 227–38. http://dx.doi.org/10.1016/0022-4049(88)90124-7.

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6

Sebastian, Bino, A. Unnikrishnan, Kannan Balakrishnan, and P. B. Ramkumar. "Mathematical Morphology on Hypergraphs Using Vertex-Hyperedge Correspondence." ISRN Discrete Mathematics 2014 (March 13, 2014): 1–6. http://dx.doi.org/10.1155/2014/436419.

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The focus of this paper is to develop computationally efficient mathematical morphology operators on hypergraphs. To this aim we consider lattice structures on hypergraphs on which we build morphological operators. We develop a pair of dual adjunctions between the vertex set and the hyperedge set of a hypergraph H, by defining a vertex-hyperedge correspondence. This allows us to recover the classical notion of a dilation/erosion of a subset of vertices and to extend it to subhypergraphs of H. This paper also studies the concept of morphological adjunction on hypergraphs for which both the inpu

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7

Clare, Pierre, Tyrone Crisp, and Nigel Higson. "ADJOINT FUNCTORS BETWEEN CATEGORIES OF HILBERT -MODULES." Journal of the Institute of Mathematics of Jussieu 17, no. 2 (2016): 453–88. http://dx.doi.org/10.1017/s1474748016000074.

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Let$E$be a (right) Hilbert module over a$C^{\ast }$-algebra$A$. If$E$is equipped with a left action of a second$C^{\ast }$-algebra$B$, then tensor product with$E$gives rise to a functor from the category of Hilbert$B$-modules to the category of Hilbert$A$-modules. The purpose of this paper is to study adjunctions between functors of this sort. We shall introduce a new kind of adjunction relation, called a local adjunction, that is weaker than the standard concept from category theory. We shall give several examples, the most important of which is the functor of parabolic induction in the tempe

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8

Casacuberta, Carles, Oriol Raventós, and Andrew Tonks. "Comparing localizations across adjunctions." Transactions of the American Mathematical Society 374, no. 11 (2021): 7811–65. http://dx.doi.org/10.1090/tran/8382.

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9

Zangurashvili, D. "Factorization Systems and Adjunctions." gmj 6, no. 2 (1999): 191–200. http://dx.doi.org/10.1515/gmj.1999.191.

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10

Ardizzoni, Alessandro, and Claudia Menini. "Adjunctions and braided objects." Journal of Algebra and Its Applications 13, no. 06 (2014): 1450019. http://dx.doi.org/10.1142/s0219498814500194.

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In this paper, we investigate the categories of braided objects, algebras and bialgebras in a given monoidal category, some pairs of adjoint functors between them and their relations. In particular, we construct a braided primitive functor and its left adjoint, the braided tensor bialgebra functor, from the category of braided objects to the one of braided bialgebras. The latter is obtained by a specific elaborated construction introducing a braided tensor algebra functor as a left adjoint of the forgetful functor from the category of braided algebras to the one of braided objects. The behavio

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11

Torii, Takeshi. "Uniqueness of monoidal adjunctions." Homology, Homotopy and Applications 26, no. 2 (2024): 259–72. http://dx.doi.org/10.4310/hha.2024.v26.n2.a13.

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12

Došen, Kosta, and Zoran Petrić. "Self-adjunctions and matrices." Journal of Pure and Applied Algebra 184, no. 1 (2003): 7–39. http://dx.doi.org/10.1016/s0022-4049(03)00084-7.

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13

D., Parks Allen. "A Note Concerning Equipotent Digraph Homomorphism Sets." International Journal of Sciences Volume 8, no. 2019-05 (2019): 47–52. https://doi.org/10.5281/zenodo.3350701.

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Functor adjunctions are fundamental to category theory and have recently found applications in the empirical sciences. In this paper a functor adjunction on a special full subcategory of the category of digraphs is borrowed from mathematical biology and used to equate cardinalities of sets of homomorphisms between various types of digraphs and associated line digraphs. These equalities are especially useful for regular digraphs and are applied to obtain homomorphism set cardinality equalities for the classes of de Bruijn digraphs and Kautz digraphs. Such digraphs play important roles in bioinf

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14

Mislove, Michael W., and Frank J. Oles. "ADJUNCTIONS BETWEEN CATEGORIES OF DOMAINS." Fundamenta Informaticae 22, no. 1,2 (1995): 93–116. http://dx.doi.org/10.3233/fi-1995-22125.

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15

Došen, Kosta, and Zoran Petrić. "Symmetric Self-adjunctions and Matrices." Algebra Colloquium 19, spec01 (2012): 1051–82. http://dx.doi.org/10.1142/s1005386712000855.

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It is shown that the multiplicative monoids of Brauer's centralizer algebras generated out of the basis are isomorphic to monoids of endomorphisms in categories where an endofunctor is adjoint to itself, and where, moreover, a kind of symmetry involving the self-adjoint functor is satisfied. As in a previous paper, of which this is a companion, it is shown that such a symmetric self-adjunction is found in a category whose arrows are matrices, and the functor adjoint to itself is based on the Kronecker product of matrices.

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16

Angelsmark, Ola. "Constraints, Adjunctions and (Co)algebras." Electronic Notes in Theoretical Computer Science 33 (2000): 3–12. http://dx.doi.org/10.1016/s1571-0661(05)80341-x.

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17

Voutsadakis, George. "Remarks on classifications and adjunctions." Rendiconti del Circolo Matematico di Palermo 54, no. 1 (2005): 50–70. http://dx.doi.org/10.1007/bf02875743.

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18

Gurski, Nick, Niles Johnson, and Angélica M. Osorno. "Extending homotopy theories across adjunctions." Homology, Homotopy and Applications 19, no. 2 (2017): 89–110. http://dx.doi.org/10.4310/hha.2017.v19.n2.a6.

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19

TOWNSEND, CHRISTOPHER. "Principal bundles as Frobenius adjunctions with application to geometric morphisms." Mathematical Proceedings of the Cambridge Philosophical Society 159, no. 3 (2015): 433–44. http://dx.doi.org/10.1017/s0305004115000444.

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AbstractUsing a suitable notion of principalG-bundle, defined relative to an arbitrary cartesian category, it is shown that principal bundles can be characterised as adjunctions that stably satisfy Frobenius reciprocity. The result extends from internal groups to internal groupoids. Since geometric morphisms can be described as certain adjunctions that are stably Frobenius, as an application it is proved that all geometric morphisms, from a localic topos to a bounded topos, can be characterised as principal bundles.

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20

Yegorychev, I. E. "ADJUNCTIONS, MONADS AND EIDOSES OF CALCULATIONS." Научное мнение, no. 11 (2018): 11–20. http://dx.doi.org/10.25807/pbh.22224378.2018.11.11.20.

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21

Furber, Robert. "Categorical Equivalences from State-Effect Adjunctions." Electronic Proceedings in Theoretical Computer Science 287 (January 31, 2019): 107–26. http://dx.doi.org/10.4204/eptcs.287.6.

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22

Blumberg, Andrew J., and Emily Riehl. "Homotopical resolutions associated to deformable adjunctions." Algebraic & Geometric Topology 14, no. 5 (2014): 3021–48. http://dx.doi.org/10.2140/agt.2014.14.3021.

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23

Caramello, Olivia, Vincenzo Marra, and Luca Spada. "General affine adjunctions, Nullstellensätze, and dualities." Journal of Pure and Applied Algebra 225, no. 1 (2021): 106470. http://dx.doi.org/10.1016/j.jpaa.2020.106470.

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24

Martin, C. E., C. A. R. Hoare, and He Jifeng. "Pre-adjunctions in order enriched categories." Mathematical Structures in Computer Science 1, no. 2 (1991): 141–58. http://dx.doi.org/10.1017/s0960129500001262.

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Category theory offers a unified mathematical framework for the study of specifications and programs in a variety of styles, such as procedural, functional and concurrent. One way that these different languages may be treated uniformly is by generalising the definitions of some standard categorical concepts. In this paper we reproduce in the generalised theory analogues of some standard theorems on isomorphism, and outline their applications to programming languages.

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25

Shen, Lili, and Walter Tholen. "Topological categories, quantaloids and Isbell adjunctions." Topology and its Applications 200 (March 2016): 212–36. http://dx.doi.org/10.1016/j.topol.2015.12.020.

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26

Anno, Rina, and Timothy Logvinenko. "On adjunctions for Fourier–Mukai transforms." Advances in Mathematics 231, no. 3-4 (2012): 2069–115. http://dx.doi.org/10.1016/j.aim.2012.06.007.

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27

Zafiris, Elias. "Rosen's modelling relations via categorical adjunctions." International Journal of General Systems 41, no. 5 (2012): 439–74. http://dx.doi.org/10.1080/03081079.2012.689466.

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28

Gibbons, Jeremy, Fritz Henglein, Ralf Hinze, and Nicolas Wu. "Relational algebra by way of adjunctions." Proceedings of the ACM on Programming Languages 2, ICFP (2018): 1–28. http://dx.doi.org/10.1145/3236781.

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29

Greenlees, J. P. C., and B. Shipley. "The Cellularization Principle for Quillen adjunctions." Homology, Homotopy and Applications 15, no. 2 (2013): 173–84. http://dx.doi.org/10.4310/hha.2013.v15.n2.a11.

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30

Mašulović, Dragan. "Pre-adjunctions and the Ramsey property." European Journal of Combinatorics 70 (May 2018): 268–83. http://dx.doi.org/10.1016/j.ejc.2018.01.006.

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31

Levy, Paul Blain. "Monads and Adjunctions for Global Exceptions." Electronic Notes in Theoretical Computer Science 158 (May 2006): 261–87. http://dx.doi.org/10.1016/j.entcs.2006.04.014.

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32

Greenlees, J. P. C., and Brooke Shipley. "Fixed point adjunctions for equivariant module spectra." Algebraic & Geometric Topology 14, no. 3 (2014): 1779–99. http://dx.doi.org/10.2140/agt.2014.14.1779.

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33

Gaitsgory, Dennis. "Functors given by kernels, adjunctions and duality." Journal of Algebraic Geometry 25, no. 3 (2016): 461–548. http://dx.doi.org/10.1090/jag/654.

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34

Carqueville, Nils, and Daniel Murfet. "Adjunctions and defects in Landau–Ginzburg models." Advances in Mathematics 289 (February 2016): 480–566. http://dx.doi.org/10.1016/j.aim.2015.03.033.

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35

Genovese, Francesco. "Adjunctions of Quasi-Functors Between DG-Categories." Applied Categorical Structures 25, no. 4 (2016): 625–57. http://dx.doi.org/10.1007/s10485-016-9470-y.

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36

Hughes, Bruce. "Products and adjunctions of manifold stratified spaces." Topology and its Applications 124, no. 1 (2002): 47–67. http://dx.doi.org/10.1016/s0166-8641(01)00236-x.

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37

Bergman, George M. "On the scarcity of contravariant left adjunctions." Algebra Universalis 24, no. 1-2 (1987): 169–85. http://dx.doi.org/10.1007/bf01188394.

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38

Hu, Po, Igor Kriz, and Petr Somberg. "On some adjunctions in equivariant stable homotopy theory." Algebraic & Geometric Topology 18, no. 4 (2018): 2419–42. http://dx.doi.org/10.2140/agt.2018.18.2419.

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39

Rivas, Exequiel. "Relating Idioms, Arrows and Monads from Monoidal Adjunctions." Electronic Proceedings in Theoretical Computer Science 275 (July 10, 2018): 18–33. http://dx.doi.org/10.4204/eptcs.275.3.

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40

Gutiérrez, Javier J. "Transfer of algebras over operads along Quillen adjunctions." Journal of the London Mathematical Society 86, no. 2 (2012): 607–25. http://dx.doi.org/10.1112/jlms/jds007.

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41

Ellerman, David. "The Heteromorphic Approach to Adjunctions: Theory and History." Mathematics 12, no. 2 (2024): 311. http://dx.doi.org/10.3390/math12020311.

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Mallios and Zafiris emphasize that adjoint functors, or adjunctions, are not only “ubiquitous” in category theory but also characterize the naturality of their approach to physical geometry. Hence, in this paper, the history and theory of adjoint functors is investigated. Where do adjoint functors come from mathematically, and how did the concept develop historically?

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42

Araújo, Manuel. "Coherence for adjunctions in a 3-category via string diagrams." Compositionality 4 (August 30, 2022): 2. http://dx.doi.org/10.32408/compositionality-4-2.

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We construct a 3-categorical presentation Adj(3,1) and define a coherent adjunction in a strict 3-category C as a map Adj(3,1)→C. We use string diagrams to show that any adjunction in C can be extended to a coherent adjunction in an essentially unique way. The results and their proofs will apply in the context of Gray 3-categories after the string diagram calculus is shown to hold in that context in an upcoming paper.

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43

Jacobs, Bart. "Semantics of the second order lambda calculus." Mathematical Structures in Computer Science 1, no. 3 (1991): 327–60. http://dx.doi.org/10.1017/s0960129500001341.

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In the literature ther are two main notins of model for the second order λ-calculus: one by Bruce, Meyer and Mitchell (the BMM-model, for short) in set-theoretical formulation and one category-theoretical by Seely. Here we generalise Seely's notion, using semifunctors and semi-adjunctions from Hayashi, and introduce λ2-algebras, λη2-algebras, λ2-models and λη-models, similarly to the untyped λ-calculus. Non-extensional abstraction of both term and type variables is described by semi-adjunctions (essentially as in Martini's thesis). We show that also for second order sume, the β-(and commutatio

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44

NAUMANN, DAVID A. "A categorical model for higher order imperative programming." Mathematical Structures in Computer Science 8, no. 4 (1998): 351–99. http://dx.doi.org/10.1017/s0960129598002552.

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The order-enriched category of monotonic predicate transformers over posets is a model of the refinement calculus of higher order imperative programs and pre-post specifications. This category is shown to be equivalent to the category of spans over ideal relations, and ideal relations are shown to be spans over monotonic functions between posets. To do this we use a skew span construction because the standard categorical span constructions are inapplicable. Axioms are given for products and coproducts of underlying posets as well as the homset as a coexponent, using inequations (for various ki

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45

KUDRYAVTSEVA, GANNA. "A DUALIZING OBJECT APPROACH TO NONCOMMUTATIVE STONE DUALITY." Journal of the Australian Mathematical Society 95, no. 3 (2013): 383–403. http://dx.doi.org/10.1017/s1446788713000323.

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AbstractThe aim of the present paper is to extend the dualizing object approach to Stone duality to the noncommutative setting of skew Boolean algebras. This continues the study of noncommutative generalizations of different forms of Stone duality initiated in recent papers by Bauer and Cvetko-Vah, Lawson, Lawson and Lenz, Resende, and also the current author. In this paper we construct a series of dual adjunctions between the categories of left-handed skew Boolean algebras and Boolean spaces, the unital versions of which are induced by dualizing objects $\{ 0, 1, \ldots , n+ 1\} $, $n\geq 0$.

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46

Schwarzweller, Christoph. "Ring and Field Adjunctions, Algebraic Elements and Minimal Polynomials." Formalized Mathematics 28, no. 3 (2020): 251–61. http://dx.doi.org/10.2478/forma-2020-0022.

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Summary In [6], [7] we presented a formalization of Kronecker’s construction of a field extension of a field F in which a given polynomial p ∈ F [X]\F has a root [4], [5], [3]. As a consequence for every field F and every polynomial there exists a field extension E of F in which p splits into linear factors. It is well-known that one gets the smallest such field extension – the splitting field of p – by adjoining the roots of p to F. In this article we start the Mizar formalization [1], [2] towards splitting fields: we define ring and field adjunctions, algebraic elements and minimal polynomia

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47

Li, Lingqiang, and Qiu Jin. "On adjunctions between Lim, SL-Top, and SL-Lim." Fuzzy Sets and Systems 182, no. 1 (2011): 66–78. http://dx.doi.org/10.1016/j.fss.2010.10.002.

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48

Riehl, Emily, and Dominic Verity. "Homotopy coherent adjunctions and the formal theory of monads." Advances in Mathematics 286 (January 2016): 802–88. http://dx.doi.org/10.1016/j.aim.2015.09.011.

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49

Gómez-Torrecillas, J., and B. Mesablishvili. "Some exact sequences associated with adjunctions in bicategories. Applications." Transactions of the American Mathematical Society 371, no. 12 (2019): 8255–95. http://dx.doi.org/10.1090/tran/7625.

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50

Frankland, Martin. "Behavior of Quillen (co)homology with respect to adjunctions." Homology, Homotopy and Applications 17, no. 1 (2015): 67–109. http://dx.doi.org/10.4310/hha.2015.v17.n1.a3.

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