{"title":"Isotropy groups of free racks and quandles","authors":"Jason Parker","doi":"10.1142/S0219498822501638","DOIUrl":"https://doi.org/10.1142/S0219498822501638","url":null,"abstract":"In this article, we characterize the (covariant) isotropy groups of free, finitely generated racks and quandles. As a consequence, we show that the usual inner automorphisms of such racks and quandles are precisely those automorphisms that are \"coherently extendible\". We then use this result to compute the global isotropy groups of the categories of racks and quandles, i.e. the automorphism groups of the identity functors of these categories.","PeriodicalId":124673,"journal":{"name":"arXiv: Category Theory","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123633643","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Computing homological residue fields in algebra and topology","authors":"Paul Balmer, James C. Cameron","doi":"10.1090/proc/15412","DOIUrl":"https://doi.org/10.1090/proc/15412","url":null,"abstract":"We determine the homological residue fields, in the sense of tensor-triangular geometry, in a series of concrete examples ranging from topological stable homotopy theory to modular representation theory of finite groups.","PeriodicalId":124673,"journal":{"name":"arXiv: Category Theory","volume":"59 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132111745","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Double Categories of Open Dynamical Systems (Extended Abstract)","authors":"D. J. Myers","doi":"10.4204/EPTCS.333.11","DOIUrl":"https://doi.org/10.4204/EPTCS.333.11","url":null,"abstract":"A (closed) dynamical system is a notion of how things can be, together with a notion of how they may change given how they are. The idea and mathematics of closed dynamical systems has proven incredibly useful in those sciences that can isolate their object of study from its environment. But many changing situations in the world cannot be meaningfully isolated from their environment -- a cell will die if it is removed from everything beyond its walls. To study systems that interact with their environment, and to design such systems in a modular way, we need a robust theory of open dynamical systems. In this extended abstract, we put forward a general definition of open dynamical system. We define two general sorts of morphisms between these systems: covariant morphisms which include trajectories, steady states, and periodic orbits; and contravariant morphisms which allow for plugging variables of some systems into parameters of other systems. We define an indexed double category of open dynamical systems indexed by their interface and use a double Grothendieck construction to construct a double category of open dynamical systems. In our main theorem, we construct covariantly representable indexed double functors from the indexed double category of dynamical systems to an indexed double category of spans. This shows that all covariantly representable structures of dynamical systems -- including trajectories, steady states, and periodic orbits -- compose according to the laws of matrix arithmetic.","PeriodicalId":124673,"journal":{"name":"arXiv: Category Theory","volume":"251 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131798340","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Module Eilenberg-Watts calculus","authors":"J. Fuchs, Gregor Schaumann, C. Schweigert","doi":"10.1090/conm/771/15509","DOIUrl":"https://doi.org/10.1090/conm/771/15509","url":null,"abstract":"The categorical formulation of the Eilenberg-Watts calculus relates, for any pair of finite categories M and N, the finite categories Fun^{le}(N,M) and Fun^{re}(N,M) of linear left or right exact functors and the Deligne product bar N boxtimes M by adjoint equivalences. We establish a variant of this calculus for the case that the finite categories M and N are module categories over a finite tensor category. This provides in particular canonical and explicitly computable equivalences between categories of left or right exact module functors (or, more generally, balanced functors) and certain twisted centers of bimodule categories.","PeriodicalId":124673,"journal":{"name":"arXiv: Category Theory","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125570869","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Category of Chain Bundles","authors":"P. G. Romeo, Riya Jose","doi":"10.1007/978-981-33-4842-4_13","DOIUrl":"https://doi.org/10.1007/978-981-33-4842-4_13","url":null,"abstract":"","PeriodicalId":124673,"journal":{"name":"arXiv: Category Theory","volume":"41 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127191283","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
M. J. A. Paniagua, A. Facchini, M. Gran, G. Janelidze
{"title":"Unusual spectral categories","authors":"M. J. A. Paniagua, A. Facchini, M. Gran, G. Janelidze","doi":"10.4171/rsmup/53","DOIUrl":"https://doi.org/10.4171/rsmup/53","url":null,"abstract":"The paper is devoted to a kind of `very non-abelian' spectral categories. Under strong conditions on a category $mathcal{X}$, we prove, among other things, that, for a given faithful localization $mathcal{C}tomathcal{X}$, we have canonical equivalences $mathrm{Spec}(mathcal{C})simmathcal{X}sim(mathrm{Category,,of,,injective,,objects,,in},, mathcal{C})$, and that $mathcal{C}$ has natural injective envelopes.","PeriodicalId":124673,"journal":{"name":"arXiv: Category Theory","volume":"68 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121378739","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"2-Dimensional Categories","authors":"Niles Johnson, Donald Yau","doi":"10.1093/OSO/9780198871378.001.0001","DOIUrl":"https://doi.org/10.1093/OSO/9780198871378.001.0001","url":null,"abstract":"\u0000 2-Dimensional Categories provides an introduction to 2-categories and bicategories, assuming only the most elementary aspects of category theory. A review of basic category theory is followed by a systematic discussion of 2-/bicategories; pasting diagrams; lax functors; 2-/bilimits; the Duskin nerve; the 2-nerve; internal adjunctions; monads in bicategories; 2-monads; biequivalences; the Bicategorical Yoneda Lemma; and the Coherence Theorem for bicategories. Grothendieck fibrations and the Grothendieck construction are discussed next, followed by tricategories, monoidal bicategories, the Gray tensor product, and double categories. Completely detailed proofs of several fundamental but hard-to-find results are presented for the first time. With exercises and plenty of motivation and explanation, this book is useful for both beginners and experts.","PeriodicalId":124673,"journal":{"name":"arXiv: Category Theory","volume":"112 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131951584","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Aspects of Categorical Recursion Theory","authors":"Pieter J. W. Hofstra, P. Scott","doi":"10.1007/978-3-030-66545-6_7","DOIUrl":"https://doi.org/10.1007/978-3-030-66545-6_7","url":null,"abstract":"","PeriodicalId":124673,"journal":{"name":"arXiv: Category Theory","volume":"50 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127553338","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Homotopy theory of algebras of substitutes and their localisation","authors":"M. Batanin, David White","doi":"10.1090/tran/8600","DOIUrl":"https://doi.org/10.1090/tran/8600","url":null,"abstract":"We study the category of algebras of substitudes (also known to be equivalent to the regular patterns of Getzler) equipped with a (semi)model structure lifted from the model structure on the underlying presheaves. We are especially interested in the case when the model structure on presheaves is a Cisinski style localisation with respect to a proper Grothendieck fundamental localiser. For example, for $mathtt{W}=mathtt{W}_{infty}$ the minimal fundamental localiser, the local objects in such a localisation are locally constant presheaves, and local algebras of substitudes are exactly algebras whose underlying presheaves are locally constant. \u0000We investigate when this localisation has nice properties. We single out a class of such substitudes which we call left localisable and show that the substitudes for $n$-operads, symmetric, and braided operads are in this class. As an application we develop a homotopy theory of higher braided operads and prove a stabilisation theorem for their $mathtt{W}_k$-localisations. This theorem implies, in particular, a generalisation of the Baez-Dolan Stabilisation Hypothesis for higher categories.","PeriodicalId":124673,"journal":{"name":"arXiv: Category Theory","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121764306","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The Category of Modules on an n-Trivial Extension: the Basic Properties","authors":"Dirar Benkhadra, D. Bennis, J. R. G. Rozas","doi":"10.1142/s1005386720000504","DOIUrl":"https://doi.org/10.1142/s1005386720000504","url":null,"abstract":"In this paper we investigate a categorical aspect of $n$-trivial extension of a ring by a family of modules. Namely, we introduce the right (resp., left) $n$-trivial extension of a category by a family of endofunctors. Among other results, projective, injective and flat objects of this category are characterized. We end the paper with two applications. We characterize when an $n$-trivial extension ring is $k$-perfect and we establish a result on the selfinjective dimension of an $n$-trivial extension ring.","PeriodicalId":124673,"journal":{"name":"arXiv: Category Theory","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133992354","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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