Adjoint functors ≈ Adjoint functors
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On the embedding of convex spaces in stratified L-convex spaces Open
Consider L being a continuous lattice, two functors from the category of convex spaces (denoted by CS) to the category of stratified L-convex spaces (denoted by SL-CS) are defined. The first functor enables us to prove that the category CS…
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Eilenberg-Watts calculus for finite categories and a bimodule Radford 𝑆⁴ theorem Open
We obtain Morita invariant versions of Eilenberg-Watts type theorems, relating Deligne products of finite linear categories to categories of left exact as well as of right exact functors. This makes it possible to switch between different …
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Lax monoidal adjunctions, two‐variable fibrations and the calculus of mates Open
We provide a calculus of mates for functors to the ‐category of ‐categories and extend Lurie's unstraightening equivalences to show that (op)lax natural transformations correspond to maps of (co)cartesian fibrations that do not necessarily…
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ADJOINT FUNCTORS BETWEEN CATEGORIES OF HILBERT -MODULES Open
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 $B$ -modul…
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On the computation of fusion over the affine Temperley–Lieb algebra Open
Fusion product originates in the algebraisation of the operator product\nexpansion in conformal field theory. Read and Saleur (2007) introduced an\nanalogue of fusion for modules over associative algebras, for example those\nappearing in t…
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Generalized Long–Moody functors Open
In this paper, we generalize the principle of the Long-Moody construction for\nrepresentations of braid groups to other groups, such as mapping class groups\nof surfaces. Namely, we introduce endofunctors over a functor category that\nenco…
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Cross-effects and the classification of Taylor towers Open
Let [math] be a homotopy functor with values in the category of spectra. We show that partially stabilized cross-effects of [math] have an action of a certain operad. For functors from based spaces to spectra, it is the Koszul dual of the …
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Induced log-concavity of equivariant matroid invariants Open
Inspired by the notion of equivariant log-concavity, we introduce the concept of induced log-concavity for a sequence of representations of a finite group. In this paper we prove the induced log-concavity of the equivariant Kazhdan–Lusztig…
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On distributivity in higher algebra I: the universal property of bispans Open
Structures where we have both a contravariant (pullback) and a covariant (pushforward) functoriality that satisfy base change can be encoded by functors out of ( $\infty$ -)categories of spans (or correspondences). In this paper, we study …
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Topological Noetherianity of polynomial functors Open
We prove that any finite-degree polynomial functor over an infinite field is topologically Noetherian. This theorem is motivated by the recent resolution, by Ananyan-Hochster, of Stillman’s conjecture; and a recent Noetherianity proof by D…
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A category theoretic approach to metaphor comprehension: Theory of indeterminate natural transformation Open
We propose the theory of indeterminate natural transformation (TINT) to investigate the dynamical creation of meaning as an association relationship between images, focusing on metaphor comprehension as an example. TINT models meaning crea…
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Equivalences from tilting theory and commutative algebra from the adjoint functor point of view Open
We give a category theoretic approach to several known equivalences from (classic) tilting theory and commutative algebra. Furthermore, we apply our main results to establish a duality theory for relative Cohen-Macaulay modules in the sens…
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$\mathbb{P}^n$-functors Open
We propose a new theory of (non-split) P^n-functors. These are F: A -> B for which the adjunction monad RF is a repeated extension of Id_A by powers of an autoequivalence H and three conditions are satisfied: the monad condition, the adjoi…
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Hedetniemi’s Conjecture and Adjoint Functors in Thin Categories Open
We survey results on Hedetniemi's conjecture which are connected to adjoint functors in the "thin" category of graphs, and expose the obstacles to extending these results.
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Structure and Semantics Open
There are many category-theoretic notions of algebraic theory, including Lawvere theories, monads, PROPs and operads. The first central notion of this thesis is a common generalisation of these, which we call a proto-theory. In order to de…
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A Hierarchical Category Model for Geometrical Product Specifications (GPS) Open
International standards for tolerancing (ISO GPS) have undergone considerable evolutionary changes to meet the demands of the modern information age. Their expanding quantity and complexity have proposed a great obstacle to their informati…
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Locally Nameless Sets Open
This paper provides a new mathematical foundation for the locally nameless representation of syntax with binders, one informed by nominal techniques. It gives an equational axiomatization of two key locally nameless operations, "variable o…
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Goodwillie calculus and Mackey functors Open
We show that the category of $n$-excisive functors from the $\infty$-category of spectra to a target stable $\infty$-category $\mathbf{E}$ is equivalent to the category of $\mathbf{E}$-valued Mackey functors on an indexing category built f…
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Proper Functors and their Rational Fixed Point Open
The rational fixed point of a set functor is well-known to capture the behaviour of finite coalgebras. In this paper we consider functors on algebraic categories. For them the rational fixed point may no longer be a subcoalgebra of the fin…
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About the equivalence between monads and monadic functors Open
Given a $\infty$-category X we exhibit the $\infty$-category of right adjoint functors with target X as a localization of the opposite of the $\infty$-category of monads on X. This localization restricts to an equivalence between the $\inf…
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Linear Representations and Frobenius Morphisms of Groupoids Open
Given a morphism of (small) groupoids with injective object map, we provide \nsu cient and necessary conditions under which the induction and co-induction functors \nbetween the categories of linear representations are naturally isomorphic…
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Dg analogues of the Zuckerman functors and the dual Zuckerman functors II Open
In the first part of this series of papers we constructed dg analogues of the Zuckerman functors over commutative rings and the dual Zuckerman functors over the field of complex numbers. In this paper we construct their derived functors in…
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Homotopy theory of Moore flows (II) Open
This paper proves that the q-model structures of Moore flows and of multipointed d-spaces are Quillen equivalent. The main step is the proof that the counit and unit maps of the Quillen adjunction are isomorphisms on the q-cofibrant object…
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Generalized stability for abstract homotopy theories Open
We show that a derivator is stable if and only if homotopy finite limits and homotopy finite colimits commute, if and only if homotopy finite limit functors have right adjoints, and if and only if homotopy finite colimit functors have left…
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Infinity‐operads and Day convolution in Goodwillie calculus Open
We prove two theorems about Goodwillie calculus and use those theorems to\ndescribe new models for Goodwillie derivatives of functors between pointed\ncompactly-generated infinity-categories. The first theorem say that the\nconstruction of…
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Planar diagrammatics of self-adjoint functors and recognizable tree series Open
A pair of biadjoint functors between two categories produces a collection of elements in the centers of these categories, one for each isotopy class of nested circles in the plane. If the centers are equipped with a trace map into the grou…
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On Diers theory of Spectrum I : Stable functors and right multi-adjoints Open
Diers developed a general theory of right multi-adjoint functors leading to a purely categorical, point-set construction of spectra. Situations of multiversal properties return sets of canonical solutions rather than a unique one. In the c…
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A classification of small homotopy functors from spectra to spectra Open
We show that every small homotopy functor from spectra to spectra is weakly equivalent to a filtered colimit of representable functors represented in cofibrant spectra. Moreover, we present this classification as a Quillen equivalence of t…
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Free incomplete Tambara functors are almost never flat Open
Free algebras are always free as modules over the base ring in classical algebra. In equivariant algebra, free incomplete Tambara functors play the role of free algebras and Mackey functors play the role of modules. Surprisingly, free inco…
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The monoidal structure on strict polynomial functors Open
The category of strict polynomial functors inherits an internal tensor product from the category of divided powers. To investigate this monoidal structure, we consider the category of representations of the symmetric group S d which admits…