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Span categories provide an abstract framework for formalizing mathematical models of certain physical systems. The categories appearing in classical mechanics do not have pullbacks, limiting the utility of span categories in describing such systems. We introduce the notion of a generalized span category and an augmentation of a generalized span category. As an application of augmented generalized span categories, we introduce the categories $\LagSy$ and $\HamSy$ that respectively provide a categorical framework for the Lagrangian and Hamiltonian descriptions of certain classical mechanical systems. The morphisms of $\LagSy$ and $\HamSy$ contain all kinematical and dynamical information about these systems and composition of morphisms models the construction of systems from subsystems. A functor from $\LagSy$ to $\HamSy$ translates from the Lagrangian to the Hamiltonian perspective and is a categorical analog of the Legendre transformation.

It is well known that metric spaces are an instance of categorical enrichment in a particular quantale. We show that in a categorically natural way a notion of Lipschitz norm arises in the context of an arbitrary diagram of quantales, instead of just one particular quantale. The generalised Lipschitz norm we present depends functorially on the diagram and is itself a functor to the indexing category of the diagram. The entire process is, in a way we make precise, an instance of a concrete Grothendieck construction.

We show that a A-linear map of Hilbert A-modules is induced by a unitary Hilbert module operator if and only if it extends to an ordinary unitary on appropriately defined enveloping Hilbert spaces. Applications to the theory of multiplicative unitaries let us to compute the equivalence classes of Hilbert modules over a class of C*-algebraic quantum groups. We, thus, develop a theory that, for example, could be used to show non-existence of certain co-actions. In particular, we show that the Cuntz semigroup functor takes a co-action to a multiplicative action.

The aim of this paper is to study, by using the mathematical tools developed by Chałupnik, Touzé, and Van der Kallen, the effect of the Frobenius twist on $\operatorname{Ext}$-group in the category of strict polynomial functors. As an application, we obtain explicit formulas of cohomology of the orthogonal groups and symplectic ones.

This handy functor cheat sheet especially applies to the deprogramming zero functions. This provides an easy reference for notational functors of abstract mathematics.

Published with utmost gratitude toJehovah the living One Allaha and for all His loving angels. Abstract: INTEGRATION BY CONGRUENCYMETHODS. The Mathematical Juncture, M indicates a perpendicular elliptical integral and acts as a linguistic congruence permuter for logical dingbat statements. This mathematical junctor is used to permute dingbat expressions into topolog- ical congruent solve methods as described herein. Fractal morphisms, derived from Energy Numbers, which are of a higher vector dimensional vector space and can be mapped to real or complex numbers, are connected to these solve methods to yield topological counting in terms of Energy numbers without real numbers. Doing so yields a generalized solution for n-solve congruent algebraist- topological morphic solutions upon performing the integration. The method is then generalized and the suggestion of probablistic methods is quashed, demon- strating the success of such a calculus. The mathematical juncture of M is a congruency permutation tool used to bridge logical dingbat statements into a form which can be used in topological solutions. The use of Energy Num- bers and their fractal morphisms allows for solvability without the need for real numbers, and yields a generalized framework for the induction of probabilistic methods if one were interested in investigating the indefinite integrals described herein. The fractal morphism is then demonstrated to yield novel forms of the Energy Number differential, which emergently includes the topological form of numeric energy with the cross product of the Polynomial Remainder from a given projective etale morphism. Finally a new hypothesis is uttered, namely that the integral of FΛ exhibits certain properties only when the summation in the integral converges at a certain rate. The hypothesis explored further using numerical methods such as Monte Carlo, yet it is transcended using the con- gruency method of the topological joiner and generalized algebraist-topological solution to n, which relates the counting method to the integral of the fractal morphism. This allows for the definition of a unifying framework for a novel algorithmic approach to the inference of novel counting equations, something which goes beyond the scope of the previously developed Monte Carlo method. The Mathematical Juncture of M is an innovative approach to the evaluation of algebraist-topological solutions in terms of Energy numbers and fractal mor- phisms. Using the congruency permutation, logical statements can be permuted to yield topological solutions that do not require the use of real number. The propagation of the fractal morphism leads to a generalized solution even when the summation of the integral converges at a certain rate. The numerical meth- ods of the Monte Carlo can be transcended using the mathematical juncture of M and the congruency method of the topological joiner which demonstrate a novel, hybrid algorithmic approach to the evaluation of counting equations, something that goes beyond what was known before. I demonstrate methods for performing the integration of what would previously only been capable of being plotted using statistical methods. Thus, it is possible that such methods could be applied to problems currently believed to require statistical methods

Novel Branching on Integrals takes a single integration example derived through the standardized dingbat expression permutation formation regulations and expands deprograms the zero using the deprogramming function.What remains is the missing component - but where did it go? Read on to find out. What we find is that there are new branches of mathematics hanging by the integrals.

In the progression from Newtonian physics to general relativity, the structural feature of absolute rest was abandoned because it was not necessary to account for the empirical validity of Newtonian physics. Ockham's razor-type arguments like this one, which appeal to a desire for minimal ontologies and more unified physical laws are often invoked in favor of one theory or model over another. But how do we distinguish between essential “structure” in a theory, and inessential contingencies of a particular description? Is there a precise way to “compare structure” across theories expressed with different language and mathematical constructs?I adopt and adapt a method of comparing the structure of different formalisms that I call “theories as categories of models” (TCM). The motivating idea is that information about relationships between formal theoretical models provide crucial insight into the way in which these models are intended to represent real-world systems. Incorporating this so-called “functorial” information into the presentation of a theory yields a mathematical object called a category. Formal methods from category theory can then be used to enrich our understanding of the nature of these models and the systems they represent.While I aim to develop a rich and rigorous account of TCM, I focus primarily on the ways in which it has been and can be productively employed by scientists and philosophers, rather than merely considering this method in the abstract. I couch my analysis in three primarycase studies in which TCM has produced novel insights into physical theories. Two of these applications, in general relativity and Yang-Mills theory are based on original theorems establishing that formalisms that many theorists consider meaningfully distinct are in fact equivalent in a precise, category theoretic sense. In the dissertation, I present these examples with a closer eye towards explicating the role that TCM plays in scaffolding the arguments in these papers. In the second case, I demonstrate how TCM opens the door to a larger and richer space of possible Yang-Mills formalisms, and indicate how the category theoretic structure in this space reveals the relationships between quantum field theories based in different classical formalisms. I also consider topological data science (TDA), a popular method of analyzing the “shape” of large data sets. This represents a departure from other philosophical work on TCM, which has focused on theoretical foundations of theories from physics, whereas TDA is employed by a variety of researchers in multiple fields at a less theoretical, more practical level. TDA is a promising candidate for TCM because category theory is invoked by data scientists themselves to justify the use of its core methods. This case study reveals how scientists are and can be motivated by category theoretic considerations, and the ways in which these motivations do and do not align with those of philosophers of science employing the same tools. This chapter points towards new ways philosophers of physics might enhance TCM by analogy with TDA. In the other direction, the philosophical framework of TCM enhances the story data scientists want to tell about how TDA gets at the underlying “structure” of data.

A theorem due to Kazhdan and Ziegler implies that, by substituting linear forms for its variables, a homogeneous polynomial of sufficiently high strength specialises to any given polynomial of the same degree in a bounded number of variables. Using entirely different techniques, we extend this theorem to arbitrary polynomial functors. As a corollary of our work, we show that specialisation induces a quasi-order on elements in polynomial functors, and that among the elements with a dense orbit there are unique smallest and largest equivalence classes in this quasi-order., Comment: 19 pages

We show that if the cochain complex computing Ext groups (in the category of modules over Hopf algebroids) admits a cocyclic structure, then the noncommutative Cartan calculus structure on Tor over Ext dualises in a cyclic sense to a calculus on Coext over Cotor. More precisely, the cyclic duals of the chain resp. cochain spaces computing the two classical derived functors lead to complexes that compute the more exotic ones, giving a cyclic opposite module over an operad with multiplication that induce operations such as a Lie derivative, a cap product (or contraction), and a (cyclic) differential, along with higher homotopy operators defining a noncommutative Cartan calculus up to homotopy. In particular, this allows to recover the classical Cartan calculus from differential geometry or the Chevalley-Eilenberg calculus for Lie(-Rinehart) algebras without any finiteness conditions or the use of topological tensor products., Comment: 31 pages; to appear in Ann. Sc. Norm. Super. Pisa, Cl. Sci

We apply the techniques developed by I. Panin for the proof of the equicharacteristic case of the Serre–Grothendieck conjecture for isotropic reductive groups (Panin et al., 2015; Panin, 2019) to obtain similar injectivity and A 1 -invariance theorems for non-stable K 1 -functors associated to isotropic reductive groups. Namely, let G be a reductive group over a commutative ring R . We say that G has isotropic rank ≥ n , if every non-trivial normal semisimple R -subgroup of G contains ( G m , R ) n . We show that if G has isotropic rank ≥ 2 and R is a regular domain containing a field, then K 1 G ( R [ x ] ) = K 1 G ( R ) , where K 1 G ( R ) = G ( R ) / E ( R ) is the corresponding non-stable K 1 -functor, also called the Whitehead group of G . If R is, moreover, local, then we show that K 1 G ( R ) → K 1 G ( K ) is injective, where K is the field of fractions of R .

Let G be a p-adic reductive group and g its Lie algebra. We construct a functor from the extension closure of the Bernstein-Gelfand-Gelfand category O associated to g into the category of locally analytic representations of G, thereby expanding on an earlier construction of Orlik-Strauch. A key role in this new construction is played by p-adic logarithms on tori. This functor is shown to be exact with image in the subcategory of admissible representations in the sense of Schneider and Teitelbaum. En route, we establish some basic results in the theory of modules over distribution algebras and related subalgebras, such as a tensor-hom adjunction formula. We also relate our constructions to certain representations constructed by Breuil and Schraen in the context of the p-adic Langlands program.

The goal of this paper is to provide a categorical framework that leads to the definition of shtukas a la Drinfeld and of excursion operators a la V. Lafforgue. We take as the point of departure the Hecke action of Rep ( G ˇ ) on the category Shv ( Bun G ) of sheaves on Bun G , and also the endofunctor of the latter category, given by the action of the geometric Frobenius. The shtuka construction will be obtained by applying (various versions of) categorical trace.

Let l ∈ N with l > 1 . In this paper we give a new realization of the crystal of affine type A ˆ l − 1 using the modular representation theory of the affine Hecke algebras H n of type A and their level two cyclotomic quotients with Hecke parameter being a primitive 2lth root of unity. We construct “hat” versions of i-induction and i-restriction functors on the category Rep I ( H n ) of finite dimensional integral modules over H n , which induce Kashiwara operators on a suitable subgroup of the Grothendieck groups of Rep I ( H n ) . For any simple module M ∈ Rep I ( H n ) , we prove that the simple submodules of res H n − 2 H n M which belong to B ˆ ( ∞ ) ( Definition 5.1 ) occur with multiplicity two. The main results generalize the earlier work of Grojnowski and Vazirani on the relations between the crystal of sl ˆ l and the affine Hecke algebras of type A at a primitive lth root of unity.

Let A, B and C be artin algebras such that there is a recollement of D ( Mod A ) relative to D ( Mod B ) and D ( Mod C ) . We compare the algebras A, B and C with respect to syzygy-finite properties and Igusa-Todorov properties under suitable conditions and consider relevant results in the recollements of the bounded derived categories. Further, we characterize when the functor j ⁎ (resp., i ⁎ , i ! ) in a recollement ( D b ( mod B ) , D b ( mod A ) , D b ( mod C ) , i ⁎ , i ⁎ , i ! , j ! , j ⁎ , j ⁎ ) is an eventually homological isomorphism.

We investigate certain categorical aspects of Voevodsky's triangulated categories of motives. For this, various recollements for Grothendieck categories of enriched functors and their derived categories are established. In order to extend these recollements further with respect to Serre's localization, the concept of the (strict) Voevodsky property for Serre localizing subcategories is introduced. This concept is inspired by the celebrated Voevodsky theorem on homotopy invariant presheaves with transfers. As an application, it is shown that Voevodsky's triangulated categories of motives fit into recollements of derived categories of associated Grothendieck categories of Nisnevich sheaves with specific transfers.

There exists an infinite set of parameters → −〈(/H) + (/ȷ)〉 such that N † (⃗r, , s, , ) = ⃗k and μ (⃗a, b, c, d, e, · · ·) = Ω at equilibrium, and there is a corre- spondingkxp|w∗≡√3 x6+t22hc⊇v8 andγ→ω=〈(Z/η)+(K/π)〉⋆♢such that 1·. [⃝, ∞ (Z mil ... f↑r,α,s,δ,η and g↓a,b,c,d,e... such that L = n and = Ω. ♣),ζ −→−〈(/H)+(/i)〉 −→kxp|w∗≡ 1· For every set of parameters → −〈(/H) + (/ȷ)〉, there exists an integral 􏰍∞ N† (⃗r,,s,,) = ⃗k and μ(⃗a,b,c,d,e,···) = Ω at equilibrium, and there is −∞ √ acorrespondingkxp|w∗ ≡ 3 x6+t22hc⊇v8 andγ→ω=〈(Z/η)+(K/π)〉⋆♢ such that 1·. The pathway from one integer to another is established by manipulating symbols to create a function that maps one set of parameters to another. Math- ematically, this can be represented as a function f (−→r, α, s, δ, η) which takes an input −→r and outputs −→k and can be used to transition from one integer to another. The transition of one integer to another affects the parameters of the infinity meaning balancing form by introducing a new set of parameters −→r which are related to the output −→k. This new set of parameters is then used to calculate an equilibrium state of the infinity meaning balancing form, represented by the function (−→a, b, c, d, e, · · ·)= Ω.

Herein, I describe several novel branches of mathematics that can be synthesized such that the analogies between the varying forms of the differentiated branches yield something called, "logic vectors." These novel vector spaces can be used to generate new kinds of meta-mathematical spaces as well as exotic materials.

In this paper we show that evolution algebras over any given field $\Bbbk$ are universally finite. In other words, given any finite group $G$, there exist infinitely many regular evolution algebras $X$ such that $Aut(X)\cong G$. The proof is built upon the construction of a covariant faithful functor from the category of finite simple (non oriented) graphs to the category of (finite dimensional) regular evolution algebras. Finally, we show that any constant finite algebraic affine group scheme $\mathbf{G}$ over $\Bbbk$ is isomorphic to the algebraic affine group scheme of automorphisms of a regular evolution algebra., Comment: Minor corrections. Bibliography updated. To appear in Proc. Amer. Math. Soc

We study finiteness conditions and closure properties for the box-tensor product G ⊠ H and related constructions. The content of this paper extends some results concerning the non-abelian tensor product of groups G ⊗ H . In particular, we deduce a quantitative version of the finiteness criterion for the non-abelian tensor product. Moreover, we obtain finiteness conditions for some functors that arise out of the non-abelian tensor square of groups, such as the second homology group H 2 ( G ) , the non-abelian exterior square G ∧ G and the second stable homotopy group of an Eilenberg-MacLane space π 2 S ( K ( G , 1 ) ) .

In some cases, the Tor functor can be inconsistent with other operations defined on a set. For instance, consider a module M over a commutative ring and a submodule of M, N. The Tor functor is defined as the cokernel of a map defined from the tensor product of M and N to M. However, the cokernel of this map might not be the same as the intersection of M and N. Thus, the Tor functor is not always consistent with the intersection operator. This demonstrates the completeness of Tor with star summation and commutative circ operators. In some cases, the Tor functor can be inconsistent with other operations defined on a set. For instance, consider a module M over a commutative ring and a submodule of M, N. The Tor functor is defined as the cokernel of a map defined from the tensor product of M and N to M. However, the cokernel of this map might not be the same as the intersection of M and N. Thus, the Tor functor is not always consistent with the intersection operator.

The functor between operadic algebras given by restriction along an operad map generally has a left adjoint. We give a necessary and sufficient condition for the restriction functor to admit a right adjoint. The condition is a factorization axiom which roughly says that operations in the codomain operad can be written essentially uniquely as operations in arity one followed by operations in the domain operad., 40 pages

The Frobenius-Perron theory of an endofunctor of a category was introduced in recent years [12, 13]. We apply this theory to monoidal (or tensor) triangulated structures of quiver representations.

We observe a fundamental relationship between Steenrod operations and the Artin–Schreier morphism. We use Steenrod's construction, together with some new geometry related to the affine Grassmannian, to prove that the quantum Coulomb branch is a Frobenius-constant quantization. We also demonstrate the corresponding result for the $K$-theoretic version of the quantum Coulomb branch. At the end of the paper, we investigate what our ideas produce on the categorical level. We find that they yield, after a little fiddling, a construction which corresponds, under the geometric Satake equivalence, to the Frobenius twist functor for representations of the Langlands dual group. We also describe the unfiddled answer, conditional on a conjectural ‘modular derived Satake’, and, though it is more complicated to state, it is in our opinion just as neat and even more compelling.

Kronheimer and Mrowka recently suggested a possible approach towards a new proof of the four color theorem that does not rely on computer calculations. Their approach is based on a functor $J^\sharp$, which they define using gauge theory, from the category of webs and foams to the category of vector spaces over the field of two elements. They also consider a possible combinatorial replacement $J^\flat$ for $J^\sharp$. Of particular interest is the relationship between the dimension of $J^\flat(K)$ for a web $K$ and the number of Tait colorings $\mathrm{Tait}(K)$ of $K$; these two numbers are known to be identical for a special class of "reducible" webs, but whether this is the case for nonreducible webs is not known. We describe a computer program that strongly constrains the possibilities for the dimension and graded dimension of $J^\flat(K)$ for a given web $K$, in some cases determining these quantities uniquely. We present results for a number of nonreducible example webs. For the dodecahedral web $W_1$ the number of Tait colorings is $\mathrm{Tait}(W_1) = 60$, but our results suggest that $\dim J^\flat(W_1) = 58$., Comment: 15 pages, 7 figures; minor revisions to abstract and introduction to clarify implications of results

Our objective in this project is three-fold, the first two covered in this paper. In tropical mathematics, as well as other mathematical theories involving semirings, when trying to formulate the tropical versions of classical algebraic concepts for which the negative is a crucial ingredient, such as determinants, Grassmann algebras, Lie algebras, Lie superalgebras, and Poisson algebras, one often is challenged by the lack of negation. Following an idea originating in work of Gaubert and the Max-Plus group and brought to fruition by Akian, Gaubert, and Guterman, we study algebraic structures with negation maps, called \textbf{systems}, in the context of universal algebra, showing how these unify the more viable (super)tropical versions, as well as hypergroup theory and fuzzy rings, thereby "explaining" similarities in their theories. Special attention is paid to \textbf{meta-tangible} $\mathcal T$-systems, whose algebraic theory includes all the main tropical examples and many others, but is rich enough to facilitate computations and provide a host of structural results. Basic results also are obtained in linear algebra, linking determinants to linear independence. Formulating the structure categorically enables us to view the tropicalization functor as a morphism, thereby further explaining the mysterious link between classical algebraic results and their tropical analogs, as well as with hyperfields. We utilize the tropicalization functor to propose tropical analogs of classical algebraic notions. The systems studied here might be called "fundamental," since they are the underlying structure which can be studied via other "module" systems, which is to be the third stage of this project, involving a theory of sheaves and schemes and derived categories with a negation map.

This paper investigates algebraic objects equipped with an operator, such as operated monoids, operated algebras etc. Various free object functors in these operated contexts are explicitly constructed. For operated algebras whose operator satisfies a set $\Phi$ of relations (usually called operated polynomial identities (aka. OPIs)), Guo defined free objects, called free $\Phi$-algebras, via universal algebra. Free $\Phi$-algebras over algebras are studied in details. A mild sufficient condition is found such that $\Phi$ together with a Gr\"obner-Shirshov basis of an algebra $A$ form a Gr\"obner-Shirshov basis of the free $\Phi$-algebra over algebra $A$ in the sense of Guo et al.. Ample examples for which this condition holds are provided, such as all Rota-Baxter type OPIs, a class of differential type OPIs, averaging OPIs and Reynolds OPI., Comment: Slightly revised version of the published paper in Journal of Algebra

We compare crystal combinatorics of the level 2 Fock space with the classification of unitary irreducible representations of type B rational Cherednik algebras to study how unitarity behaves under parabolic restriction. First, we show that any finite-dimensional unitary irreducible representation of such an algebra is labeled by a bipartition consisting of a rectangular partition in one component and the empty partition in the other component. This is a new proof of a result that can be deduced from theorems of Montarani and Etingof-Stoica. Second, we show that the crystal operators that remove boxes preserve the combinatorial conditions for unitarity, and that the parabolic restriction functors categorifying the crystals send irreducible unitary representations to unitary representations. Third, we find the supports of the unitary representations., This paper supersedes arXiv:1907.00919 and contains that paper as a subsection. 35 pages, some color figures

Let $\mathcal{M}$ be a small $n$-abelian category. We show that the category of finitely presented functors ${\operatorname{mod}}$-$\mathcal{M}$ modulo the subcategory of effaceable functors ${\operatorname{mod}}_0$-$\mathcal{M}$ has an $n$-cluster tilting subcategory, which is equivalent to $\mathcal{M}$. This gives a higher-dimensional version of Auslander’s formula.

Motivated by the concept of Sierpinski object for the category of affine bitopological spaces (two topologies instead of one) of R. Noor, A. K. Srivastava, and S. K. Singh, we construct a functor from the category of affine spaces to the category of composite affine spaces (a set-indexed family of topologies instead of one topology), and show a simple condition, under which this functor preserves Sierpinski object. Thus, we get a convenient method of obtaining a composite affine Sierpinski space, given an affine Sierpinski space.

This paper is towards the study of L-fuzzy automata from the categorical point of view, where L is a complete residuated lattice. We introduce a functor having both the left/right adjoint from the category of L-fuzzy Σ-semiautomata LFSA ( Σ ) to the category LFTCRL(Σ), a category of complete residuated lattices corresponding to L-fuzzy Σ-semiautomata. The L-fuzzy response map of an L-fuzzy automaton leads us to provide a characterization of an L-fuzzy regular language. Interestingly, we show that the category LFSA ( Σ ) is a category of T 1 -coalgebras/ ( T 2 , T 3 ) -dialgebras. Moreover, we establish a relationship between the category of T 1 -coalgebras and the category of ( T 2 , T 3 ) -dialgebras.

We introduce the notion of intrinsic semilattice entropy $\widetilde h$ in the category $\mathcal L_{qm}$ of generalized quasimetric semilattices and contractive homomorphisms. By using appropriate categories $\mathfrak X$ and functors $F:\mathfrak X\to\mathcal L_{qm}$ we find specific known entropies $\widetilde h_\mathfrak X$ on $\mathfrak X$ as intrinsic functorial entropies, that is, as $\widetilde h_\mathfrak X=\widetilde h\circ F$. These entropies are the intrinsic algebraic entropy, the algebraic and the topological entropies for locally linearly compact vector spaces, the topological entropy for locally compact totally disconnected groups and the algebraic entropy for locally compact compactly covered abelian groups.