Gluing derived equivalences together

Let I be a small indexing category, G: I op ! Cat be a functor and BG 2 Cat denote the Grothendieck construction on G. We define and study Quillen pairs between the category of diagrams of simplicial sets (resp. categories) indexed on BG and the category of I-diagrams over N(G) (resp. G). As an application we obtain a Quillen equivalence between the categories of presheaves of simplicial sets (resp. groupoids) on a stack M and presheaves of simplicial sets (resp. groupoids) over M. The motivation for this paper was the study of homotopy theory of (pre)sheaves on a stack. Since the site associated to a stack M is a Grothedieck construction this led us to an investigation of the homotopy theory of diagrams indexed on a category which is itself a Grothendieck construction (of a diagram of small categories). The body of the paper is concerned with analyzing various Quillen pairs between diagram categories. These adjunctions are of general interest and we present some examples not related to the theory of stacks. We conclude the paper with the applications to stacks. Stacks were introduced in algebraic geometry in order to parametrize families of objects when the presence of automorphisms prevented representability by a scheme or even a sheaf [A, DM, Gi]. Recently stacks have come to play an important role in algebraic topology. Complex oriented cohomology theories give rise to stacks over the moduli stack of formal groups and in certain situations, conversely, stacks over the moduli stack of formal groups give rise to spectra [G, R2, GHMR, B]. One fundamental example is the spectrum of topological modular forms [Hp] which is associated to the moduli stack of elliptic curves. Classically, stacks were defined as categories fibered in groupoids over a site C which satisfy descent [DM, Definition 4.1]. In [H] we show that a category fibered in groupoids F over C is a stack if and only if the assignment satisfies the homotopy sheaf

This chapter defines the Grothendieck construction for a lax functor into the category of small categories. It then proves that, for such a pseudofunctor, its Grothendieck construction is its lax colimit. Most of the rest of the chapter contains a detailed proof of the Grothendieck Construction Theorem, which states that the Grothendieck construction is part of a 2-equivalence. A generalization of the Grothendieck construction that applies to an indexed bicategory is also discussed.

We extend some classical results – such as Quillen’s Theorem A, the Grothendieck construction, Thomason’s theorem and the characterisation of homotopically cofinal functors – from the homotopy theory of small categories to polynomial monads and their algebras. As an application we give a categorical proof of the Dwyer–Hess and Turchin results concerning the explicit double delooping of spaces of long knots.

Enriched Lawvere theories are a generalization of Lawvere theories that allow\nus to describe the operational semantics of formal systems. For example, a\ngraph enriched Lawvere theory describes structures that have a graph of\noperations of each arity, where the vertices are operations and the edges are\nrewrites between operations. Enriched theories can be used to equip systems\nwith operational semantics, and maps between enriching categories can serve to\ntranslate between different forms of operational and denotational semantics.\nThe Grothendieck construction lets us study all models of all enriched theories\nin all contexts in a single category. We illustrate these ideas with the\nSKI-combinator calculus, a variable-free version of the lambda calculus.\n

Cartesian modules over representations of small categories

Consider the first-order theory of a category.d It has a sort of objects, and a sort of arrows (so we may think of it as a small category). We show that, assuming the principle of unique substitutions, the setoids inside a type theoretic universe provide a model for this first-order theory. We also show that the principle of unique substitutions is not derivable in type theory, but that it is strictly weaker than the principle of unique identity proofs.

Categorical Foundations for K-theory

We study the generalized right ample identity, introduced by the author in a previous paper. Let [Formula: see text] be a reduced [Formula: see text]-Fountain semigroup which satisfies the congruence condition. We can associate with [Formula: see text] a small category [Formula: see text] whose set of objects is identified with the set [Formula: see text] of idempotents and its morphisms correspond to elements of [Formula: see text]. We prove that [Formula: see text] satisfies the generalized right ample identity if and only if every element of [Formula: see text] induces a homomorphism of left [Formula: see text]-actions between certain classes of generalized Green’s relations. In this case, we interpret the associated category [Formula: see text] as a discrete form of a Peirce decomposition of the semigroup algebra. We also give some natural examples of semigroups satisfying this identity.

Let k be a commutative ring, $\mathcal {A}$ and $\mathcal {B}$ – two k-linear categories with an action of a group G. We introduce the notion of a standard G-equivalence from $\mathcal {K}_{p}^{\mathrm {b}}\mathcal {B}$ to $\mathcal {K}_{p}^{\mathrm {b}}\mathcal {A}$ , where $\mathcal {K}_{p}^{\mathrm {b}}\mathcal {A}$ is the homotopy category of finitely generated projective $\mathcal {A}$ -complexes. We construct a map from the set of standard G-equivalences to the set of standard equivalences from $\mathcal {K}_{p}^{\mathrm {b}}\mathcal {B}$ to $\mathcal {K}_{p}^{\mathrm {b}}\mathcal {A}$ and a map from the set of standard G-equivalences from $\mathcal {K}_{p}^{\mathrm {b}}\mathcal {B}$ to $\mathcal {K}_{p}^{\mathrm {b}}\mathcal {A}$ to the set of standard equivalences from $\mathcal {K}_{p}^{\mathrm {b}}(\mathcal {B}/G)$ to $\mathcal {K}_{p}^{\mathrm {b}}(\mathcal {A}/G)$ , where $\mathcal {A}/G$ denotes the orbit category. We investigate the properties of these maps and apply our results to the case where $\mathcal {A}=\mathcal {B}=R$ is a Frobenius k-algebra and G is the cyclic group generated by its Nakayama automorphism ν. We apply this technique to obtain the generating set of the derived Picard group of a Frobenius Nakayama algebra over an algebraically closed field.

Let C be a small category. In this paper, we mainly study the category of modules M od‐ R on ringed sites ( C , R ) . We firstly reprove the Theorem A of the paper [Wu, M. and Xu,F., Skew category algebras and modules on ringed finite sites. J. A. 631, 2023], then we characterize M od‐ R in terms of the torsion modules on Gr ( R ) , where Gr ( R ) is the linear Grothendieck construction of R . Finally, we investigate the hereditary torsion pairs, TTF triples and Abelian recollements of M od‐ R . When C is finite, the complete classifications of all these are given, respectively.

The global Cauchy problem for the non linear Klein-Gordon equation-II

Representability of monoids by endomorphisms of algebras from a given class C is often decided by a category-theoretical approach: a full embedding of a binding category B into C is constructed. Since every category of algebras is isomorphic to a full subcategory of any binding category and each small category occurs as a full subcategory of B, every monoid is then isomorphic to the endomorphism monoid of an algebra in C. Furthermore, by the Hedrlfn-KuSera Theorem [19], under the set-theoretical axiom (M), saying that there is only a set of strongly measurable cardinals, any concrete category can be fully embedded into any binding category B. There are numerous binding varieties of algebras. Thus, for instance, Hedrl in and Pultr [7] proved that a category of algebras of a given type zl is binding ff and only if the sum of Zl is bigger than 1. The list includes semigroups [8], commutative rings with unit [3, 4], lattices with (0, 1)-homomorphisms [6], and a locally finite lattice variety [2], to name a few. This leads to a problem of characterization of binding varieties. Two non-structural characterizations are given in Rosicl@ [23] where it is shown that under (M) every concrete complete, cocomplete, locally and colocally small category is binding if and only if a certain two-object category can be fully embedded into it, and in Sichler [26], where the representability of a finite category similarly decides whether a variety of unary algebras is binding. A structural theorem appears in Pultr and Sichler [21] where all varieties of idempotent unary algebras with two operations which are binding are described by identities. The aim of this paper is a characterization of binding (and almost-binding) varieties among those definable by identities in basic operations only. We prove: let V be a variety of algebras of type A given by identities containing only basic operations. Then V is almost-binding if and only if the sum of arities of operations in A is bigger than 1 and it is binding if and only if it is

On the cohomology rings of small categories

The group dihedral homology of an algebra over a field with characteristic zero was introduced by Tsygan (1983). The dihedral homology and cohomology of an algebra with involution over commutative ring with identity, associated with the small category, were studied by Krasauskas et al. (1988), Loday (1987), and Lodder (1993). The aim of this work is concerned with dihedral and reflexive (co)homology of small pre‐additive category. We also define the free product of involutive algebras associated with this category and study its dihedral homology group. Finally, following Perelygin (1990), we show that a small pre‐additive category is Morita equivalence.

We construct a cofibrantly generated Quillen model structure on the category of small n-fold categories and prove that it is Quillen equivalent to the standard model structure on the category of simplicial sets. An n-fold functor is a weak equivalence if and only if the diagonal of its n-fold nerve is a weak equivalence of simplicial sets. This is an n-fold analogue to Thomason's Quillen model structure on Cat. We introduce an n-fold Grothendieck construction for multisimplicial sets, and prove that it is a homotopy inverse to the n-fold nerve. As a consequence, we completely prove that the unit and counit of the adjunction between simplicial sets and n-fold categories are natural weak equivalences.