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Register a new userHigher weak (co)limits, adjoint functor theorems, and higher Brown representability
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We prove general adjoint functor theorems for weakly (co)complete $n$-categories. This class of $n$-categories includes the homotopy $n$-categories of (co)complete $infty$-categories -- in particular, these $n$-categories do not admit all small (co)limits in general. We also introduce Brown representability for (homotopy) $n$-categories and prove a Brown representability theorem for localizations of compactly generated $n$-categories. This class of $n$-categories includes the homotopy $n$-categories of presentable $infty$-categories if $n geq 2$ and the homotopy $n$-categories of stable presentable $infty$-categories for any $n geq 1$.
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Adjoint functor theorems give necessary and sufficient conditions for a functor to admit an adjoint. In this paper we prove general adjoint functor theorems for functors between $infty$-categories. One of our main results is an $infty$-categorical generalization of Freyds classical General Adjoint Functor Theorem. As an application of this result, we recover Luries adjoint functor theorems for presentable $infty$-categories. We also discuss the comparison between adjunctions of $infty$-categories and homotopy adjunctions, and give a treatment of Brown representability for $infty$-categories based on Hellers purely categorical formulation of the classical Brown representability theorem.
We survey Lawvere theories at the level of infinity categories, as an alternative framework for higher algebra (rather than infinity operads). From a pedagogical perspective, they make many key definitions and constructions less technical. They are also better-suited than operads for equivariant homotopy theory and its relatives. Our main result establishes a universal property for the infinity category of Lawvere theories, which completely characterizes the relationship between a Lawvere theory and its infinity category of models. Many familiar properties of Lawvere theories follow directly. As a consequence, we prove that the Burnside category is a classifying object for additive categories, as promised in an earlier paper, and as part of a more general correspondence between enriched Lawvere theories and module Lawvere theories.
We generalize toposic Galois theory to higher topoi. We show that locally constant sheaves in a locally (n-1)-connected n-topos are equivalent to representations of its fundamental pro-n-groupoid, and that the latter can be described in terms of Galois torsors. We also show that finite locally constant sheaves in an arbitrary infinity-topos are equivalent to finite representations of its fundamental pro-infinity-groupoid. Finally, we relate the fundamental pro-infinity-groupoid of 1-topoi to the construction of Artin and Mazur and, in the case of the etale topos of a scheme, to its refinement by Friedlander.
For topological spaces $X$ and $Y$, a (not necessarily continuous) function $f:X rightarrow Y$ naturally induces a functor from the category of closed subsets of $X$ (with morphisms given by inclusions) to the category of closed subsets of $Y$. The function $f$ also naturally induces a functor from the category of closed subsets of $Y$ to the category of closed subsets of $X$. Our aim in this expository note is to show that the function $f$ is continuous if and only if the first of the above two functors is a left adjoint to the second. We thereby obtain elementary examples of adjoint pairs (apparently) not part of the standard introductory treatments of this subject.
We investigate the notion of involutive weak globular $omega$-categories via Jacque Penons approach. In particular, we give the constructions of a free self-dual globular $omega$-magma, of a free strict involutive globular $omega$-category, over an $omega$-globular set, and a contraction between them. The monadic definition of involutive weak globular $omega$-categories is given as usual via algebras for the monad induced by a certain adjunction. In our case, the adjunction is obtained from the free functor that associates to every $omega$-globular set the above contraction. Some examples of involutive weak globular $omega$-categories are also provided.
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