Characterizing Van Kampen Squares via Descent Data

Fibred Amalgamation, Descent Data, and Van Kampen Squares in Topoi

Co-tabulations, Bicolimits and Van-Kampen Squares in Collagories

We previously defined collagories essentially as “distributive allegories without zero morphisms”. Collagories are sufficient for accommodating the relation-algebraic approach to graph transformation, and closely correspond to the adhesive categories important for the categorical DPO approach to graph transformation. Heindel and Sobociński have recently characterised the Van-Kampen colimits use...

A Markov Chain Theory Approach to Characterizing the Minimax Optimality of Stochastic Gradient Descent (for Least Squares)

This work provides a simplified proof of the statistical minimax optimality of (iterate averaged) stochastic gradient descent (SGD), for the special case of least squares. This result is obtained by analyzing SGD as a stochastic process and by sharply characterizing the stationary covariance matrix of this process. The finite rate optimality characterization captures the constant factors and ad...

Van Kampen theorems for toposes

In this paper we introduce the notion of an extensive 2-category, to be thought of as a “2-category of generalized spaces”. We consider an extensive 2-category K equipped with a binary-product-preserving pseudofunctor C : K op → CAT, which we think of as specifying the “coverings” of our generalized spaces. We prove, in this context, a van Kampen theorem which generalizes and refines one of Bro...

Van Kampen diagrams are bicolimits in Span

In adhesive categories, pushouts along monomorphisms are Van Kampen (vk) squares, a special case of a more general notion called vk-diagram. Other examples of vk-diagrams include coproducts in extensive categories and strict initial objects. Extensive and adhesive categories characterise useful exactness properties of, respectively, coproducts and pushouts along monos and have found several app...