Milind Gunjal

Milind Gunjal

Ph.D., Mathematics

Publications

Monoidal 2-Categories and K-Theory of Waldhausen Categories

Published in ProQuest, 2025

Abstract: This dissertation explores the interplay between higher categorical structures and algebraic K-theory, focusing on refining Waldhausen’s K-theory via monoidal bicategories and Picard 2-groupoids. We study algebraic models for connected n-types of K-theory spaces, with emphasis on the 2-type and 3-type. We construct monoidal bicategories from Waldhausen categories and lift the K-theory to a functor that encodes multiplicative and coherence data. Using stabilized quadratic modules, we model the connected 3-type and define a universal 2-determinant functor valued in a Picard 2-groupoid. In the second part, we analyze symmetric monoidal bicategories using categorical extensions and biextensions. Extending previous work by Breen, we relate symmetric structures to alternating biextensions and compute associated cohomological invariants via quadratic functors. We provide a full cocycle-level analysis of the classifying invariants in Eilenberg–MacLane cohomology, giving explicit quadratic functors that capture the higher coherence conditions associated with the symmetry structure. These results advance the structural understanding of algebraic K-theory and reveal new connections between homotopical algebra, higher monoidal categories, and cohomological classification of categorical symmetries.

Download here

Symmetric Monoidal Bicategories and Biextensions

Published in arXiv, 2024

Abstract: We study monoidal 2-categories and bicategories in terms of categorical extensions and the cohomological data they determine in appropriate cohomology theories with coefficients in Picard groupoids. In particular, we analyze the hierarchy of possible commutativity conditions in terms of progressive stabilization of these data. We also show that monoidal structures on bicategories give rise to biextensions of a pair of (abelian) groups by a Picard groupoid, and that the progressive vanishing of obstructions determined by the tower of commutative structures corresponds to appropriate symmetry conditions on these biextensions. In the fully symmetric case, which leads us fully into the stable range, we show how our computations can be expressed in terms of the cubical Q-construction underlying MacLane (co)homology.

Download here

Vector Bundles from an Algebraic Viewpoint

Published in Indian Institute of Science Education and Research, Kolkata, 2019

Abstract: In this Master’s thesis, we will give a brief introduction to projective modules and vector bundles and discuss the correspondence between modules over the ring of continuous functions (C(X)) and vector bundles over X. We will also introduce covering space theory to get a better understanding of the correspondence between algebraic and topological objects. A detailed discussion about the compactifications is also presented. We will also establish the notion of localization of vector bundles in terms of localization of C(X)-modules at a point using Swan’s theorem. We will discuss some examples of Swan’s theorem. Finally, we motivate the statement of the Quillen-Suslin theorem.

Download here