Research overview
Formal category theory
I am interested in exploring the dualities of constructions on virtual double categories (fc-multicategories) or even fc-polycategories, if the context necessitates. A primary example of this is that the functor $\mathbf{Mod}$ respects the horizontal duality of virtual double categories. I also aim to develop higher-dimensional analogue of virtual double categories to provide a natural framework for encoding the tricategory of pseudomonoids and pseudomodules in $\mathbf{Cat}$ or other monoidal bicategories.
Bi/tri-categorical structures
To any involutive category $(\mathcal{C},T\colon \mathcal{C} \rightarrow \mathcal{C},\alpha \colon T^2 \Rightarrow 1)$, one can assign a $1$-category of relative involutions $0\mathbf{Inv}(\mathcal{C},T,\alpha)$, whose objects are generalized involutions of the form $Tx \rightarrow x \in \mathcal{C}$. These generalized involutions and their morphisms turn out to be pretty useful in defining the dualities of strict $n$-categories. More generally, similar constructions exist in bicategories or tricategories and could be potentially used to provide more precise descriptions of higher dualities.
Abstract homotopy theory
Slice model categories provide model slice $(\infty,1)$-categories. In some important cases, these slice model categories also happen to be enriched slice model categories. While there are accounts of enriched slice-over construction for model categories, the theory for slice-under construction was not well developed. I develop a complete description for both slice-over and slice-under construction for enriched model categories. I aim to use this framework to characterize generalized join and slice constructions in the future.
Papers & Preprints
Relative involutions and inverses in monoidal dualities and slice constructions
Varinderjit Mann · Preprint, 2026 [arXiv]
Revisiting colimits in $\mathbf{Cat}$ and homotopy Category
Varinderjit Mann · Preprint, 2026 [arXiv]
Talks
Microcosm principle, dualities, and virtual double categories
AMS Fall Central Sectional Meeting, University of Minnesota Duluth — October 2026 [slides]
Enriched slice categories and model structures
Topology Seminar, Binghamton University — March 2026 [slides]
Enriched slice categories
BUGCAT Conference, Binghamton University — November 2025
Relative categories from infinity categories
BUGCAT Conference, Binghamton University — October 2024