arXiv subjects

Amit Kuber

Publications and source records attributed to Amit Kuber.

22 records · Page 2Linked to original sources

arXiv2016

Definable Categories

We introduce the notion of a definable category--a category equivalent to a full subcategory of a locally finitely presentable category that is closed under products, directed colimits and pure subobjects. Definable subcategories are precisely the finite-injectivity classes. We prove a $2$-duality between the $2$-category of small exact categories and the $2$-category of definable categories, and provide a new proof of its additive version. We further introduce a third vertex of the $2$-category of regular toposes and show that the diagram of $2$-(anti-)equivalences between three $2$-categories commutes, the corresponding additive triangle is well-known.

math.CT↗

arXiv2015

Taxotopy Theory of Posets I: van Kampen Theorems

Given functors $F,G:\mathcal C\to\mathcal D$ between small categories, when is it possible to say that $F$ can be "continuously deformed" into $G$ in a manner that is not necessarily reversible? In an attempt to answer this question in purely category-theoretic language, we use adjunctions to define a `taxotopy' preorder $\preceq$ on the set of functors $\mathcal C\to\mathcal D$, and combine this data into a `fundamental poset' $(Λ(\mathcal C,\mathcal D),\preceq)$. The main objects of study in this paper are the fundamental posets $Λ(\mathbf 1,P)$ and $Λ(\mathbb Z,P)$ for a poset $P$, where $\mathbf 1$ is the singleton poset and $\mathbb Z$ is the ordered set of integers; they encode the data about taxotopy of points and chains of $P$ respectively. Borrowing intuition from homotopy theory, we show that a suitable cone construction produces `null-taxotopic' posets and prove two forms of van Kampen theorem for computing fundamental posets via covers of posets.

math.CT↗

arXiv2013

On the Grothendieck ring of varieties

Let $\operatorname{K}_0(\operatorname{Var}_k)$ denote the Grothendieck ring of $k$-varieties over an algebraically closed field $k$. Larsen and Lunts asked if two $k$-varieties having the same class in $\operatorname{K}_0 (\operatorname{Var}_k)$ are piecewise isomorphic. Gromov asked if a birational self-map of a $k$-variety can be extended to a piecewise automorphism. We show that these two questions are equivalent over any algebraically closed field. If these two questions admit a positive answer, then we prove that its underlying abelian group is a free abelian group. Furthermore, if $\mathfrak B$ denotes the multiplicative monoid of birational equivalence classes of irreducible $k$-varieties then we also prove that the associated graded ring of the Grothendieck ring is the monoid ring $\mathbb Z[\mathfrak B]$.

math.AG↗

arXiv2013

Grothendieck Rings of Theories of Modules

The model-theoretic Grothendieck ring of a first order structure, as defined by Krajicěk and Scanlon, captures some combinatorial properties of the definable subsets of finite powers of the structure. In this paper we compute the Grothendieck ring, $K_0(M_\mathcal R)$, of a right $R$-module $M$, where $\mathcal R$ is any unital ring. As a corollary we prove a conjecture of Prest that $K_0(M)$ is non-trivial, whenever $M$ is non-zero. The main proof uses various techniques from the homology theory of simplicial complexes.

math.LO↗