arXiv subjects

Özgün Ünlü

Publications and source records attributed to Özgün Ünlü.

7 recordsLinked to original sources

arXiv2025

Simplicial lists in operad theory I

We define a category $\mathsf{List}$ whose objects are sets and morphisms are mappings which assign to an element in the domain an ordered sequence (list) of elements in the codomain. We introduce and study a category of simplicial objects $\mathsf{slist}$ whose objects are functors $Δ^{op} \to \mathsf{List}$, which we call simplicial lists, and morphisms are natural transformations which have functions as components. We demonstrate that $\mathsf{sList}$ supports the combinatorics of (non-symmetric) operads by constructing a fully-faithful nerve functor $N^l : \mathsf{Operad} \to \mathsf{sList}$ from the category of operads. This leads to a reasonable model for the theory of non-symmetric $\infty$-operads. We also demonstrate that $\mathsf{sList}$ has the structure of a presheaf category. In particular, we study a subcategory $\mathsf{sList}_{\text{op}}$ of operadic simplicial lists, in which the nerve functor takes values. The latter category is also a presheaf category over a base whose objects may be interpreted as levelled trees. We construct a coherent nerve functor which outputs an $\infty$-operad for each operad enriched in Kan complexes. We also define homology groups of simplicial lists and study first properties.

math.AT↗

arXiv2024

The Euler Characteristic of Finite Categories

We associate a rational number $χ(\mathcal{A})$ to every category $\mathcal{A}$ whose object and morphism sets are finite. We show that the assignment $χ$ is additive under disjoint union and it preserves products. Hence we consider $χ$ as an Euler measure on a family of categories where this assignment also obeys a version of the inclusion--exclusion principle.

math.CT↗

arXiv2022

Minimal models of some differential graded modules

Minimal models of chain complexes associated with free torus actions on spaces have been extensively studied in the literature. In this paper, we discuss these constructions using the language of operads. The main goal of this paper is to define a new Koszul operad that has projections onto several of the operads used in these minimal model constructions.

math.AT↗

arXiv2021

Stabilization and costabilization with respect to an action of a monoidal category

We study actions of monoidal categories on objects in a suitably enriched $2$-category, and applications in stable homotopy theory. Given a monoidal category $\mathcal{I}$ and an $\mathcal{I}$-object $\mathcal{A}$, the (co)stabilization of $\mathcal{A}$ is obtained by universally forcing the $\mathcal{I}$-action to be reversible so that every object of $\mathcal{I}$ acts on $\mathcal{A}$ by auto-equivalences. We introduce a notion of $\mathcal{I}$-equivariance for morphisms between $\mathcal{I}$-objects and give constructions of stabilization and costabilization in terms of weak ends and coends in an enriched $2$-category of $\mathcal{I}$-objects and $\mathcal{I}$-equivariant morphisms. We observe that the stabilization of a relative category with respect to an action coincides with the usual notion of stabilization in stable homotopy theory when the action is defined by loop space functors. We show that several examples that exist in the literature, including various categories of spectra, fit into our setting after fixing $\mathcal{A}$ and the $\mathcal{I}$-action on it. In particular, categories of sequential spectra, coordinate free spectra, genuine equivariant spectra, parameterized spectra indexed by vector bundles are obtained in terms of weak ends in the $2$-category of relative categories. On the other hand, the costabilization of a relative category with respect to an action gives a stable relative category akin to a version Spanier-Whitehead category. In particular, we establish a form of duality between constructions of stable homotopy categories by spectra and by Spanier-Whitehead like categories.

math.CT↗

arXiv2018

Carlsson's rank conjecture and a conjecture on square-zero upper triangular matrices

Let $k$ be an algebraically closed field and $A$ the polynomial algebra in $r$ variables with coefficients in $k$. In case the characteristic of $k$ is $2$, Carlsson conjectured that for any $DG$-$A$-module $M$ of dimension $N$ as a free $A$-module, if the homology of $M$ is nontrivial and finite dimensional as a $k$-vector space, then $2^r\leq N$. Here we state a stronger conjecture about varieties of square-zero upper-triangular $N\times N$ matrices with entries in $A$. Using stratifications of these varieties via Borel orbits, we show that the stronger conjecture holds when $N < 8$ or $r < 3$ without any restriction on the characteristic of $k$. As a consequence, we attain a new proof for many of the known cases of Carlsson's conjecture and give new results when $N > 4$ and $r = 2$.

math.AC↗

arXiv2018

Continuous Sensitivity and Reversibility

Let $n$ be a positive integer and $f$ a differentiable function from a convex subset $C$ of the Euclidean space $\mathbb{R}^n$ to a smooth manifold. We define an invariant of $f$ via counting certain threshold functions associated to $f$. We call this invariant the continuous sensitivity of $f$ and denote it by $\mathrm{cs}_{C}(f)$. This invariant is a real number between $0$ and $n$ and measures how sensitive $f$ is to change in its input variables. For example, if $f$ is a constant function then $\mathrm{cs}_{C}(f)=0$. On the other extreme, if $\mathrm{cs}_{C}(f)=n$ then $f$ is one-to-one on $C$. This last statement is important for reversibility problems. To say that a function is reversible one can write an explicit inverse of the function. However, this is not always easy. Even a multilinear function can have a complicated inverse function. Here we give tools to compute continuous sensitivity which makes it possible to answer reversibility problems without finding explicit inverse functions.

math.CO↗

arXiv2018

Semigroup actions on sets and the Burnside ring

In this paper we discuss some enlargements of the category of sets with semigroup actions and equivariant functions. We show that these enlarged categories possess two idempotent endofunctors. In the case of groups these enlarged categories are equivalent to the usual category of group actions and equivariant functions, and these idempotent endofunctors reverse a given action. For a general semigroup we show that these enlarged categories admit homotopical category structures defined by using these endofunctors and show that up to homotopy these categories are equivalent to the usual category of sets with semigroup actions. We finally construct the Burnside ring of a monoid by using homotopical structure of these categories, so that when the monoid is a group this definition agrees with the usual definition, and we show that when the monoid is commutative, its Burnside ring is equivalent to the Burnside ring of its Gröthendieck group.

math.AT↗