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Clemens Berger

Publications and source records attributed to Clemens Berger.

At least 19 recordsLinked to original sources
arXiv2026

Locally involutive semigroups

We introduce locally involutive semigroups and embed them into the category of ordered groupoids. This embedding restricts to a correspondence between quasi-involutive semigroups and ordered groupoids with mediator, extending the classical ESN-correspondence between inverse semigroups and inductive groupoids. An important subcategory of locally involutive semigroups is formed by left involutive semigroups because the classifying topos of an inverse semigroup S is equivalent to the category of left involutive semigroups \'etale over S [4]. We recover this equivalence from a general adjointness and use the latter to determine when a left involutive semigroup \'etale over S is actually an involutive semigroup. Any left involutive semigroup \'etale over S embeds into an involutive S-algebra as we call it. The underlying semigroup of this algebra is involutive.

math.GR↗
arXiv2023

On the profinite fundamental group of a connected Grothendieck topos

We show that finite (i.e. locally finite and decomposition-finite) objects of a connected Grothendieck topos span a Boolean pretopos with an essentially unique Galois point. The automorphism group of this point carries a profinite topology whose classifying topos is equivalent to the given Grothendieck topos if the latter is finitely generated. This leads to an intrinsic definition of the fundamental group of any connected Grothendieck topos.

math.CT↗
arXiv2023

Moment categories and operads

A moment category is endowed with a distinguished set of split idempotents, called moments, which can be transported along morphisms. Equivalently, a moment category is a category with an active/inert factorisation system fulfilling two simple axioms. These axioms imply that the moments of a fixed object form a monoid, actually a left regular band. Each locally finite unital moment category defines a specific type of operad which records the combinatorics of partitioning moments into elementary ones. In this way the notions of symmetric, non-symmetric and $n$-operad correspond to unital moment structures on $Γ$, $Δ$ and $Θ_n$ respectively. There is an analog of the plus construction of Baez-Dolan taking a unital moment category $\mathbb{C}$ to a unital hypermoment category $\mathbb{C}^+$. Under this construction, $\mathbb{C}$-operads get identified with $\mathbb{C}^+$-monoids, i.e. presheaves on $\mathbb{C}^+$ satisfying strict Segal conditions. We show that the plus construction of Segal's category $Γ$ embeds into the dendroidal category $Ω$ of Moerdijk-Weiss.

math.CT↗
arXiv2023

Stone duality for spectral sheaves and the patch monad

We establish a duality between global sheaves on spectral spaces and right distributive bands. This is a sheaf-theoretical extension of classical Stone duality between spectral spaces and bounded distributive lattices. The topology of a spectral space admits a refinement, the so-called patch topology, giving rise to a patch monad on sheaves over a fixed spectral space. Under the duality just mentioned the algebras of this patch monad are shown to correspond to distributive skew lattices.

math.CT↗
arXiv2022

Trees, graphs and aggregates: a categorical perspective on combinatorial surface topology, geometry, and algebra

Taking a Feynman categorical perspective, several key aspects of the geometry of surfaces are deduced from combinatorial constructions with graphs. This provides a direct route from combinatorics of graphs to string topology operations via topology, geometry and algebra. In particular, the inclusion of trees into graphs and the dissection of graphs into aggregates yield a concise formalism for cyclic and modular operads as well as their polycyclic and surface type generalizations. The latter occur prominently in two-dimensional topological field theory and in string topology. The categorical viewpoint allows us to use left Kan extensions of Feynman operations as an efficient computational tool. The computations involve the study of certain categories of structured graphs which are expected to be of independent interest.

math.AT↗
arXiv2018

Comprehensive factorisation systems

We establish a correspondence between consistent comprehension schemes and complete orthogonal factorisation systems. The comprehensive factorisation of a functor between small categories arises in this way. Similar factorisation systems exist for the categories of topological spaces, simplicial sets, small multicategories and Feynman categories. In each case comprehensive factorisation induces a natural notion of universal covering, leading to a Galois-type definition of fundamental group for based objects of the category.

math.CT↗
arXiv2017

Gabriel-Morita theory for excisive model categories

We develop a Gabriel-Morita theory for strong monads on pointed monoidal model categories. Assuming that the model category is excisive, i.e. the derived suspension functor is conservative, we show that if the monad T preserves cofibre sequences up to homotopy and has a weakly invertible strength, then the category of T-algebras is Quillen equivalent to the category of T(I)-modules where I is the monoidal unit. This recovers Schwede's theorem on connective stable homotopy over a pointed Lawvere theory as special case.

math.AT↗
arXiv2017

Homotopy theory for algebras over polynomial monads

We study the existence and left properness of transferred model structures for "monoid-like" objects in monoidal model categories. These include genuine monoids, but also all kinds of operads as for instance symmetric, cyclic, modular, higher operads, properads and PROP's. All these structures can be realised as algebras over polynomial monads. We give a general condition for a polynomial monad which ensures the existence and (relative) left properness of a transferred model structure for its algebras. This condition is of a combinatorial nature and singles out a special class of polynomial monads which we call tame polynomial. Many important monads are shown to be tame polynomial.

math.CT↗
arXiv2016

Central reflections and nilpotency in exact Mal'tsev categories

We study nilpotency in the context of exact Mal'tsev categories taking central extensions as the primitive notion. This yields a nilpotency tower which is analysed from the perspective of Goodwillie's functor calculus. We show in particular that the reflection into the subcategory of $n$-nilpotent objects is the universal endofunctor of degree $n$ if and only if every $n$-nilpotent object is $n$-folded. In the special context of a semi-abelian category, an object is $n$-folded precisely when its Higgins commutator of length $n+1$ vanishes.

math.CT↗
arXiv2013

On the homotopy theory of enriched categories

We give sufficient conditions for the existence of a Quillen model structure on small categories enriched in a given monoidal model category. This yields a unified treatment for the known model structures on simplicial, topological, dg- and spectral categories. Our proof is mainly based on a fundamental property of cofibrant enriched categories on two objects, stated below as the Interval Cofibrancy Theorem.

math.AT↗
arXiv2011

Monads with arities and their associated theories

After a review of the concept of "monad with arities" we show that the category of algebras for such a monad has a canonical dense generator. This is used to extend the correspondence between finitary monads on sets and Lawvere's algebraic theories to a general correspondence between monads and theories for a given category with arities. As application we determine arities for the free groupoid monad on involutive graphs and recover the symmetric simplicial nerve characterisation of groupoids.

math.CT↗
arXiv2009

The lattice path operad and Hochschild cochains

We introduce two coloured operads in sets -- the lattice path operad and a cyclic extension of it -- closely related to iterated loop spaces and to universal operations on cochains. As main application we present a formal construction of an $E_2$-action (resp. framed $E_2$-action) on the Hochschild cochain complex of an associative (resp. symmetric Frobenius) algebra.

math.AT↗
arXiv2008

On an extension of the notion of Reedy category

We extend the classical notion of a Reedy category so as to allow non-trivial automorphisms. Our extension includes many important examples occuring in topology such as Segal's category Gamma, or the total category of a crossed simplicial group such as Connes' cyclic category Lambda. For any generalized Reedy category R and any cofibrantly generated model category E, the functor category E^R is shown to carry a canonical model structure of Reedy type.

math.AT↗
arXiv2008

On the derived category of an algebra over an operad

We present a general construction of the derived category of an algebra over an operad and establish its invariance properties. A central role is played by the enveloping operad of an algebra over an operad.

math.AT↗
arXiv2008

Resolution of coloured operads and rectification of homotopy algebras

We provide general conditions under which the algebras for a coloured operad in a monoidal model category carry a Quillen model structure, and prove a Comparison Theorem to the effect that a weak equivalence between suitable such operads induces a Quillen equivalence between their categories of algebras. We construct an explicit Boardman-Vogt style cofibrant resolution for coloured operads, thereby giving a uniform approach to algebraic structures up to homotopy over coloured operads. The Comparison Theorem implies that such structures can be rectified.

math.AT↗
arXiv2006

Iterated wreath product of the simplex category and iterated loop spaces

Generalising Segal's approach to 1-fold loop spaces, the homotopy theory of $n$-fold loop spaces is shown to be equivalent to the homotopy theory of reduced $Θ_n$-spaces, where $Θ_n$ is an iterated wreath product of the simplex category $Δ$. A sequence of functors from $Θ_n$ to $Γ$ allows for an alternative description of the Segal-spectrum associated to a $Γ$-space. In particular, each Eilenberg-MacLane space $K(π,n)$ has a canonical reduced $Θ_n$-set model.

math.AT↗
arXiv2006

The Boardman-Vogt resolution of operads in monoidal model categories

We extend the W-construction of Boardman and Vogt to operads of an arbitrary monoidal model category with suitable interval, and show that it provides a cofibrant resolution for well-pointed sigma-cofibrant operads. The standard simplicial resolution of Godement as well as the cobar-bar chain resolution are shown to be particular instances of this generalised W-construction.

math.AT↗