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Kristóf Kanalas

Publications and source records attributed to Kristóf Kanalas.

9 recordsLinked to original sources
arXiv2026

Positively closed $Sh(B)$-valued models

We study positively closed and strongly positively closed topos-valued models of coherent theories. Positively closed is a global notion (it is defined in terms of all possible outgoing homomorphisms), while strongly positively closed is a local notion (it only concerns the definable sets inside the model). For $\mathbf{Set}$-valued models of coherent theories they coincide. We prove that if $\mathcal{E}=Sh(B,τ_{coh})$ for a complete Boolean algebra, then positively closed but not strongly positively closed $\mathcal{E}$-valued models of coherent theories exist, yet, there is an alternative local property which characterizes positively closed $\mathcal{E}$-valued models. A large part of our discussion is given in the context of infinite quantifier geometric logic, dealing with the fragment $L^g_{κκ}$ where $κ$ is weakly compact.

math.CT↗
arXiv2025

Pure maps are strict monomorphisms

We prove that $i)$ if $\mathcal{A}$ is $λ$-accessible and it is axiomatizable in (finitary) coherent logic then $λ$-pure maps are strict monomorphisms and $ii)$ if there is a proper class of strongly compact cardinals and $\mathcal{A}$ is $λ$-accessible then for some $μ\vartriangleright λ$ every $μ$-pure map is a strict monomorphism.

math.CT↗
arXiv2025

$Sh(B)$-valued models of $(κ,κ)$-coherent categories

A basic technique in model theory is to name the elements of a model by introducing new constant symbols. We describe the analogous construction in the language of syntactic categories/ sites. As an application we identify $\mathbf{Set}$-valued regular functors on the syntactic category with a certain class of topos-valued models (we will refer to them as "$Sh(B)$-valued models"). For the coherent fragment $L_{ωω}^g \subseteq L_{ωω}$ this was proved by Jacob Lurie, our discussion gives a new proof, together with a generalization to $L_{κκ}^g$ when $κ$ is weakly compact. We present some further applications: first, a $Sh(B)$-valued completeness theorem for $L_{κκ}^g$ ($κ$ is weakly compact), second, that $\mathcal{C}\to \mathbf{Set} $ regular functors (on coherent categories with disjoint coproducts) admit an elementary map to a product of coherent functors.

math.CT↗
arXiv2025

Every theory is eventually of presheaf type

We give a detailed and self-contained introduction to the theory of $λ$-toposes and prove the following: 1) A $λ$-separable $λ$-topos has enough $λ$-points. 2) The classifying $λ$-topos of a $κ$-site $(\mathcal{C},E)$ is a presheaf topos (assuming $κ\vartriangleleft λ=λ^{<λ}$, $|\mathcal{C}|,|E|<λ$).

math.CT↗
arXiv2022

Positive model theory of interpretations

We prove analogues of model theory results for $\mathcal{C}\to \mathcal{D}$ coherent functors, including variants of the omitting types theorem and some results on ultraproduct constructions. We introduce a distributive lattice valued invariant of $\mathcal{C}\to \mathbf{Set}$ coherent functors that vanishes precisely on positively closed models, then we study its functorial properties.

math.CT↗
arXiv2022

A (2,1)-model structure for conceptual completeness

We prove the (2,1)-categorical analogue of the small object argument and give a (2,1)-model structure on the category of small coherent categories, coherent functors and natural isomorphisms. It is induced by a higher dimensional example of a reflective factorisation system, determined by the full subcategory of pretoposes. We prove it to be right proper and the generating trivial cofibrations are described. Whitehead's theorem gives conceptual completeness.

math.CT↗
arXiv2021

Generic countably infinite groups

Countably infinite groups (with a fixed underlying set) constitute a Polish space $G$ with a suitable metric, hence the Baire category theorem holds in $G$. We study isomorphism invariant subsets of $G$, which we call group properties. We say that the generic countably infinite group is of property $P$ if $P$ is comeager in $G$. We prove that every group property with the Baire property is either meager or comeager. We show that there is a comeager elementary equivalence class in $G$ but every isomorphism class is meager. We prove that the generic group is algebraically closed, simple, not finitely generated and not locally finite. We show that in the subspace of Abelian groups the generic group is isomorphic to the unique countable, divisible torsion group that contains every finite Abelian group. We sketch the model-theoretic setting in which many of our results can be generalized. We briefly discuss a connection with infinite games.

math.LO↗
arXiv2021

The (2,1)-category of small coherent categories

There is a well-known correspondence between coherent theories (and their interpretations) and coherent categories (resp. functors), hence the (2,1)-category $\mathbf{Coh_{\sim}}$ (of small coherent categories, coherent functors and all natural isomorphisms) is of logical interest. We prove that this category admits all small 2-limits and 2-colimits (in the ($\infty $,1)-categorical sense), and prove a 2-categorical small object argument to provide weak factorisation systems for coherent functors.

math.CT↗