Applied Categorical Structures – Official Blog

by Voutsadakis, George

Algebraic systems play in the theory of algebraizability of π-institutions the role that algebras play in the theory of algebraizable sentential logics. In this same sense, ℐ-algebraic systems are to a π-institution ℐ what
$\mathcal{S}$
-algebras are to a sentential logic
$\mathcal{S}$
. More precisely, an (ℐ,N)-algebraic system is the sentence functor reduct of an N′-reduced (N,N′)-full model of a π-institution ℐ. Algebraic systems are formally introduced and their relationship with full models and with bilogical morphisms is investigated.

DOI: 10.1007/s10485-005-5797-5
Print publication date: 6/1/2005
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by Barto, Libor

The quasicategory ℚ of all set functors (i.e. endofunctors of the category
$\mathbb{SET}$
of all sets and mappings) and all natural transformations has a terminal object – the constant functor C₁. We construct here the terminal (or at least the smallest weakly terminal object, which is rigid) in some important subquasicategories of ℚ – in the quasicategory
$\mathbb{F}$
of faithful connected set functors and all natural transformations, and in the quasicategories
$\mathbb{B}^{(\kappa)}$
of all set functors and natural transformations which preserve filters of points (up to cardinality κ).

DOI: 10.1007/s10485-005-5796-6
Print publication date: 6/1/2005
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by Lack, Stephen; Paoli, Simona

We study internal structures in the category of algebras for an operad, and show that these themselves admit an operadic description. The main case of interest is where the operad is on an abelian category, and the internal structures in question are those of internal category, internaln-category, or internal (cubical) n-tuple category. This allows an operadic treatment of crossed modules and other crossed structures.

DOI: 10.1007/s10485-005-2959-4
Print publication date: 6/1/2005
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by Lack, Stephen

We investigate limits in the 2-category of strict algebras and lax morphisms for a 2-monad. This includes both the 2-category of monoidal categories and monoidal functors as well as the 2-category of monoidal categories and opomonoidal functors, among many other examples.

DOI: 10.1007/s10485-005-2958-5
Print publication date: 6/1/2005
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by Bullejos, M.; Faro, E.; Blanco, V.

The notion of geometric nerve of a 2-category (Street, J. Pure Appl. Algebra (1987), 283–335) provides a full and faithful functor if regarded as defined on the category of 2-categories and lax 2-functors. Furthermore, lax 2-natural transformations between lax 2-functors give rise to homotopies between the corresponding simplicial maps. These facts allow us to prove a representation theorem of the general non-abelian cohomology of groupoids (classifying non-abelian extensions of groupoids) by means of homotopy classes of simplicial maps.

DOI: 10.1007/s10485-005-2957-6
Print publication date: 6/1/2005
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by Stubbe, Isar

Applying (enriched) categorical structures we define the notion of ordered sheaf on a quantaloid
$\mathcal{Q}$
, which we call ‘
$\mathcal{Q}$
-order’. This requires a theory of semicategories enriched in the quantaloid
$\mathcal{Q}$
, that admit a suitable Cauchy completion. There is a quantaloid
$\mathsf{Idl}(\mathcal{Q})$
of
$\mathcal{Q}$
-orders and ideal relations, and a locally ordered category
$\mathsf{Ord}(\mathcal{Q})$
of
$\mathcal{Q}$
-orders and monotone maps; actually,
$\mathsf{Ord}(\mathcal{Q})=\mathsf{Map}(\mathsf{Idl}(\mathcal{Q}))$
. In particular is
$\mathsf{Ord}(\Omega)$
, with Ω a locale, the category of ordered objects in the topos of sheaves on Ω. In general
$\mathcal{Q}$
-orders can equivalently be described as Cauchy complete categories enriched in the split-idempotent completion of
$\mathcal{Q}$
. Applied to a locale Ω this generalizes and unifies previous treatments of (ordered) sheaves on Ω in terms of Ω-enriched structures.

DOI: 10.1007/s10485-004-7421-5
Print publication date: 6/1/2005
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