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Let $𝒜$ and $ℬ$ be abelian categories with enough projective and injective objects, and $T : 𝒜 \to ℬ$ a left exact additive functor. Then one has a comma category $(ℬ ↓ T)$ . It is shown that if $T : 𝒜 \to ℬ$ is $𝒳$ -exact, then $(^{⊥} 𝒳, 𝒳)$ is a (hereditary) cotorsion pair in $𝒜$ and $(^{⊥} 𝒴, 𝒴)$ ) is a (hereditary) cotorsion pair in $ℬ$ if and only if $((\binom{^{⊥} 𝒳}{^{⊥} 𝒴})), 〈 𝐡 (𝒳, 𝒴) 〉)$ is a (hereditary) cotorsion pair in $(ℬ ↓ T)$ and $𝒳$ and $𝒴$ are closed under extensions. Furthermore, we characterize when special preenveloping classes in abelian categories $𝒜$ and $ℬ$ can induce special preenveloping classes...

Decompositions of the category of noncommutative sets and Hochschild and cyclic homology

Jolanta Słomińska (2003)

Open Mathematics

In this note we show that the main results of the paper [PR] can be obtained as consequences of more general results concerning categories whose morphisms can be uniquely presented as compositions of morphisms of their two subcategories with the same objects. First we will prove these general results and then we will apply it to the case of finite noncommutative sets.

Des propriétés de finitude des foncteurs polynomiaux

Aurélien Djament (2016)

Fundamenta Mathematicae

We study finiteness properties, especially the noetherian property, the Krull dimension and a variation of finite presentation, in categories of polynomial functors (notion introduced by Djament and Vespa) from a small symmetric monoidal category whose unit is an initial object to an abelian category. We prove in particular that the category of polynomial functors from the category of free abelian groups ℤⁿ with split monomorphisms to abelian groups is "almost" locally noetherian. We also give an...

Descent for compact 0-dimensional spaces.

Janelidze, George, Sobral, Manuela (2008)

Theory and Applications of Categories [electronic only]

Duality for CCD lattices.

Marmolejo, Francisco, Rosebrugh, Robert, Wood, R.J. (2009)

Theory and Applications of Categories [electronic only]

Embeddings into categories with fixed points in representations

Jiří Adámek, Václav Koubek, Jan Reiterman (1981)

Czechoslovak Mathematical Journal

Epimorphisms and semilattices of semigroups.

J. Crawley (1980)

Semigroup forum

Every small $𝒮 l$ -enriched category is Morita equivalent to an $𝒮 l$ -monoid.

Mesablishvili, Bachuki (2004)

Theory and Applications of Categories [electronic only]

Exponentiability in homotopy slices of TOP and pseudo-slices of CAT.

Niefield, Susan (2007)

Theory and Applications of Categories [electronic only]

Exponentiability in lax slices of $𝒯 o p$ .

Niefield, Susan (2006)

Theory and Applications of Categories [electronic only]

Exponential Objects

Marco Riccardi (2015)

Formalized Mathematics

In the first part of this article we formalize the concepts of terminal and initial object, categorical product [4] and natural transformation within a free-object category [1]. In particular, we show that this definition of natural transformation is equivalent to the standard definition [13]. Then we introduce the exponential object using its universal property and we show the isomorphism between the exponential object of categories and the functor category [12].

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