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We raise the question of when a simplicial object in a catetgory is deemed contractible. The literature offers three definitions. One is the existence of an “extra degeneracy”, indexed by $- 1$ , which does not quite live up to the name. This can be strengthened to a “strong extra degeneracy". Another possibility is that it be homotopic to a constant simplicial object. Despite claims in the literature to the contrary, we show that all three are distinct concepts with strong extra degeneracy implies extra...

Copure injective resolutions, flat resolvents and dimensions

Edgar E. Enochs, Jenda M. G. Overtoun (1993)

Commentationes Mathematicae Universitatis Carolinae

In this paper, we show the existence of copure injective preenvelopes over noetherian rings and copure flat preenvelopes over commutative artinian rings. We use this to characterize $n$ -Gorenstein rings. As a consequence, if the full subcategory of strongly copure injective (respectively flat) modules over a left and right noetherian ring $R$ has cokernels (respectively kernels), then $R$ is $2$ -Gorenstein.

Cosimplicial resolutions and homotopy spectral sequences in model categories.

Bousfield, A.K. (2003)

Geometry & Topology

Cotorsion pairs in comma categories

Yuan Yuan, Jian He, Dejun Wu (2024)

Czechoslovak Mathematical Journal

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...

Crossed complexes, and free crossed resolutions for amalgamated sums and HNN-extensions of groups.

Brown, R., Moore, E.J., Porter, T., Wensley, C.D. (2002)

Georgian Mathematical Journal

Crossed extensions of algebras and Hochschild cohomology.

Baues, Hans-Joachim, Minian, Elias Gabriel (2002)

Homology, Homotopy and Applications

Crossed modules as $G$ -categories

Robert Gordon (1993)

Cahiers de Topologie et Géométrie Différentielle Catégoriques

Crossed modules in $C a t$ and a Brown-Spencer theorem for $2$ -categories

Timothy Porter (1985)

Cahiers de Topologie et Géométrie Différentielle Catégoriques

Crossed n-folds extensions of groups and cohomology.

Johannes Huebschmann (1980)

Commentarii mathematici Helvetici

Crossed squares and 2-crossed modules of commutative algebras.

Arvasi, Zekeri̇ya (1997)

Theory and Applications of Categories [electronic only]

Cubical Abelian groups with connections are equivalent to chain complexes.

Brown, Ronald, Higgins, Philip J. (2003)

Homology, Homotopy and Applications

Cubical sets and their site.

Grandis, Marco, Mauri, Luca (2003)

Theory and Applications of Categories [electronic only]

Cup $i$ -produit sur les algèbres graduées avec symétries et algèbres de Gerstenhaber

Arwa Abbassi (2013)

Annales mathématiques Blaise Pascal

Dans ce papier, on définit, dans le cadre des algèbres graduées avec symétries la notion de cup $i$ -produit introduite par Steenrod dans [11]. En utilisant le cup 1-produit, on montre que la cohomologie associée à une algèbre graduée avec symétries est une algèbre de Gerstenhaber.

Cup products, duality and periodicity for generalized group cohomology

Marton Harris (1970)

Fundamenta Mathematicae

Cyclic cohomology of (extended) Hopf algebras

M. Khalkhali, B. Rangipour (2003)

Banach Center Publications

We review recent progress in the study of cyclic cohomology of Hopf algebras, extended Hopf algebras, invariant cyclic homology, and Hopf-cyclic homology with coefficients, starting with the pioneering work of Connes-Moscovici.

Cyclic homology, a survey

Jean-Louis Loday (1986)

Banach Center Publications

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