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1. CMB Online first

Luo, Xiu-Hua
 Exact Morphism category and Gorenstein-projective representations Let $Q$ be a finite acyclic quiver, $J$ be an ideal of $kQ$ generated by all arrows in $Q$, $A$ be a finite-dimensional $k$-algebra. The category of all finite-dimensional representations of $(Q, J^2)$ over $A$ is denoted by $\operatorname{rep}(Q, J^2, A)$. In this paper, we introduce the category $\operatorname{exa}(Q,J^2,A)$, which is a subcategory of $\operatorname{rep}{}(Q,J^2,A)$ of all exact representations. The main result of this paper explicitly describes the Gorenstein-projective representations in $\operatorname{rep}{}(Q,J^2,A)$, via the exact representations plus an extra condition. As a corollary, $A$ is a self-injective algebra, if and only if the Gorenstein-projective representations are exactly the exact representations of $(Q, J^2)$ over $A$. Keywords:representations of a quiver over an algebra, exact representations, Gorenstein-projective modulesCategory:18G25

2. CMB 2014 (vol 57 pp. 721)

Bruillard, Paul; Galindo, César; Hong, Seung-Moon; Kashina, Yevgenia; Naidu, Deepak; Natale, Sonia; Plavnik, Julia Yael; Rowell, Eric C.
 Classification of Integral Modular Categories of Frobenius--Perron Dimension $pq^4$ and $p^2q^2$ We classify integral modular categories of dimension $pq^4$ and $p^2q^2$, where $p$ and $q$ are distinct primes. We show that such categories are always group-theoretical except for categories of dimension $4q^2$. In these cases there are well-known examples of non-group-theoretical categories, coming from centers of Tambara-Yamagami categories and quantum groups. We show that a non-group-theoretical integral modular category of dimension $4q^2$ is equivalent to either one of these well-known examples or is of dimension $36$ and is twist-equivalent to fusion categories arising from a certain quantum group. Keywords:modular categories, fusion categoriesCategory:18D10

3. CMB 2013 (vol 57 pp. 318)

Huang, Zhaoyong
 Duality of Preenvelopes and Pure Injective Modules Let $R$ be an arbitrary ring and $(-)^+=\operatorname{Hom}_{\mathbb{Z}}(-, \mathbb{Q}/\mathbb{Z})$ where $\mathbb{Z}$ is the ring of integers and $\mathbb{Q}$ is the ring of rational numbers, and let $\mathcal{C}$ be a subcategory of left $R$-modules and $\mathcal{D}$ a subcategory of right $R$-modules such that $X^+\in \mathcal{D}$ for any $X\in \mathcal{C}$ and all modules in $\mathcal{C}$ are pure injective. Then a homomorphism $f: A\to C$ of left $R$-modules with $C\in \mathcal{C}$ is a $\mathcal{C}$-(pre)envelope of $A$ provided $f^+: C^+\to A^+$ is a $\mathcal{D}$-(pre)cover of $A^+$. Some applications of this result are given. Keywords:(pre)envelopes, (pre)covers, duality, pure injective modules, character modulesCategories:18G25, 16E30

4. CMB 2013 (vol 57 pp. 506)

Galindo, César
 On Braided and Ribbon Unitary Fusion Categories We prove that every braiding over a unitary fusion category is unitary and every unitary braided fusion category admits a unique unitary ribbon structure. Keywords:fusion categories, braided categories, modular categoriesCategories:20F36, 16W30, 18D10

5. CMB 2011 (vol 55 pp. 319)

Jardine, J. F.
 The Verdier Hypercovering Theorem This note gives a simple cocycle-theoretic proof of the Verdier hypercovering theorem. This theorem approximates morphisms $[X,Y]$ in the homotopy category of simplicial sheaves or presheaves by simplicial homotopy classes of maps, in the case where $Y$ is locally fibrant. The statement proved in this paper is a generalization of the standard Verdier hypercovering result in that it is pointed (in a very broad sense) and there is no requirement for the source object $X$ to be locally fibrant. Keywords:simplicial presheaf, hypercover, cocycleCategories:14F35, 18G30, 55U35

6. CMB 2010 (vol 53 pp. 425)

Chapoton, Frédéric
 Free Pre-Lie Algebras are Free as Lie Algebras We prove that the $\mathfrak{S}$-module $\operatorname{PreLie}$ is a free Lie algebra in the category of $\mathfrak{S}$-modules and can therefore be written as the composition of the $\mathfrak{S}$-module $\operatorname{Lie}$ with a new $\mathfrak{S}$-module $X$. This implies that free pre-Lie algebras in the category of vector spaces, when considered as Lie algebras, are free on generators that can be described using $X$. Furthermore, we define a natural filtration on the $\mathfrak{S}$-module $X$. We also obtain a relationship between $X$ and the $\mathfrak{S}$-module coming from the anticyclic structure of the $\operatorname{PreLie}$ operad. Categories:18D50, 17B01, 18G40, 05C05

7. CMB 2009 (vol 52 pp. 273)

MacDonald, John; Scull, Laura
 Amalgamations of Categories We consider the pushout of embedding functors in $\Cat$, the category of small categories. We show that if the embedding functors satisfy a 3-for-2 property, then the induced functors to the pushout category are also embeddings. The result follows from the connectedness of certain associated slice categories. The condition is motivated by a similar result for maps of semigroups. We show that our theorem can be applied to groupoids and to inclusions of full subcategories. We also give an example to show that the theorem does not hold when the property only holds for one of the inclusion functors, or when it is weakened to a one-sided condition. Keywords:category, pushout, amalgamationCategories:18A30, 18B40, 20L17

8. CMB 2008 (vol 51 pp. 310)

Witbooi, P. J.
 Relative Homotopy in Relational Structures The homotopy groups of a finite partially ordered set (poset) can be described entirely in the context of posets, as shown in a paper by B. Larose and C. Tardif. In this paper we describe the relative version of such a homotopy theory, for pairs $(X,A)$ where $X$ is a poset and $A$ is a subposet of $X$. We also prove some theorems on the relevant version of the notion of weak homotopy equivalences for maps of pairs of such objects. We work in the category of reflexive binary relational structures which contains the posets as in the work of Larose and Tardif. Keywords:binary reflexive relational structure, relative homotopy group, exact sequence, locally finite space, weak homotopy equivalenceCategories:55Q05, 54A05;, 18B30

9. CMB 2008 (vol 51 pp. 81)

Kassel, Christian
 Homotopy Formulas for Cyclic Groups Acting on Rings The positive cohomology groups of a finite group acting on a ring vanish when the ring has a norm one element. In this note we give explicit homotopies on the level of cochains when the group is cyclic, which allows us to express any cocycle of a cyclic group as the coboundary of an explicit cochain. The formulas in this note are closely related to the effective problems considered in previous joint work with Eli Aljadeff. Keywords:group cohomology, norm map, cyclic group, homotopyCategories:20J06, 20K01, 16W22, 18G35

10. CMB 2007 (vol 50 pp. 440)

Raghuram, A.
 A KÃ¼nneth Theorem for $p$-Adic Groups Let $G_1$ and $G_2$ be $p$-adic groups. We describe a decomposition of ${\rm Ext}$-groups in the category of smooth representations of $G_1 \times G_2$ in terms of ${\rm Ext}$-groups for $G_1$ and $G_2$. We comment on ${\rm Ext}^1_G(\pi,\pi)$ for a supercuspidal representation $\pi$ of a $p$-adic group $G$. We also consider an example of identifying the class, in a suitable ${\rm Ext}^1$, of a Jacquet module of certain representations of $p$-adic ${\rm GL}_{2n}$. Categories:22E50, 18G15, 55U25

11. CMB 2007 (vol 50 pp. 182)

Chapoton, Frédéric
 On the Coxeter Transformations for Tamari Posets A relation between the anticyclic structure of the dendriform operad and the Coxeter transformations in the Grothendieck groups of the derived categories of modules over the Tamari posets is obtained. Categories:18D50, 18E30, 06A11

12. CMB 2006 (vol 49 pp. 407)

Jardine, J. F.
 Intermediate Model Structures for Simplicial Presheaves This note shows that any set of cofibrations containing the standard set of generating projective cofibrations determines a cofibrantly generated proper closed model structure on the category of simplicial presheaves on a small Grothendieck site, for which the weak equivalences are the local weak equivalences in the usual sense. Categories:18G30, 18F20, 55U35

13. CMB 2004 (vol 47 pp. 321)

Bullejos, M.; Cegarra, A. M.
 Classifying Spaces for Monoidal Categories Through Geometric Nerves The usual constructions of classifying spaces for monoidal categories produce CW-complexes with many cells that, moreover, do not have any proper geometric meaning. However, geometric nerves of monoidal categories are very handy simplicial sets whose simplices have a pleasing geometric description: they are diagrams with the shape of the 2-skeleton of oriented standard simplices. The purpose of this paper is to prove that geometric realizations of geometric nerves are classifying spaces for monoidal categories. Keywords:monoidal category, pseudo-simplicial category,, simplicial set, classifying space, homotopy typeCategories:18D10, 18G30, 55P15, 55P35, 55U40

14. CMB 2003 (vol 46 pp. 429)

Sastry, Pramathanath; Tong, Yue Lin L.
 The Grothendieck Trace and the de Rham Integral On a smooth $n$-dimensional complete variety $X$ over ${\mathbb C}$ we show that the trace map ${\tilde\theta}_X \colon\break H^n (X,\Omega_X^n) \to {\mathbb C}$ arising from Lipman's version of Grothendieck duality in \cite{ast-117} agrees with $$(2\pi i)^{-n} (-1)^{n(n-1)/2} \int_X \colon H^{2n}_{DR} (X,{\mathbb C}) \to {\mathbb C}$$ under the Dolbeault isomorphism. Categories:14F10, 32A25, 14A15, 14F05, 18E30

15. CMB 2002 (vol 45 pp. 180)

Connolly, Francis X.; Prassidis, Stratos
 On the Exponent of the ${\nk}_0$-Groups of Virtually Infinite Cyclic Groups It is known that the $K$-theory of a large class of groups can be computed from the $K$-theory of their virtually infinite cyclic subgroups. On the other hand, Nil-groups appear to be the obstacle in calculations involving the $K$-theory of the latter. The main difficulty in the calculation of Nil-groups is that they are infinitely generated when they do not vanish. We develop methods for computing the exponent of ${\nk}_0$-groups that appear in the calculation of the $K_0$-groups of virtually infinite cyclic groups. Categories:18F25, 19A31

16. CMB 2001 (vol 44 pp. 459)

Kahl, Thomas
 LS-catÃ©gorie algÃ©brique et attachement de cellules Nous montrons que la A-cat\'egorie d'un espace simplement connexe de type fini est inf\'erieure ou \'egale \a $n$ si et seulement si son mod\ele d'Adams-Hilton est un r\'etracte homotopique d'une alg\ebre diff\'erentielle \a $n$ \'etages. Nous en d\'eduisons que l'invariant $\Acat$ augmente au plus de 1 lors de l'attachement d'une cellule \`a un espace. We show that the A-category of a simply connected space of finite type is less than or equal to $n$ if and only if its Adams-Hilton model is a homotopy retract of an $n$-stage differential algebra. We deduce from this that the invariant $\Acat$ increases by at most 1 when a cell is attached to a space. Keywords:LS-category, strong category, Adams-Hilton models, cell attachmentsCategories:55M30, 18G55

17. CMB 2000 (vol 43 pp. 162)

Foth, Philip
 Moduli Spaces of Polygons and Punctured Riemann Spheres The purpose of this note is to give a simple combinatorial construction of the map from the canonically compactified moduli spaces of punctured complex projective lines to the moduli spaces $\P_r$ of polygons with fixed side lengths in the Euclidean space $\E^3$. The advantage of this construction is that one can obtain a complete set of linear relations among the cycles that generate homology of $\P_r$. We also classify moduli spaces of pentagons. Categories:14D20, 18G55, 14H10

18. CMB 2000 (vol 43 pp. 138)

Boyd, C.
 Exponential Laws for the Nachbin Ported Topology We show that for $U$ and $V$ balanced open subsets of (Qno) Fr\'echet spaces $E$ and $F$ that we have the topological identity $$\bigl( {\cal H}(U\times V), \tau_\omega \bigr) = \biggl( {\cal H} \Bigl( U; \bigl( {\cal H}(V), \tau_\omega \bigr) \Bigr), \tau_\omega \biggr).$$ Analogous results for the compact open topology have long been established. We also give an example to show that the (Qno) hypothesis on both $E$ and $F$ is necessary. Categories:46G20, 18D15, 46M05

19. CMB 2000 (vol 43 pp. 3)

 Resolutions of Associative and Lie Algebras Certain canonical resolutions are described for free associative and free Lie algebras in the category of non-associative algebras. These resolutions derive in both cases from geometric objects, which in turn reflect the combinatorics of suitable collections of leaf-labeled trees. Keywords:resolutions, homology, Lie algebras, associative algebras, non-associative algebras, Jacobi identity, leaf-labeled trees, associahedronCategories:18G10, 05C05, 16S10, 17B01, 17A50, 18G50

20. CMB 1997 (vol 40 pp. 39)

Zhao, Dongsheng
 On projective $Z$-frames This paper deals with the projective objects in the category of all $Z$-frames, where the latter is a common generalization of different types of frames. The main result obtained here is that a $Z$-frame is ${\bf E}$-projective if and only if it is stably $Z$-continuous, for a naturally arising collection ${\bf E}$ of morphisms. Categories:06D05, 54D10, 18D15