There are many interesting situations in which algebraic structure can be described by operads [1, 12, 13, 14, 17, 20, 27, 32, 33, 34, 35]. Let (C,⊗, k) be a symmetric monoidal closed category (Section 2) with all small limits and colimits. It is possible to define two types of operads (Definition 6.1) in this setting, as well as algebras and modules over these operads. One type, called Σ-opera...

Let k be a commutative ring. E∞ k-algebras are associative and commutative k-algebras up to homotopy, as codified in the action of an E∞ operad; A∞ k-algebras are obtained by ignoring permutations. Using a particularly well-behaved E∞ algebra, we explain an associative and commutative operadic tensor product that effectively hides the operad: an A∞ algebra or E∞ algebra A is defined in terms of...

Let k be a commutative ring and let C be the operad of differential graded k-modules obtained as the singular k-chains of the linear isometries operad [4, §V.9]. We show that the category of C-algebras is a proper closed model category. We use the amenable description of the coproduct in this category [4, V.3.4] to analyze the coproduct of and develop a homotopy theory for algebras over an arbi...

It is proved that for a commutative noetherian ring with dualizing complex the homotopy category of projective modules is equivalent, as a triangulated category, to the homotopy category of injective modules. Restricted to compact objects, this statement is a reinterpretation of Grothendieck’s duality theorem. Using this equivalence it is proved that the (Verdier) quotient of the category of ac...

We define equivariant periodic cyclic homology for bornological quantum groups. Generalizing corresponding results from the group case, we show that the theory is homotopy invariant, stable and satisfies excision in both variables. Along the way we prove Radfords formula for the antipode of a bornological quantum group. Moreover we discuss anti-Yetter-Drinfeld modules and establish an analogue ...

Homotopy type theory is a recent research area connecting type theory with homotopy theory by interpreting types as spaces. In particular, one can prove and mechanize type-theoretic analogues of homotopy-theoretic theorems, yielding “synthetic homotopy theory”. Here we consider the Seifert–van Kampen theorem, which characterizes the loop structure of spaces obtained by gluing. This is useful in...

In this paper we introduce a generalization of M-small modules and discuss about the torsion theory cogenerated by this kind of modules in category . We will use the structure of the radical of a module in  and get some suitable results about this class of modules. Also the relation between injective hull in  and this kind of modules will be investigated in this article.   For a module  we show...