For every stable model category M with a certain extra structure, we produce an associated model structure on the pro-category pro-M and a spectral sequence, analogous to the Atiyah-Hirzebruch spectral sequence, with reasonably good convergence properties for computing in the homotopy category of pro-M. Our motivating example is the category of pro-spectra. The extra structure referred to above...

In this article, we introduce the concept of co-covering which is the dual of covering concept. Then, we prove several theorems being similar to the theorems that have been de-veloped for the covering concept. For example, we provide the lifting criterion for co-coverings of a topological space X, which helps us to classify them by subgroups of the group of all homeomorphisms of X.

The definition of a category of (T,V)-algebras, where V is a unital commutative quantale and T is a Set-monad, requires the existence of a certain lax extension of T. In this article, we present a general construction of such an extension. This leads to the formation of two categories of (T,V)-algebras: the category Alg(T,V) of canonical (T,V)-algebras, and the category Alg(T′,V) of op-canonica...

If S is an order-adjoint monad, that is, a monad on Set that factors through the category of ordered sets with left adjoint maps, then any monad morphism τ : S → T makes T orderadjoint, and the Eilenberg-Moore category of T is monadic over the category of monoids in the Kleisli category of S.

Let T be a Tannakian category with a fiber functor ω : T → VerC, where VerC denotes the category of finite dimensional C-vector spaces. An object t ∈ T is called reducible if there exist non-zero objects x, y ∈ T such that t = x ⊕ y. An object is called irreducible if it is not reducible. If moreover every object x of T can be written uniquely as a sum of irreducible objects x = x1 ⊕ x2 ⊕ . . ....

We define the Chow $t$-structure on $\infty$-category of motivic spectra $\mathcal{SH}(k)$ over an arbitrary base field $k$. identify heart this $\mathcal{SH}(k)^{c\heartsuit}$ when exponential characteristic $k$ is inverted. Restricting to cellular subcategory, we $\mathcal{SH}(k)^{\mathrm{cell},c\heartsuit}$ as category even graded $\mathrm{MU}_{2*}\mathrm{MU}$-comodules. Furthermore, show th...

We study categorical models for the unitless fragment of multiplicative linear logic. We find that the appropriate notion of model is a special kind of pro-monoidal category. Since the theory of promonoidal categories has not been developed very thoroughly, at least in the published literature, we need to develop it here. The most natural way to do this – and the simplest, once the (substantial...