## Reedy fibrant and cofibrant objects

Let C be a Reedy category, M be a model category. Then consider Fun(C,M), the category of diagrams from C to M equipped with the Reedy model structure. I wonder if there is a simple characterization of fibrant, cofibrant objects in this category. Let 0,1 be the initial and terminal object in Fun(C,M). It seems … Read more

## Universal enveloping algebra functor preserves quasi-isomorphism

Let k be a field of characteristic 0. Let DGAk denote the category of DG algebras and DGLAk denote the category of DG Lie algebras. It is well known that there are model structures on them that the weak equivalences are the quasi-isomorphisms and the fibrations are the maps which are degreewise surjective. We have … Read more

## Is there a schema category for hyperstructures?

I am completely fascinated by Niels Baas’ notion of hyperstructures, chiefly because I can see how such gadgets could be used in modeling both biological and social systems, or other evolutionary systems (the basic intuition being that they describe complex systems obtained by “glueing” lower cells of arbitrary shapes). Now, here is my question: there … Read more

## Model structure on spaces with local coefficients

Is there a model structure on the category of topological spaces equipped with a local system (i.e. a functor from the fundamental groupoid to the category of abelian groups), such that the weak equivalences are the isomorphisms in homology with local coefficients? If so, is there a reference in which the model structure is established? … Read more

## Conditions on a Quillen functor so that its comonad is homotopy-full

I am looking for an answer to the following question: Let F:C→D be a left Quillen functor between combinatorial model categories; let ˜F⊣˜G be the induced adjunction between the homotopy categories. Which assumptions[1] on F ensure that the comonad ˜F˜G:Ho(D)→Ho(D) is a full functor? [1] I already know that F=1 works. 🙂 Answer AttributionSource : … Read more

## Which models are available for the motivic homotopy category SHS1(k)SH^{S^1}(k)

The motivic S1-stable homotopy category SHS1(k) (where k is a field that is often assumed to be perfect; yet one can probably take quite general base schemes here) is “intermediate” between the unstable motivic category H(k) and SH(k), and it appears to be rather “unpopular”. What are the main references for it and for Quillen … Read more

## π0\pi_0 in arbitrary category of simplicial objects

Let C be a category (let it be pointed and cocomplete) such that the category of simplicial objects sC is a model category. In particular, I’m interested in two cases: C is an “algebraic” category with some notion of exactness, which satisfy conditions of Quillen’s theorem II.4 C is a model category itself – in … Read more

## category of simplicial filters

I am looking for references discussing the category Filt of filters (in the sense of set theory, details below), its simplicial category sFilt and its full subcategory Top↪sFilt of topological spaces. What is known about sFilt and the embedding Top↪sFilt? Is there a model structure on sFilt compatible with the embedding? Was it discussed in … Read more

## Building conilpotent coalgebras from co-square-zero-extensions

Let K be a field of char. 0. Given a chain complex X over K denote E(X) the co-square-zero-extension on X, i.e. the cocommutative non-counital dg-coalgebra structure on X with zero-comultiplication. We think of the E(X) as the most basic conilpotent cocommutative non-counital dg-coalgebras. Is it true that every conilpotent cocommutative non-counital dg-coalgebra can be … Read more

## Does a fibrant simplicial set give fibrant diagram

If Y is a fibrant simplicial set and Δ∙ is the cosimplicial simplicial set, is Y(Δ∙) (i.e. nth simplicial set is n↦Y(Δn)=Hom(Δn,Y)) a (Reedy) fibrant bisimplicial set? Answer AttributionSource : Link , Question Author : Girish , Answer Author : Community