The flat model structure on complexes of sheaves

Abstract

Abstract. Let Ch(O) be the category of chain complexes of O-modules on a topological space T (where O is a sheaf of rings on T ). We put a Quillen model structure on this category in which the cofibrant objects are built out of flat modules. More precisely, these are the dg-flat complexes. Dually, the fibrant objects will be called dg-cotorsion complexes. We show that this model structure is monoidal, solving the previous problem of not having any monoidal model structure on Ch(O). As a corollary, we have a general framework for doing homological algebra in the category Sh(O) of O-modules. I.e., we have a natural way to define the functors Ext and Tor in Sh(O).

Balanced pairs on triangulated categories

“…Assume that (X , Y ) is a cotorsion pair in ModR, that is Y = {M : Ext 1 R (X, M ) = 0 for every X ∈ X } and X = {N : Ext 1 R (N, Y ) = 0 for every Y ∈ Y }. We introduce the following classes, which comes from [7]:…”

Cartan-Eilenberg Gorenstein Flat Complexes

“…for all i ≥ 1 with F −1 = C. Then Ext 1 (F, K i ) = 0 for any flat complex F , and it is easy to see that K i is exact for all i ≥ 0. Thus it follows from [10, Proposition 4.3.3 (1)] and [11,Theorem 3.12] that all K i are C-E cotorsion complexes for i ≥ 0, and so Ext 1 (G, K i ) = 0 for any C-E flat complex G by [4, Theorem 9.4]. We note that the sequence Recall that a complex P is finitely generated if, in case P = λ∈ P λ with P λ subcomplexes of P , then there exists a finite subset F ⊆ such that P = λ∈F P λ .…”

The politics of user-platform relationships: Co-scripting live-streaming on Twitch.tv

“…For example, Gillespie shows how YouTube, through its infrastructure, dealt with troubling sexual content in three ways: introducing a new standard to remove inappropriate videos; assigning certain videos to an 'adult' category; and algorithmically demoting these videos from 'Most Viewed', 'Top Favourited' and other browsing pages. However, Gillespie's most important point is that YouTube uses the term 'platform' to neutralise contradictions: 'Whatever possible tension there is between being a platform for empowering individual users and being a robust marketing platform and being a platform for major studio content is elided in the versatility of the term and the powerful appeal of the idea behind it' [9]. Consequently, the investigations of platforms should untangle diverging interests, and critically investigate the naturalizing efforts made by platform owners.…”