Pro-objects in the homotopy category

1969 ◽  
pp. 20-24
Author(s):  
M. Artin ◽  
B. Mazur
Keyword(s):  
Homotopy Category

scholarly journals Path homology theory of edge-colored graphs

2021 ◽  
Vol 19 (1) ◽  
pp. 706-723
Author(s):  
Yuri V. Muranov ◽  
Anna Szczepkowska
Keyword(s):  
Spectral Sequence ◽  
Edge Coloring ◽  
Homology Theory ◽  
Homotopy Category ◽  
Homology Groups ◽  
Colored Graphs ◽  
Natural Filtration ◽  
Definition Of ◽  
Exact Sequences

Abstract In this paper, we introduce the category and the homotopy category of edge-colored digraphs and construct the functorial homology theory on the foundation of the path homology theory provided by Grigoryan, Muranov, and Shing-Tung Yau. We give the construction of the path homology theory for edge-colored graphs that follows immediately from the consideration of natural functor from the category of graphs to the subcategory of symmetrical digraphs. We describe the natural filtration of path homology groups of any digraph equipped with edge coloring, provide the definition of the corresponding spectral sequence, and obtain commutative diagrams and braids of exact sequences.

scholarly journals The coherent homotopy category over a fixed space is a category of fractions

1991 ◽  
Vol 40 (3) ◽  
pp. 265-274 ◽  
Author(s):  
K.A. Hardie ◽  
K.H. Kamps ◽  
T. Porter
Keyword(s):  
Homotopy Category ◽  
Fixed Space ◽  
Category Of Fractions

scholarly journals The projective stable category of a coherent scheme

2018 ◽  
Vol 149 (1) ◽  
pp. 15-43 ◽  
Author(s):  
Sergio Estrada ◽  
James Gillespie
Keyword(s):  
Homotopy Category ◽  
Model Structure ◽  
Stable Category ◽  
Coherent Sheaves ◽  
Chain Complexes

We define the projective stable category of a coherent scheme. It is the homotopy category of an abelian model structure on the category of unbounded chain complexes of quasi-coherent sheaves. We study the cofibrant objects of this model structure, which are certain complexes of flat quasi-coherent sheaves satisfying a special acyclicity condition.

The Formalism of l-adic Sheaves

2019 ◽  
pp. 62-128
Author(s):  
Dennis Gaitsgory ◽  
Jacob Lurie
Keyword(s):  
Commutative Algebra ◽  
Mathematical Object ◽  
Product Formula ◽  
Homotopy Category ◽  
Triangulated Category

The ℓ-adic product formula discussed in Chapter 4 will need to make use of analogous structures, which are simply not visible at the level of the triangulated category Dℓ(X). This chapter attempts to remedy the situation by introducing a mathematical object Shvℓ (X), which refines the triangulated category Dℓ (X). This object is not itself a category but instead is an example of an ∞-category, which is referred to as the ∞-category of ℓ-adic sheaves on X. The triangulated category Dℓ (X) can be identified with the homotopy category of Shvℓ (X); in particular, the objects of Dℓ (X) and Shvℓ (X) are the same. However, there is a large difference between commutative algebra objects of Dℓ (X) and commutative algebra objects of the ∞-category Shvℓ (X). We can achieve (b') by viewing the complex B as a commutative algebra of the latter sort.

scholarly journals Pure derived and pure singularity categories

2019 ◽  
Vol 19 (06) ◽  
pp. 2050117
Author(s):  
Tianya Cao ◽  
Wei Ren
Keyword(s):  
Global Dimension ◽  
Derived Categories ◽  
Derived Category ◽  
Homotopy Category ◽  
Projective Modules ◽  
Strong Assumption ◽  
Singularity Category

Firstly, we compare the bounded derived categories with respect to the pure-exact and the usual exact structures, and describe bounded derived category by pure-projective modules, under a fairly strong assumption on the ring. Then, we study Verdier quotient of bounded pure derived category modulo the bounded homotopy category of pure-projective modules, which is called a pure singularity category since we show that it reflects the finiteness of pure-global dimension of rings. Moreover, invariance of pure singularity in a recollement of bounded pure derived categories is studied.

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