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Construction junctions are diamonds. They show where multiple structures must coexist on the same carrier and satisfy compatibility conditions.

Canonical static concept record

Graded object

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Summary

A family of objects indexed by degrees, usually integers or natural numbers.

Record metadata

Canonical name: Graded object
Concept ID: graded_object
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Homological algebra
Domains: Homological algebra, Category theory, Algebra
Alternate terminology: No alternate terminology is currently authored for this record.

Carrier(s)

objects $A_{n}$ in a category

Data

degree index

Concept sources

Wikipedia — Graded vector space — Graded vector space · expository · source ID wp-graded-object
nLab — Graded object — Graded object · reference · source ID nlab-graded-object
Stacks Project — Derived Categories (PDF) — Derived Categories (PDF) · reference text · source ID stacks-derived-text

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

Cohomology ring Graded object

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: story_cohomology_ring_to_graded_object_induced_forgotten
Relation type: Induced / forgotten induced-forgotten
Direction: source → target
Endpoint roles: source: Yields by induction / forgetting; target: Obtained by induction / forgetting from
Authored annotation: has an underlying graded object

Authored explanation

The cup product equips the direct sum of cohomology groups with a graded ring structure; forgetting multiplication leaves a graded object.

How to interpret this relation type

Pass canonically from a stronger object to structure it determines, or forget part of the data while retaining a valid weaker structure. The carrier may change under a canonical induced construction.

Relation sources

Hatcher — Algebraic Topology — Algebraic Topology · textbook · source ID hatcher-algebraic-topology
Wikipedia — Graded vector space — Graded vector space · expository · source ID wp-graded-object
nLab — Graded object — Graded object · reference · source ID nlab-graded-object

Indexed family Graded object

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: e_family_graded
Relation type: Add data add-data
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: equip the index set with degree indexing

Authored explanation

Identify or replace the index set by a degree set, usually $\mathbb{Z}$ or $\mathbb{N}$ .

How to interpret this relation type

Equip an existing carrier or structured object with additional chosen data, when such compatible data exists. Use a construction junction when several independently meaningful inputs must coexist on the same carrier or interact compatibly.

Relation sources

Stacks Project — Graded objects — Graded objects · reference · source ID stacks-graded-objects

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.

Graded object Chain complex

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: e_graded_chain
Relation type: Add data add-data
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: + differential $d$ of degree $-1$

Authored explanation

Add boundary maps with $d^{2}=0$ .

How to interpret this relation type

Relation sources

Stacks Project — Complexes — Complexes · reference · source ID stacks-complexes
Wikipedia — Chain complex — Chain complex · expository · source ID wp-chain-complex
nLab — Chain complex — Chain complex · reference · source ID nlab-chain-complex
Stacks Project — Derived Categories (PDF) — Derived Categories (PDF) · reference text · source ID stacks-derived-text

Graded object Cochain complex

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: e_graded_cochain
Relation type: Add data add-data
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: + differential $d$ of degree $+1$

Authored explanation

Add coboundary maps with $d^{2}=0$ .

How to interpret this relation type

Relation sources

Stacks Project — Complexes — Complexes · reference · source ID stacks-complexes
Wikipedia — Chain complex — Chain complex · expository · source ID wp-chain-complex
nLab — Chain complex — Chain complex · reference · source ID nlab-chain-complex
Stacks Project — Derived Categories (PDF) — Derived Categories (PDF) · reference text · source ID stacks-derived-text

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Graded object

Summary

Record metadata

Carrier(s)

Data

Concept sources

Incoming relations (arrows to this concept)

Cohomology ring Graded object

Authored explanation

How to interpret this relation type

Relation sources

Indexed family Graded object

Authored explanation

How to interpret this relation type

Relation sources

Outgoing relations (arrows from this concept)

Graded object Chain complex

Authored explanation

How to interpret this relation type

Relation sources

Graded object Cochain complex

Authored explanation

How to interpret this relation type

Relation sources