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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

Simplicial object

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Summary

A contravariant functor from the simplex category $\Delta$ to a category.

Record metadata

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

Carrier(s)

category $C$
objects $X_{n}$ of $C$ for $n\ge 0$

Data

face and degeneracy morphisms

Axioms / constraints

simplicial identities

Notes

Equivalently, a functor $\Delta^{\mathrm{op}}\to C$ . The ambient category is part of the specification.

Concept sources

Wikipedia — Cosimplicial object — Cosimplicial object · expository · source ID wp-cosimplicial-object
nLab — Simplicial object — Simplicial object · reference · source ID nlab-simplicial-object
Stacks Project — Simplicial objects and chain complexes — Simplicial objects and chain complexes · reference · source ID stacks-simplicial-objects
Riehl — Category Theory in Context — Category Theory in Context · textbook · source ID riehl-category-context

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

category + simplicial diagram data(construction junction)Simplicial object

Permalink to relation

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

Relation ID: j_simplicial_object__out
Relation type: Combine compatible structures combine-compatible
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: impose simplicial identities

Authored explanation

Every $X_{n}$ is an object of the same category $C$ ; add face and degeneracy morphisms in $C$ satisfying the simplicial identities.

How to interpret this relation type

Feed several structures into a construction junction and impose compatibility between them.

Relation sources

Stacks Project — Simplicial objects and chain complexes — Simplicial objects and chain complexes · reference · source ID stacks-simplicial-objects
nLab — Simplicial object — Simplicial object · reference · source ID nlab-simplicial-object
Riehl — Category Theory in Context — Category Theory in Context · textbook · source ID riehl-category-context

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.