Reedy category¶

In mathematics, especially category theory, a Reedy category is a category R that has a structure so that the functor category from R to a model category M would also get the induced model category structure.

Version: v1 · 2026-09-28 · History
Domain-specific #: 11708
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomains: Category Theory, Homotopy Theory → Mathematics

Core Idea¶

Reedy category is treated here as the recurring category theory identity summarized by this source-grounded definition: In mathematics, especially category theory, a Reedy category is a category R that has a structure so that the functor category from R to a model category M would also get the induced model category structure.

In mathematics, especially category theory, a Reedy category is a category R that has a structure so that the functor category from R to a model category M would also get the induced model category structure. A prototypical example is the simplex category or its opposite. It was introduced by Christopher Reedy in his unpublished manuscript.

A Reedy category consists of the following data: a category R, two wide (lluf) subcategories R_-, R_+ and a functorial factorization of each map into a map in R_- followed by a map in R_+ that are subject to the condition: for some total preordering (degree), the nonidentity maps in R_-, R_+ lower or raise degrees. A Reedy model structure is a canonical model-category structure placed on the functor category M^R when R is a Reedy category and M is a model category. An Eilenberg–Zilber category is a variant of a Reedy category.

For Reedy category, the abstraction is narrower than the article's general subject matter: a positive case must preserve In mathematics, especially category theory, a Reedy category is a category R that has a structure so that the functor category from R to a model category M would also get the induced model category structure. Retaining only the name, a familiar example, or a downstream effect is insufficient. The specialist roles and tests remain anchored in category theory, which is why this identity is domain-specific rather than prime.

Structural Signature¶

Sig role-phrases:

Defining carrier — A Reedy category consists of the following data: a category R, two wide (lluf) subcategories R_-, R_+ and a functorial factorization of each map into a map in R_- followed by a map in R_+ that are subject to the condition: for some total preordering (degree), the nonidentity maps in R_-, R_+ lower or raise degrees.
Constitutive relation — It was introduced by Christopher Reedy in his unpublished manuscript.
Operating condition — Note some authors such as nlab require each factorization to be unique.
Recognition evidence — A Reedy model structure is a canonical model-category structure placed on the functor category M^R when R is a Reedy category and M is a model category.
Admissible variation — An Eilenberg–Zilber category is a variant of a Reedy category.
Characteristic consequence — In mathematics, especially category theory, a Reedy category is a category R that has a structure so that the functor category from R to a model category M would also get the induced model category structure.
Failure boundary — A prototypical example is the simplex category or its opposite.

What It Is Not¶

Not the whole field of category theory. The node requires the specific identity stated by In mathematics, especially category theory, a Reedy category is a category R that has a structure so that the functor category from R to a model category M would also get the induced model category structure.
Not an over-broad reading. Note some authors such as nlab require each factorization to be unique.
Not an over-broad reading. A Reedy model structure is a canonical model-category structure placed on the functor category M^R when R is a Reedy category and M is a model category.
Not an over-broad reading. A Reedy category consists of the following data: a category R, two wide (lluf) subcategories R_-, R_+ and a functorial factorization of each map into a map in R_- followed by a map in R_+ that are subject to the condition: for some total preordering (degree), the nonidentity maps in R_-, R_+ lower or raise degrees.
Not automatically Simplex category. Retrieval proximity does not establish equivalence; the two identities must be compared by carrier, operation, and failure boundary.

Scope of Application¶

Reedy category applies literally inside category theory wherever the source-defined carrier and relation can be established. Its documented habitats include:

Definition. Note some authors such as nlab require each factorization to be unique.
Reedy model structure. A Reedy model structure is a canonical model-category structure placed on the functor category M^R when R is a Reedy category and M is a model category.
Definition. A Reedy category consists of the following data: a category R, two wide (lluf) subcategories R_-, R_+ and a functorial factorization of each map into a map in R_- followed by a map in R_+ that are subject to the condition: for some total preordering (degree), the nonidentity maps in R_-, R_+ lower or raise degrees.
Eilenberg–Zilber category. An Eilenberg–Zilber category is a variant of a Reedy category.
Documented setting. In mathematics, especially category theory, a Reedy category is a category R that has a structure so that the functor category from R to a model category M would also get the induced model category structure.
Documented setting. A prototypical example is the simplex category or its opposite.

Outside category theory, the name should be retained only when these same operational conditions survive; otherwise the comparison belongs to the broader parent Pattern or should be marked as analogy.

Clarity¶

A clear use of Reedy category names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is In mathematics, especially category theory, a Reedy category is a category R that has a structure so that the functor category from R to a model category M would also get the induced model category structure. The strongest recognition evidence in the frozen account is: A Reedy model structure is a canonical model-category structure placed on the functor category M^R when R is a Reedy category and M is a model category. A report should distinguish that evidence from a proxy, consequence, or common implementation. It should also state the qualification Note some authors such as nlab require each factorization to be unique. so that a reader can reproduce the classification rather than infer it from topical resemblance.

Manages Complexity¶

Reedy category compresses multiple category theory details into a stable diagnostic relation. The source shows both the central mechanism—it was introduced by Christopher Reedy in his unpublished manuscript.—and the practical consequence—in mathematics, especially category theory, a Reedy category is a category R that has a structure so that the functor category from R to a model category M would also get the induced model category structure. This compression makes cases comparable while leaving parameters, conventions, exceptions, and evidential quality explicit. It is lossy by design: local history and implementation details may be omitted only when they do not alter the defining relation.

Abstract Reasoning¶

Type the carrier. Identify the category theory entities to which the claim applies.
State the relation. Use the source-grounded identity: In mathematics, especially category theory, a Reedy category is a category R that has a structure so that the functor category from R to a model category M would also get the induced model category structure.
Check operation and conditions. Note some authors such as nlab require each factorization to be unique.
Demand recognition evidence. A Reedy model structure is a canonical model-category structure placed on the functor category M^R when R is a Reedy category and M is a model category.
Test variation. Change an implementation or setting while preserving an Eilenberg–Zilber category is a variant of a Reedy category.
Run the collapse test. Remove the defining operation; if the label still seems equally apt, only a topic or correlate was retained.
Reduce cautiously. When the specialist conditions cannot be carried, route the residual comparison to Pattern.

Knowledge Transfer¶

Within the home domain. Knowledge about Reedy category transfers literally when a new case preserves the same carrier type, relation, and recognition test. Note some authors such as nlab require each factorization to be unique. A Reedy model structure is a canonical model-category structure placed on the functor category M^R when R is a Reedy category and M is a model category.

Beyond the home domain. No canonical parent is asserted for Reedy category. An outside case receives the specialist name only when the same typed roles and rejection conditions can be filled literally; otherwise the comparison remains an analogy pending later graph densification.

Examples¶

Canonical¶

Note some authors such as nlab require each factorization to be unique. This case is canonical because it supplies a concrete carrier and lets the defining relation be checked rather than merely named.

Mapped back: carrier → the entities in the documented case; operation → In mathematics, especially category theory, a Reedy category is a category R that has a structure so that the functor category from R to a model category M would also get the induced model category structure; recognition evidence → A Reedy model structure is a canonical model-category structure placed on the functor category M^R when R is a Reedy category and M is a model category

Applied / In Practice¶

A Reedy model structure is a canonical model-category structure placed on the functor category M^R when R is a Reedy category and M is a model category. The applied case shows how the identity is used under a second setting or qualification while keeping the same operative relation.

Mapped back: changed setting → Reedy model structure; invariant → In mathematics, especially category theory, a Reedy category is a category R that has a structure so that the functor category from R to a model category M would also get the induced model category structure; boundary → the case exits the class when note some authors such as nlab require each factorization to be unique

Structural Tensions¶

T1 — Stable identity versus admissible variation. Note some authors such as nlab require each factorization to be unique. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Which changes preserve the defining relation, and which replace it?

T2 — Recognition versus proxy. A Reedy model structure is a canonical model-category structure placed on the functor category M^R when R is a Reedy category and M is a model category. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the cited evidence establish the identity or only a correlated sign?

T3 — Definition versus implementation. A Reedy category consists of the following data: a category R, two wide (lluf) subcategories R_-, R_+ and a functorial factorization of each map into a map in R_- followed by a map in R_+ that are subject to the condition: for some total preordering (degree), the nonidentity maps in R_-, R_+ lower or raise degrees. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Is the observed implementation constitutive, optional, or merely common?

T4 — Scope versus overextension. An Eilenberg–Zilber category is a variant of a Reedy category. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Can every claimed application fill the same typed roles without metaphor?

T5 — Transfer versus domain accent. A Reedy category consists of the following data: a category R, two wide (lluf) subcategories R_-, R_+ and a functorial factorization of each map into a map in R_- followed by a map in R_+ that are subject to the condition: for some total preordering (degree), the nonidentity maps in R_-, R_+ lower or raise degrees. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the receiving case instantiate Reedy category literally, co-instantiate Pattern, or only resemble it?

T6 — Autonomy versus reduction. It was introduced by Christopher Reedy in his unpublished manuscript. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: What does Reedy category distinguish that the broader parent Pattern leaves together?

Structural–Framed Character¶

Reedy category is mixed or framed-leaning. Its structural side is the repeatable organization summarized by In mathematics, especially category theory, a Reedy category is a category R that has a structure so that the functor category from R to a model category M would also get the induced model category structure. Its framed side is the category theory vocabulary that fixes the carrier, evidence, exceptions, and admissible transformations.

Evaluative weight: the identity can be stated descriptively even when applications carry practical stakes. Human-practice dependence: the source-grounded carrier determines whether the relation exists independently or is constituted by a practice. Institutional origin: disciplinary conventions stabilize the name and test. Vocabulary portability: Note some authors such as nlab require each factorization to be unique. Import versus recognition: literal transfer requires the same mechanism; shape alone is analogy.

Its portable skeleton is Pattern. Its character: a recurring specialist identity whose thin organization can be abstracted, while its operational meaning remains domain-bound.

Structural Core vs. Domain Accent¶

What is skeletal. In mathematics, especially category theory, a Reedy category is a category R that has a structure so that the functor category from R to a model category M would also get the induced model category structure. The stable skeleton is the typed relation expressed in that definition and the entry's recognition and collapse tests. The source identifies these operative conditions: A Reedy category consists of the following data: a category R, two wide (lluf) subcategories R-, R+ and a functorial factorization of each map into a map in R- followed by a map in R+ that are subject to the condition: for some total preordering (degree), the nonidentity maps in R-, R+ lower or raise degrees. It was introduced by Christopher Reedy in his unpublished manuscript. It further constrains recognition and variation through: Note some authors such as nlab require each factorization to be unique. A Reedy model structure is a canonical model-category structure placed on the functor category M^R when R is a Reedy category and M is a model category.

What is domain-bound. category theory supplies the operative entities, technical vocabulary, warrants, and exceptions that make Reedy category literal. Its documented scope includes the condition that Note some authors such as nlab require each factorization to be unique. Another bounded application condition is that A Reedy model structure is a canonical model-category structure placed on the functor category M^R when R is a Reedy category and M is a model category. These are not decorative examples; they determine which carrier and evidence can fill the abstraction's roles.

Why no parent is asserted. Removing those specialist details does not currently yield one live catalog node that is a necessary genus for every instance. The entry is therefore approved as unparented rather than attached by topical resemblance. Its collapse evidence remains specific—An Eilenberg–Zilber category is a variant of a Reedy category.—and future graph densification may discover a defensible relation only if it preserves that boundary.

This entry is a kind of Category.

Approved unparented node. No current live node supplies a defensible necessary genus or structural prerequisite for Reedy category. The reviewed identity is: In mathematics, especially category theory, a Reedy category is a category R that has a structure so that the functor category from R to a model category M would also get the induced model category structure. The accelerated suggestion was declined because topical or lexical similarity does not establish hierarchy; the node is admitted without a parent pending later graph densification.
Related reasoning operations. Evidence, representation, comparison, classification, transformation, or evaluation may participate in particular cases, but participation does not make any one of them a necessary parent of every instance.

Relationships to Other Abstractions¶

Current abstraction Reedy category Domain-specific

Parents (1) — more general patterns this builds on

Reedy category is a kind of Category Prime

Reedy category is a domain-specific kind of category under its frozen identity and differentia. Complete-catalog comparison found the corresponding live broader identity.

Hierarchy paths (3) — routes to 3 parentless roots

Reedy category → Category → Associativity → Invariance

Show alternative paths (2)

Neighborhood in Abstraction Space¶

Reedy category sits in a moderately populated region (57^th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Formal Grammar & Syntactic Structure (16 abstractions)

Nearest neighbors

Adequate subcategory — 0.85
Profunctor — 0.85
Kripke–Platek set theory with urelements — 0.85
Conjunctive grammar — 0.85
Subfunctor — 0.85

Computed from structural-signature embeddings · 2026-10-08

Not to Be Confused With¶

Pattern. The parent omits the specialist differentia. Tell: Can the case establish In mathematics, especially category theory, a Reedy category is a category R that has a structure so that the functor category from R to a model category M would also get the induced model category structure?
Simplex category. The category Δ of nonempty finite ordinals [n] and order-preserving maps, whose functors into or out of another category define simplicial and cosimplicial objects. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Homotopy Category. A category that retains objects while replacing maps by homotopy classes or, more generally, formally inverting a designated class of weak equivalences. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Localizing Subcategory. A Serre subcategory whose exact quotient functor admits a right-adjoint section, so the objects it annihilates define a recoverable Gabriel localization of an abelian category. Tell: Which entry's carrier, operation, and failure condition are satisfied?
A measurement, proxy, or consequence. Those may provide evidence without being the identity. Tell: Would Reedy category remain present if the detector or downstream effect changed?
A metaphorical analogue. A similar shape outside category theory lacks the specialist mechanism. Tell: Do the native roles transfer literally, or only the parent Pattern?

References¶

Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Reedy_category (revision 1347481306).
Preserved source candidate: https://math.mit.edu/~psh/
Preserved source candidate: https://ncatlab.org/nlab/show/Reedy+category
Preserved source candidate: https://mathoverflow.net/questions/176983/the-definition-of-reedy-category
Preserved source candidate: https://cisinski.app.uni-regensburg.de/CatLR.pdf
Preserved source candidate: http://pantodon.jp/index.rb?body=Reedy_category

The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.