Coherency (homotopy theory)

In some situations, isomorphisms need to be chosen in a coherent way. Often, this can be achieved by choosing canonical isomorphisms. But in some cases, such as prestacks, there can be several canonical isomorphisms and there might not be an obvious choice among them.

In practice, coherent isomorphisms arise by weakening equalities; e.g., strict associativity may be replaced by associativity via coherent isomorphisms. For example, via this process, one gets the notion of a weak 2-category from that of a strict 2-category.

Replacing coherent isomorphisms by equalities is usually called strictification or rectification.

A coherence condition is a collection of conditions requiring that various compositions of elementary morphisms are equal. Typically the elementary morphisms are part of the data of the category. A coherence theorem states that, in order to be assured that all these equalities hold, it suffices to check a small number of identities.

Part of the data of a monoidal category is a chosen morphism

$\backslash alpha\_\{A,B,C\}$, called the associator:

: $\backslash alpha\_\{A,B,C\}\; \backslash colon\; (A\backslash otimes\; B)\backslash otimes\; C\; \backslash rightarrow\; A\backslash otimes(B\backslash otimes\; C)$

for each triple of objects $A,\; B,\; C$ in the category. Using compositions of these $\backslash alpha\_\{A,B,C\}$, one can construct a morphism

: $(\; (\; A\_N\; \backslash otimes\; A\_\{N-1\}\; )\; \backslash otimes\; A\_\{N-2\}\; )\; \backslash otimes\; \backslash cdots\; \backslash otimes\; A\_1)\; \backslash rightarrow\; (\; A\_N\; \backslash otimes\; (\; A\_\{N-1\}\; \backslash otimes\; \backslash cdots\; \backslash otimes\; (\; A\_2\; \backslash otimes\; A\_1)\; ).$

Actually, there are many ways to construct such a morphism as a composition of various $\backslash alpha\_\{A,B,C\}$. One coherence condition that is typically imposed is that these compositions are all equal.{{harv|Kelly|1964|loc=Introduction}}

Typically one proves a coherence condition using a coherence theorem, which states that one only needs to check a few equalities of compositions in order to show that the rest also hold. In the above example, one only needs to check that, for all quadruples of objects $A,B,C,D$, the following diagram commutes.

Image:Monoidal category pentagon.svg

Any pair of morphisms from $(\; (\; \backslash cdots\; (\; A\_N\; \backslash otimes\; A\_\{N-1\}\; )\; \backslash otimes\; \backslash cdots\; )\; \backslash otimes\; A\_2\; )\; \backslash otimes\; A\_1)$ to $(\; A\_N\; \backslash otimes\; (\; A\_\{N-1\}\; \backslash otimes\; (\; \backslash cdots\; \backslash otimes\; (\; A\_2\; \backslash otimes\; A\_1)\; \backslash cdots\; )\; )$ constructed as compositions of various $\backslash alpha\_\{A,B,C\}$ are equal.

Two simple examples that illustrate the definition are as follows. Both are directly from the definition of a category.

Let {{nowrap|f : A → B}} be a morphism of a category containing two objects A and B. Associated with these objects are the identity morphisms {{nowrap|1_A : A → A}} and {{nowrap|1_B : B → B}}. By composing these with f, we construct two morphisms:

:{{nowrap|f o 1_A : A → B}}, and

:{{nowrap|1_B o f : A → B}}.

Both are morphisms between the same objects as f. We have, accordingly, the following coherence statement:

:{{nowrap|1= f o 1_A = f{{Hair space}} = 1_B o f}}.

Let {{nowrap|f : A → B}}, {{nowrap|g : B → C}} and {{nowrap|h : C → D}} be morphisms of a category containing objects A, B, C and D. By repeated composition, we can construct a morphism from A to D in two ways:

:{{nowrap|(h o g) o f : A → D}}, and

:{{nowrap|h o (g o f) : A → D}}.

We have now the following coherence statement:

:{{nowrap|1= (h o g) o f = h o (g o f)}}.

In these two particular examples, the coherence statements are theorems for the case of an abstract category, since they follow directly from the axioms; in fact, they are axioms. For the case of a concrete mathematical structure, they can be viewed as conditions, namely as requirements for the mathematical structure under consideration to be a concrete category, requirements that such a structure may meet or fail to meet.

Mac Lane's coherence theorem states, roughly, that if diagrams of certain types commute, then diagrams of all types commute.{{harvnb|Mac Lane|1978|loc=Chapter VII, Section 2}} A simple proof of that theorem can be obtained using the permutoassociahedron, a polytope whose combinatorial structure appears implicitly in Mac Lane's proof.See {{harvnb|Kapranov|1993}} and {{harvnb|Reiner|Ziegler|1994}}

There are several generalizations of Mac Lane's coherence theorem.See, for instance [https://ncatlab.org/nlab/show/coherence+theorem coherence theorem (nlab)] Each of them has the rough form that "every weak structure of some sort is equivalent to a stricter one".{{harvnb|Shulman|2012|loc=Section 1}}

{{expand section|date=September 2019}}

• Coherence condition
• Canonical isomorphism
• 2-category
• Pseudoalgebra
• Tricategory

{{reflist}}

{{refbegin}}

• {{cite journal

| last1=Cordier | first1=Jean-Marc

| last2=Porter | first2=Timothy

| title=Homotopy coherent category theory

| journal=Transactions of the American Mathematical Society

| doi=10.1090/S0002-9947-97-01752-2 | doi-access=free

| volume=349

| issue=1

| date=1997

| pages=1–54}}

• § 5. of {{cite journal

| last1=Mac Lane | first1=Saunders | authorlink1=Saunders Mac Lane

| title=Topology and Logic as a Source of Algebra (Retiring Presidential Address)

| journal=Bulletin of the American Mathematical Society

| volume=82

| issue=1

| date=January 1976

| pages=1–40

| doi=10.1090/S0002-9904-1976-13928-6 | doi-access=free}}

• {{cite book

| last1=Mac Lane | first1=Saunders | authorlink1=Saunders Mac Lane

| orig-year=1971

| title=Categories for the working mathematician

| series=Graduate texts in mathematics

| publisher=Springer-Verlag

| date=1978

| volume=5 | doi=10.1007/978-1-4757-4721-8 | doi-access=free| isbn=978-1-4419-3123-8 }}

• Ch. 5 of {{cite book | last1=Kamps | first1=Klaus Heiner | last2=Porter | first2=Timothy | title=Abstract Homotopy and Simple Homotopy Theory | doi=10.1142/2215 | publisher=World Scientific | date=April 1997 | isbn=9810216025}}
• {{cite journal|first=Mike |last=Shulman |title=Not every pseudoalgebra is equivalent to a strict one |journal=Advances in Mathematics |volume=229 |number=3 |year=2012 |pages=2024–2041 |arxiv=1005.1520 |doi=10.1016/j.aim.2011.01.010 |doi-access=free}}
• {{cite journal

| first1=Mikhail M. | last1=Kapranov | authorlink1=Mikhail Kapranov

| title=The permutoassociahedron, Mac Lane's coherence theorem and asymptotic zones for the KZ equation

| journal=Journal of Pure and Applied Algebra

| volume=85

| issue=2

| date=1993

| pages=119–142

| doi=10.1016/0022-4049(93)90049-Y | doi-access=}}

• {{cite journal

| last1=Reiner | first1=Victor

| last2=Ziegler | first2=Günter M. | authorlink2=Günter M. Ziegler

| title=Coxeter-associahedra

| journal=Mathematika

| volume=41

| issue=2

| date=1994

| pages=364–393

| doi=10.1112/S0025579300007452}}

• {{Eom| title = Homotopy coherence | author-last1 = Porter | author-first1 = Tim| oldid = 55488}}
• {{cite journal |last1=Cordier |first1=Jean-Marc |title=Sur la notion de diagramme homotopiquement cohérent |journal=Cahiers de Topologie et Géométrie Différentielle Catégoriques |date=1982 |volume=23 |issue=1 |pages=93–112 |url=http://eudml.org/doc/91292 |issn=1245-530X}}
• {{cite book |doi=10.1017/9781108588737 |title=Higher Categories and Homotopical Algebra |date=2019 |last1=Cisinski |first1=Denis-Charles |isbn=978-1-108-58873-7 }}
• {{cite book |doi=10.4171/047-1/1 |chapter=Lectures on tensor categories |title=Quantum Groups |series=IRMA Lectures in Mathematics and Theoretical Physics |date=2008 |last1=Calaque |first1=Damien |last2=Etingof |first2=Pavel |volume=12 |pages=1–38 |arxiv=math/0401246 |isbn=978-3-03719-047-0 }}
• {{cite journal |doi=10.1016/0021-8693(64)90018-3 |title=On MacLane's conditions for coherence of natural associativities, commutativities, etc |date=1964 |last1=Kelly |first1=G.M |journal=Journal of Algebra |volume=1 |issue=4 |pages=397–402 }}
• {{cite book |last1=Kelly |first1=G. M. |last2=Laplaza |first2=M. |last3=Lewis |first3=G. |last4=Mac Lane |first4=Saunders|doi=10.1007/BFb0059553 |title=Coherence in Categories |series=Lecture Notes in Mathematics |date=1972 |volume=281 |isbn=978-3-540-05963-9 }}
• {{cite journal |doi=10.1016/0022-4049(86)90005-8 |title=A universal property of the convolution monoidal structure |date=1986 |last1=Im |first1=Geun Bin |last2=Kelly |first2=G.M. |journal=Journal of Pure and Applied Algebra |volume=43 |pages=75–88 }}
• {{cite book |doi=10.1007/978-1-4612-0783-2_11 |chapter=Tensor Categories |title=Quantum Groups |series=Graduate Texts in Mathematics |date=1995 |last1=Kassel |first1=Christian |volume=155 |pages=275–293 |isbn=978-1-4612-6900-7 }}
• {{cite book |doi=10.1007/BFb0059555 |chapter=Coherence for distributivity |title=Coherence in Categories |series=Lecture Notes in Mathematics |date=1972 |last1=Laplaza |first1=Miguel L. |volume=281 |pages=29–65 |isbn=978-3-540-05963-9 }}
• {{cite journal |doi=10.1006/aima.1999.1881 |doi-access=free |title=A Coherent Approach to Pseudomonads |date=2000 |last1=Lack |first1=Stephen |journal=Advances in Mathematics |volume=152 |issue=2 |pages=179–202 }}
• {{cite journal |url=https://hdl.handle.net/1911/62865 |hdl=1911/62865 |title=Natural Associativity and Commutativity |date=October 1963 |last1=MacLane |first1=Saunders|journal=Rice Institute Pamphlet - Rice University Studies }}
• {{cite book |author-link=Saunders Mac Lane |last=Mac Lane |first=Saunders |date=1971 |title=Categories for the working mathematician |series=Graduate texts in mathematics |volume=4 |publisher=Springer |chapter=7. Monoids §2 Coherence |pages=161–165 |title-link=Categories for the Working Mathematician |doi=10.1007/978-1-4612-9839-7_8 |isbn=9781461298397 |chapter-url=https://link.springer.com/chapter/10.1007/978-1-4612-9839-7_8}}
• {{cite journal |doi=10.1016/0022-4049(85)90087-8 |title=Coherence for bicategories and indexed categories |date=1985 |last1=MacLane |first1=Saunders |last2=Paré |first2=Robert |journal=Journal of Pure and Applied Algebra |volume=37 |pages=59–80 }}
• {{cite journal |doi=10.1016/0022-4049(89)90113-8 |title=A general coherence result |date=1989 |last1=Power |first1=A.J. |journal=Journal of Pure and Applied Algebra |volume=57 |issue=2 |pages=165–173 }}
• {{cite journal |url=https://eudml.org/doc/91637 |title=The syntax of coherence |journal=Cahiers de Topologie et Géométrie Différentielle Catégoriques |date=2000 |volume=41 |issue=4 |pages=255–304 |last1=Yanofsky |first1=Noson S. }}

{{refend}}

• {{cite journal |doi=10.1090/S0002-9904-1976-13928-6 |title=Topology and logic as a source of algebra |date=1976 |last1=Mac Lane |first1=Saunders |journal=Bulletin of the American Mathematical Society |volume=82 |pages=1–40 |doi-access=free }}

• https://ncatlab.org/nlab/show/homotopy+coherent+diagram
• https://unapologetic.wordpress.com/2007/07/01/the-strictification-theorem/
• {{cite arXiv |eprint=2109.01249 |last1=Malkiewich |first1=Cary |last2=Ponto |first2=Kate |title=Coherence for bicategories, lax functors, and shadows |date=2021 |class=math.CT }}

Category:Category theory

Category:Homotopy theory