Coherency (homotopy theory)

{{Jargon|date=September 2024}}{{Short description|Standard that diagrams must satisfy up to isomorphism}} In mathematics, specifically in homotopy theory and (higher) category theory, '''coherency''' is the standard that equalities or diagrams must satisfy when they hold "up to homotopy" or "up to isomorphism".

Often, more than one way of defining a mapping between mathematical objects might be considered "natural". Then the question might arise, which way to choose? Coherency implies that it doesn't matter which way is chosen, because all the alternative definitions are equivalent. The equivalence is often manifest in a commutative diagram.

The adjectives such as "pseudo-" and "lax-" are used to refer to the fact equalities are weakened in coherent ways; e.g., pseudo-functor, pseudoalgebra.

== Coherent isomorphism == 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. === Weak 2-category === In a weak 2-category the composition of 1-morphisms does not satisfy associativity as an equation, however for each triple <math>A \mathrel{\stackrel f \longrightarrow} B \mathrel{\stackrel g \longrightarrow} C \mathrel{\stackrel h \longrightarrow} D</math>, there are 2-morphisms

File:Associativity coherence isomorphisms.svg

in <math>\mathcal{B} (A,D)</math> this is called '''associativity coherence isomorphisms'''.<ref name=Leinster2004>{{harvnb|Leinster|2004|loc=Definition 1.5.1}}</ref> <ref>{{harvnb|associator in nLab}}</ref>

For each 1-cell <math>A \mathrel{\stackrel f \longrightarrow} B</math>, isomorphism

File:Unit coherence isomorphisms.svg

in <math>\mathcal{B} (A,B)</math> this is called '''unit coherence isomorphisms'''.{{R|Leinster2004}}

== Coherence condition == 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 and the coherence conditions appear as conditions in definitions. 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 <math>\alpha_{A,B,C}</math>, called the ''associator'':

: <math>\alpha_{A,B,C} \colon (A\otimes B)\otimes C \rightarrow A\otimes(B\otimes C)</math>

for each triple of objects <math>A, B, C</math> in the category. Using compositions of these <math>\alpha_{A,B,C}</math>, one can construct a morphism

: <math>( ( A_N \otimes A_{N-1} ) \otimes A_{N-2} ) \otimes \cdots \otimes A_1) \rightarrow ( A_N \otimes ( A_{N-1} \otimes \cdots \otimes ( A_2 \otimes A_1) ). </math>

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

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 <math>A,B,C,D</math>, the following diagram commutes. 640px|center

Any pair of morphisms from <math> ( ( \cdots ( A_N \otimes A_{N-1} ) \otimes \cdots ) \otimes A_2 ) \otimes A_1) </math> to <math> ( A_N \otimes ( A_{N-1} \otimes ( \cdots \otimes ( A_2 \otimes A_1) \cdots ) ) </math> constructed as compositions of various <math>\alpha_{A,B,C}</math> are equal.

===Further examples===

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

====Identity==== 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''}}.

====Associativity of composition==== 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.

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

There are several generalizations of Mac Lane's coherence theorem.<ref>See, for instance [https://ncatlab.org/nlab/show/coherence+theorem coherence theorem (nlab)]</ref> Each of them has the rough form that "every weak structure of some sort is equivalent to a stricter one".<ref>{{harvnb|Shulman|2012|loc=Section 1}}</ref> A coherence theorem theorem for weak 4-categories does not yet exist.<ref>{{harvnb|Crans|2000|loc=2.2 Dimension 4}}</ref>

== Homotopy coherence == {{expand section|date=September 2019}} === example === *The quasicategories give a model for ∞-categories where diagrams are automatically homotopy coherent.<ref>{{harvnb|Riehl|2018}}</ref>

===Vogt’s theorem === Let A be a small category and <math>B_s</math> a locally Kan and complete (or co-complete) S-category.

Vogt’s theorem<ref>{{harvnb|Cordier|Porter|1986}}</ref><ref>{{harvnb|Cordier|Porter|1988}}</ref> on coherent diagrams one has an equivalence of categories

<math>\mathrm{Ho}(B^A ) \simeq \mathrm{Coh}(A,B_s).</math>

== Notes == {{reflist}}

== References == {{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 book |arxiv=0908.3347 |doi=10.1007/978-3-642-12821-9_4 |chapter=A Survey of Graphical Languages for Monoidal Categories |title=New Structures for Physics |series=Lecture Notes in Physics |date=2010 |last1=Selinger |first1=P. |volume=813 |pages=289–355 |isbn=978-3-642-12820-2 }} *{{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. }} *{{cite book |last1=Leinster |first1=Tom |author-link=Tom Leinster |title=Higher Operads, Higher Categories |date=22 July 2004 |publisher=Cambridge University Press |isbn=978-0-521-53215-0 |url={{Google books|K8nPLSAzhAcC|page=50|plainurl=yes}} |language=en}} *{{cite journal |last1=Cordier |first1=Jean Marc |last2=Porter |first2=Timothy |title=Maps between homotopy coherent diagrams |journal=Topology and its Applications |date=April 1988 |volume=28 |issue=3 |pages=255–275 |doi=10.1016/0166-8641(88)90046-6}} *{{cite journal |last1=Cordier |first1=Jean-Marc |last2=Porter |first2=Timothy |title=Vogt's theorem on categories of homotopy coherent diagrams |journal=Mathematical Proceedings of the Cambridge Philosophical Society |date=1986 |volume=100 |issue=1 |pages=65–90 |doi=10.1017/S0305004100065877 |bibcode=1986MPCPS.100...65C }} *{{cite journal |last1=Crans |first1=S. |title=On braiding, syllapses and symmetries |journal=Cahiers de Topologie et Géométrie Différentielle Catégoriques |date=2000 |volume=41|issue=1|pages=2–74|url=https://www.numdam.org/item/CTGDC_2000__41_1_2_0.pdf}} -The 32nd reference in this paper is repost in Notes on Tetracategories. {{refend}}

== Further reading == *{{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 }}

== External links == *{{cite web|title=homotopy coherent diagram|url=https://ncatlab.org/nlab/show/homotopy+coherent+diagram|website=ncatlab.org}} *{{cite web|title=associator|url=https://ncatlab.org/nlab/show/associator|website=ncatlab.org|ref={{harvid|associator in nLab}}}} *{{cite web|title=simplicial foundations for homotopy coherence|url=https://ncatlab.org/timporter/show/simplicial+foundations+for+homotopy+coherence|website=ncatlab.org}} *{{cite web |last1=Armstrong |first1=John |title=The “Strictification” Theorem |url=https://unapologetic.wordpress.com/2007/07/01/the-strictification-theorem/ |website=The Unapologetic Mathematician|date=1 June 2007 }} *{{cite journal |arxiv=2109.01249 |last1=Malkiewich |first1=Cary |last2=Ponto |first2=Kate |title=Coherence for bicategories, lax functors, and shadows |date=2022 |journal=Theory and Applications of Categories |volume=38 |issue=12 | pages=328-373 |url=http://www.tac.mta.ca/tac/volumes/38/12/38-12abs.html}} *{{cite web |last1=Porter |first1=Timothy |title=The Crossed Menagerie |url=https://ncatlab.org/nlab/files/menagerie14.pdf}} *{{cite arXiv |last1=Riehl |first1=Emily |title=Homotopy coherent structures |date=2018 |class=math.CT |eprint=1801.07404}} *{{Eom |title=Higher-dimensional category |author-last1=Street |author-first1=Ross |oldid=55484}} *{{cite web |last1=Trimble |first1=Todd |title=Notes on tetracategories |url=http://math.ucr.edu/home/baez/trimble/tetracategories.html|date=2006}} Category:Category theory Category:Homotopy theory