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.
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.
Coherence theoremedit
Mac Lane's coherence theorem states, roughly, that if diagrams of certain types commute, then diagrams of all types commute.[1] A simple proof of that theorem can be obtained using the permutoassociahedron, a polytope whose combinatorial structure appears implicitly in Mac Lane's proof.[2]
There are several generalizations of Mac Lane's coherence theorem.[3] Each of them has the rough form that "every weak structure of some sort is equivalent to a stricter one".[4]
Homotopy coherenceedit
This section needs expansion. You can help by adding to it. (September 2019)
§ 5. of Mac Lane, Saunders (January 1976). "Topology and Logic as a Source of Algebra (Retiring Presidential Address)". Bulletin of the American Mathematical Society. 82 (1): 1–40. doi:10.1090/S0002-9904-1976-13928-6.
Mac Lane, Saunders (1978) [1971]. Categories for the working mathematician. Graduate texts in mathematics. Springer-Verlag. doi:10.1007/978-1-4757-4721-8.
Ch. 5 of Kamps, Klaus Heiner; Porter, Timothy (April 1997). Abstract Homotopy and Simple Homotopy Theory. World Scientific. doi:10.1142/2215. ISBN9810216025.
Shulman, Mike (2012). "Not every pseudoalgebra is equivalent to a strict one". Advances in Mathematics. 229 (3): 2024–2041. arXiv:1005.1520. doi:10.1016/j.aim.2011.01.010.
Kapranov, Mikhail M. (1993). "The permutoassociahedron, Mac Lane's coherence theorem and asymptotic zones for the KZ equation". Journal of Pure and Applied Algebra. 85 (2): 119–142. doi:10.1016/0022-4049(93)90049-Y.
coherency, homotopy, theory, mathematics, specifically, homotopy, theory, higher, category, theory, coherency, standard, that, equalities, diagrams, must, satisfy, when, they, hold, homotopy, isomorphism, adjectives, such, pseudo, used, refer, fact, equalities. 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 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 Contents 1 Coherent isomorphism 2 Coherence theorem 3 Homotopy coherence 4 See also 5 Notes 6 References 7 External linksCoherent isomorphism editIn 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 Coherence theorem editMac Lane s coherence theorem states roughly that if diagrams of certain types commute then diagrams of all types commute 1 A simple proof of that theorem can be obtained using the permutoassociahedron a polytope whose combinatorial structure appears implicitly in Mac Lane s proof 2 There are several generalizations of Mac Lane s coherence theorem 3 Each of them has the rough form that every weak structure of some sort is equivalent to a stricter one 4 Homotopy coherence editThis section needs expansion You can help by adding to it September 2019 See also editCoherence condition Canonical isomorphismNotes edit Mac Lane 1978 Chapter VII Section 2 See Kapranov 1993 and Reiner amp Ziegler 1994 See for instance coherence theorem nlab Shulman 2012 Section 1References editCordier Jean Marc Porter Timothy 1997 Homotopy coherent category theory Transactions of the American Mathematical Society 349 1 1 54 doi 10 1090 S0002 9947 97 01752 2 5 of Mac Lane Saunders January 1976 Topology and Logic as a Source of Algebra Retiring Presidential Address Bulletin of the American Mathematical Society 82 1 1 40 doi 10 1090 S0002 9904 1976 13928 6 Mac Lane Saunders 1978 1971 Categories for the working mathematician Graduate texts in mathematics Springer Verlag doi 10 1007 978 1 4757 4721 8 Ch 5 of Kamps Klaus Heiner Porter Timothy April 1997 Abstract Homotopy and Simple Homotopy Theory World Scientific doi 10 1142 2215 ISBN 9810216025 Shulman Mike 2012 Not every pseudoalgebra is equivalent to a strict one Advances in Mathematics 229 3 2024 2041 arXiv 1005 1520 doi 10 1016 j aim 2011 01 010 Kapranov Mikhail M 1993 The permutoassociahedron Mac Lane s coherence theorem and asymptotic zones for the KZ equation Journal of Pure and Applied Algebra 85 2 119 142 doi 10 1016 0022 4049 93 90049 Y Reiner Victor Ziegler Gunter M 1994 Coxeter associahedra Mathematika 41 2 364 393 doi 10 1112 S0025579300007452 External links edithttps ncatlab org nlab show homotopy coherent diagram nbsp This topology related article is a stub You can help Wikipedia by expanding it vte Retrieved from https en wikipedia org w index php title Coherency homotopy theory amp oldid 1171082350 Coherent isomorphism, wikipedia, wiki, book, books, library,
