Photo:1
Photo:2
Photo:3
Photo:4
Photo:1 Photo:2 Photo:3 Photo:4 |
| Formal definition | |
| 2>
A monoidal category is a category equipped with
a bifunctor called the tensor product or monoidal product,
an object I called the unit object or identity object,
three natural isomorphisms subject to certain coherence conditions expressing the fact that the tensor operation
is associative: there is a natural isomorphism α, called associator, with components ,
has I as left and right identity: there are two natural isomorphisms λ and ρ, respectively called left and right unitor, with components and .
The coherence conditions for these natural transformations are:
for all A, B, C and D in , the diagram
commutes;
for all A and B in , the diagram
commutes;
It follows from these three conditions that any such diagram (i.e. a diagram whose morphisms are built using α, λ, ρ, identities and tensor product) commutes: this is Mac Lane's "coherence theorem".
A strict monoidal category is one for which the natural isomorphisms α, λ and ρ are identities. Every monoidal category is monoidally equivalent to a strict monoidal category.
[edit] Tags:Category,Associative,Natural Isomorphism,Left,Right Identity,Coherence Conditions,Tensor Product,Natural Isomorphisms,Diagram,Commutes,Mac Lane's,Coherence Theorem,Equivalent,Monoid, | |
| Examples | |
| 2>
Any category with finite products is monoidal with the product as the monoidal product and the terminal object as the unit. Such a category is sometimes called a cartesian monoidal category.
Any category with finite coproducts is monoidal with the coproduct as the monoidal product and the initial object as the unit.
R-Mod, the category of modules over a commutative ring R, is a monoidal category with the tensor product of modules ⊗R serving as the monoidal product and the ring R (thought of as a module over itself) serving as the unit. As special cases one has:
K-Vect, the category of vector spaces over a field K, with the one-dimensional vector space K serving as the unit.
Ab, the category of abelian groups, with the group of integers Z serving as the unit.
For any commutative ring R, the category of R-algebras is monoidal with the tensor product of algebras as the product and R as the unit.
The category of pointed spaces is monoidal with the smash product serving as the product and the pointed 0-sphere (a two-point discrete space) serving as the unit.
The category of all endofunctors on a category C is a strict monoidal category with the composition of functors as the product and the identity functor as the unit.
Bounded-above meet semilattices are strict symmetric monoidal categories: the product is meet and the identity is the top element.
[edit] Tags:Vector Spaces,Abelian Groups,Products,Terminal Object,Coproducts,Initial Object,Category Of Modules,Commutative Ring,Tensor Product Of Modules,Category Of Vector Spaces,Field,Category Of Abelian Groups,Integers,Tensor Product Of Algebras,Category Of Pointed Spaces,Smash Product,0-sphere,Endofunctors,Bounded-above Meet Semilattices, | |
| Free strict monoidal category | |
| 2>
For every category C, the free strict monoidal category Σ(C) can be constructed as follows:
its objects are lists (finite sequences) A1, ..., An of objects of C;
there are arrows between two objects A1, ..., Am and B1, ..., Bn only if m = n, and then the arrows are lists (finite sequences) of arrows f1: A1 → B1, ..., fn: An → Bn of C;
the tensor product of two objects A1, ..., An and B1, ..., Bm is the concatenation A1, ..., An, B1, ..., Bm of the two lists, and, similarly, the tensor product of two morphisms is given by the concatenation of lists.
This operation Σ mapping category C to Σ(C) can be extended to a strict 2-monad on Cat.
[edit] Tags:Free, | |
| See also | |
| 2>
Many monoidal categories have additional structure such as braiding, symmetry or closure: the references describe this in detail.
Monoidal functors are the functors between monoidal categories which preserve the tensor product and monoidal natural transformations are the natural transformations, between those functors, which are "compatible" with the tensor product.
There is a general notion of monoid object in a monoidal category, which generalizes the ordinary notion of monoid. In particular, a strict monoidal category can be seen as a monoid object in the category of categories Cat (equipped with the monoidal structure induced by the cartesian product).
A monoidal category can also be seen as the category B(□, □) of a bicategory B with only one object, denoted □.
Rigid categories are monoidal categories in which duals with nice properties exist.
Autonomous categories are monoidal categories in which inverses exist.
A category C enriched in a monoidal category M replaces the notion of a set of morphisms between pairs of objects in C with the notion of an M-object of morphisms between every two objects in C.
[edit] Tags:Monoid Object,Monoidal Functors,Monoidal Natural Transformations, | |
| References | |
| 2>
Joyal, André; Street, Ross (1993). "Braided Tensor Categories". Advances in Mathematics 102, 20–78.
Kelly, G. Max (1964). "On MacLane's Conditions for Coherence of Natural Associativities, Commutativities, etc." Journal of Algebra 1, 397–402
Kelly, G. Max (1982). Basic Concepts of Enriched Category Theory. London Mathematical Society Lecture Note Series No. 64. Cambridge University Press. http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf.
Mac Lane, Saunders (1963). "Natural Associativity and Commutativity". Rice University Studies 49, 28–46.
Mac Lane, Saunders (1998), Categories for the Working Mathematician (2nd ed.). New York: Springer-Verlag.
Category theory portal
Retrieved from "http://en.wikipedia.org/w/index.php?title=Monoidal_category&oldid=459840395"
Categories: Monoidal categories
Personal tools
Log in / create account
Namespaces
Article
Talk
Variants
Views
Read
Edit
View history
Actions
Search
Navigation
Main page
Contents
Featured content
Current events
Random article
Donate to Wikipedia
Interaction
Help
About Wikipedia
Community portal
Recent changes
Contact Wikipedia
Toolbox
What links here
Related changes
Upload file
Special pages
Permanent link
Cite this page
Print/export
Create a bookDownload as PDFPrintable version
Languages
Deutsch
Español
Français
Nederlands
日本語
粵語
中文
This page was last modified on 9 November 2011 at 18:52.
Text is available under the Creative Commons Attribution-ShareAlike License;
additional terms may apply.
See Terms of use for details.
Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization.Contact us
Privacy policy
About Wikipedia
Disclaimers
Mobile view
if ( window.isMSIE55 ) fixalpha();
if ( window.mediaWiki ) {
mw.loader.load(["mediawiki.user", "mediawiki.util", "mediawiki.page.ready", "mediawiki.legacy.wikibits", "mediawiki.legacy.ajax", "mediawiki.legacy.mwsuggest", "ext.gadget.wmfFR2011Style", "ext.vector.collapsibleNav", "ext.vector.collapsibleTabs", "ext.vector.editWarning", "ext.vector.simpleSearch", "ext.UserBuckets", "ext.articleFeedback.startup", "ext.articleFeedbackv5.startup", "ext.markAsHelpful"]);
}
if ( window.mediaWiki ) {
mw.user.options.set({"ccmeonemails":0,"cols":80,"date":"default","diffonly":0,"disablemail":0,"disablesuggest":0,"editfont":"default","editondblclick":0,"editsection":1,"editsectiononrightclick":0,"enotifminoredits":0,"enotifrevealaddr":0,"enotifusertalkpages":1,"enotifwatchlistpages":0,"extendwatchlist":0,"externaldiff":0,"externaleditor":0,"fancysig":0,"forceeditsummary":0,"gender":"unknown","hideminor":0,"hidepatrolled":0,"highlightbroken":1,"imagesize":2,"justify":0,"math":1,"minordefault":0,"newpageshidepatrolled":0,"nocache":0,"noconvertlink":0,"norollbackdiff":0,"numberheadings":0,"previewonfirst":0,"previewontop":1,"quickbar":5,"rcdays":7,"rclimit":50,"rememberpassword":0,"rows":25,"searchlimit":20,"showhiddencats":false,"showjumplinks":1,"shownumberswatching":1,"showtoc":1,"showtoolbar":1,"skin":"vector","stubthreshold":0,"thumbsize":4,"underline":2,"uselivepreview":0,"usenewrc":0,"watchcreations":1,"watchdefault":0,"watchdeletion":0,"watchlistdays":3,"watchlisthideanons":0,
"watchlisthidebots":0,"watchlisthideliu":0,"watchlisthideminor":0,"watchlisthideown":0,"watchlisthidepatrolled":0,"watchmoves":0,"wllimit":250,"flaggedrevssimpleui":1,"flaggedrevsstable":0,"flaggedrevseditdiffs":true,"flaggedrevsviewdiffs":false,"vector-simplesearch":1,"useeditwarning":1,"vector-collapsiblenav":1,"usebetatoolbar":1,"usebetatoolbar-cgd":1,"wikilove-enabled":1,"variant":"en","language":"en","searchNs0":true,"searchNs1":false,"searchNs2":false,"searchNs3":false,"searchNs4":false,"searchNs5":false,"searchNs6":false,"searchNs7":false,"searchNs8":false,"searchNs9":false,"searchNs10":false,"searchNs11":false,"searchNs12":false,"searchNs13":false,"searchNs14":false,"searchNs15":false,"searchNs100":false,"searchNs101":false,"searchNs108":false,"searchNs109":false,"gadget-wmfFR2011Style":1});;mw.user.tokens.set({"editToken":"+\\","watchToken":false});;mw.loader.state({"user.options":"ready","user.tokens":"ready"});
/* cache key: enwiki:resourceloader:filter:minify-js:4:b41a86ec4e0fe8329bc3ce917e792339 */
}
Tags:Mathematics,Category Theory,Enriched Category, | |
zote monety |