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href="/nlab/show/semigroup">semigroup</a>, <a class="existingWikiWord" href="/nlab/show/quasigroup">quasigroup</a></li> <li><a class="existingWikiWord" href="/nlab/show/nonassociative+algebra">nonassociative algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/associative+unital+algebra">associative unital algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/commutative+algebra">commutative algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/Jordan+algebra">Jordan algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/Leibniz+algebra">Leibniz algebra</a>, <a class="existingWikiWord" href="/nlab/show/pre-Lie+algebra">pre-Lie algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/Poisson+algebra">Poisson algebra</a>, <a class="existingWikiWord" href="/nlab/show/Frobenius+algebra">Frobenius algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/lattice">lattice</a>, <a class="existingWikiWord" href="/nlab/show/frame">frame</a>, <a class="existingWikiWord" href="/nlab/show/quantale">quantale</a></li> <li><a class="existingWikiWord" href="/nlab/show/Boolean+ring">Boolean ring</a>, <a class="existingWikiWord" href="/nlab/show/Heyting+algebra">Heyting algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/commutator">commutator</a>, <a class="existingWikiWord" href="/nlab/show/center">center</a></li> <li><a class="existingWikiWord" href="/nlab/show/monad">monad</a>, <a class="existingWikiWord" href="/nlab/show/comonad">comonad</a></li> <li><a class="existingWikiWord" href="/nlab/show/distributive+law">distributive law</a></li> </ul> <h2 id="group_theory">Group theory</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/group">group</a>, <a class="existingWikiWord" href="/nlab/show/normal+subgroup">normal subgroup</a></li> <li><a class="existingWikiWord" href="/nlab/show/action">action</a>, <a class="existingWikiWord" href="/nlab/show/Cayley%27s+theorem">Cayley's theorem</a></li> <li><a class="existingWikiWord" href="/nlab/show/centralizer">centralizer</a>, <a class="existingWikiWord" href="/nlab/show/normalizer">normalizer</a></li> <li><a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a>, <a class="existingWikiWord" href="/nlab/show/cyclic+group">cyclic group</a></li> <li><a class="existingWikiWord" href="/nlab/show/group+extension">group extension</a>, <a class="existingWikiWord" href="/nlab/show/Galois+extension">Galois extension</a></li> <li><a class="existingWikiWord" href="/nlab/show/algebraic+group">algebraic group</a>, <a class="existingWikiWord" href="/nlab/show/formal+group">formal group</a></li> <li><a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a>, <a class="existingWikiWord" href="/nlab/show/quantum+group">quantum group</a></li> </ul> <h2 id="ring_theory">Ring theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/ring">ring</a>, <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+ring">local ring</a>, <a class="existingWikiWord" href="/nlab/show/Artinian+ring">Artinian ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Noetherian+ring">Noetherian ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/skewfield">skewfield</a>, <a class="existingWikiWord" href="/nlab/show/field">field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integral+domain">integral domain</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ideal">ideal</a>, <a class="existingWikiWord" href="/nlab/show/prime+ideal">prime ideal</a>, <a class="existingWikiWord" href="/nlab/show/maximal+ideal">maximal ideal</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ore+localization">Ore localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+extension">group extension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/central+simple+algebra">central simple algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derivation">derivation</a>, <a class="existingWikiWord" href="/nlab/show/Ore+extension">Ore extension</a></p> </li> </ul> <h2 id="module_theory">Module theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/module">module</a>, <a class="existingWikiWord" href="/nlab/show/bimodule">bimodule</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a>, <a class="existingWikiWord" href="/nlab/show/linear+algebra">linear algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/matrix">matrix</a>, <a class="existingWikiWord" href="/nlab/show/eigenvalue">eigenvalue</a>, <a class="existingWikiWord" href="/nlab/show/trace">trace</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/determinant">determinant</a>, <a class="existingWikiWord" href="/nlab/show/quasideterminant">quasideterminant</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/representation+theory">representation theory</a>, <a class="existingWikiWord" href="/nlab/show/Schur+lemma">Schur lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extension+of+scalars">extension of scalars</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/restriction+of+scalars">restriction of scalars</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Frobenius+reciprocity">Frobenius reciprocity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Morita+equivalence">Morita equivalence</a>, <a class="existingWikiWord" href="/nlab/show/Morita+context">Morita context</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wedderburn-Artin+theorem">Wedderburn-Artin theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a>, <a class="existingWikiWord" href="/nlab/show/additive+category">additive category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a>, <a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a></p> </li> </ul> <h2 id=""><a class="existingWikiWord" href="/nlab/show/gebra+theory">Gebras</a></h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/coalgebra">coalgebra</a>, <a class="existingWikiWord" href="/nlab/show/coring">coring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bialgebra">bialgebra</a>, <a class="existingWikiWord" href="/nlab/show/Hopf+algebra">Hopf algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/comodule">comodule</a>, <a class="existingWikiWord" href="/nlab/show/Hopf+module">Hopf module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yetter-Drinfeld+module">Yetter-Drinfeld module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/associative+bialgebroid">associative bialgebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dual+gebra">dual gebra</a>, <a class="existingWikiWord" href="/nlab/show/cotensor+product">cotensor product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hopf-Galois+extension">Hopf-Galois extension</a></p> </li> </ul> </div></div> <h4 id="category_theory">Category theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></strong></p> <h2 id="sidebar_concepts">Concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category">category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cat">Cat</a></p> </li> </ul> <h2 id="sidebar_universal_constructions">Universal constructions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+construction">universal construction</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/representable+functor">representable functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor">adjoint functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/limit">limit</a>/<a class="existingWikiWord" href="/nlab/show/colimit">colimit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weighted+limit">weighted limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/end">end</a>/<a class="existingWikiWord" href="/nlab/show/coend">coend</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extension</a></p> </li> </ul> </li> </ul> <h2 id="sidebar_theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+construction">Grothendieck construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor+theorem">adjoint functor theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monadicity+theorem">monadicity theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+lifting+theorem">adjoint lifting theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gabriel-Ulmer+duality">Gabriel-Ulmer duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freyd-Mitchell+embedding+theorem">Freyd-Mitchell embedding theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relation+between+type+theory+and+category+theory">relation between type theory and category theory</a></p> </li> </ul> <h2 id="sidebar_extensions">Extensions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf+and+topos+theory">sheaf and topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a></p> </li> </ul> <h2 id="sidebar_applications">Applications</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/applications+of+%28higher%29+category+theory">applications of (higher) category theory</a></li> </ul> <div> <p> <a href="/nlab/edit/category+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definitions'>Definitions</a></li> <ul> <li><a href='#traditional_magmas_semigroups_groups_rngs'>Traditional (magmas, semigroups, groups, rngs)</a></li> <li><a href='#identitypreserving_monoids_rings'>Identity-preserving (monoids, rings)</a></li> <li><a href='#general'>General</a></li> </ul> <li><a href='#examples'>Examples</a></li> <li><a href='#related_concepts'>Related concepts</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>In a restrictive sense, a homomorphism is a <a class="existingWikiWord" href="/nlab/show/function">function</a> between (the <a class="existingWikiWord" href="/nlab/show/underlying+sets">underlying sets</a>) of two <a class="existingWikiWord" href="/nlab/show/algebras">algebras</a> that preserves the <a class="existingWikiWord" href="/nlab/show/algebraic+structure">algebraic structure</a>.</p> <p>More generally, a homomorphism is a function between <a class="existingWikiWord" href="/nlab/show/structured+sets">structured sets</a> that preserves whatever <a class="existingWikiWord" href="/nlab/show/structure">structure</a> there is around. Even more generally, ‘homomorphism’ is just a synonym for ‘<a class="existingWikiWord" href="/nlab/show/morphism">morphism</a>’ in any <a class="existingWikiWord" href="/nlab/show/category">category</a>, the structured sets being generalised to arbitrary <a class="existingWikiWord" href="/nlab/show/object">object</a>s.</p> <p><em>Note: The word “homomorphism” has also traditionally been used for what we call a (weak) <a class="existingWikiWord" href="/nlab/show/2-functor">2-functor</a> between <a class="existingWikiWord" href="/nlab/show/bicategories">bicategories</a></em>.</p> <h2 id="definitions">Definitions</h2> <h3 id="traditional_magmas_semigroups_groups_rngs">Traditional (magmas, semigroups, groups, rngs)</h3> <p>Traditionally, a <strong>homomorphism</strong> between two <a class="existingWikiWord" href="/nlab/show/magmas">magmas</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/function">function</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex"> \phi\colon A \to B </annotation></semantics></math></div> <p>of the underlying <a class="existingWikiWord" href="/nlab/show/set">set</a>s that respects the binary operation in that for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>a</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>a</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">a_1, a_2</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><msub><mi>a</mi> <mn>1</mn></msub><mo>⋅</mo><msub><mi>a</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mi>ϕ</mi><mo stretchy="false">(</mo><msub><mi>a</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>⋅</mo><mi>ϕ</mi><mo stretchy="false">(</mo><msub><mi>a</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \phi(a_1 \cdot a_2) = \phi(a_1) \cdot \phi(a_2) \,. </annotation></semantics></math></div> <p>This definition gives us the correct notion of <strong>magma homomorphism</strong>, <strong><a class="existingWikiWord" href="/nlab/show/semigroup">semigroup</a> homomorphism</strong>, and <strong><a class="existingWikiWord" href="/nlab/show/group">group</a> homomorphism</strong>, but it is actually a bit of a coincidence that it works for groups. It does <em>not</em> give the correct definition of <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a> homomorphism, since it doesn't properly treat the <a class="existingWikiWord" href="/nlab/show/identity+elements">identity elements</a>. (However, the correct notion of monoid <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> can still be constructed from this inadequate definition of homomorphism.)</p> <p>A <strong>rng homomorphism</strong> is a function between <a class="existingWikiWord" href="/nlab/show/rngs">rngs</a> that is a homomorphism for both the additive group and the multiplicative semigroup. (For <a class="existingWikiWord" href="/nlab/show/rings+with+identity">rings with identity</a>, this is again inadequate.)</p> <h3 id="identitypreserving_monoids_rings">Identity-preserving (monoids, rings)</h3> <p>A <strong>homomorphism</strong> between two <a class="existingWikiWord" href="/nlab/show/monoids">monoids</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> is a semigroup homomorphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex"> \phi\colon A \to B </annotation></semantics></math></div> <p>of the underlying <a class="existingWikiWord" href="/nlab/show/semigroup">semigroup</a>s that preserves <a class="existingWikiWord" href="/nlab/show/identity+elements">identity elements</a> in that we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><msub><mn>1</mn> <mi>A</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mn>1</mn> <mi>B</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \phi(1_A) = 1_B \,. </annotation></semantics></math></div> <p>This definition of <strong>monoid homomorphism</strong> is a special case (via <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a>) of the definition of <a class="existingWikiWord" href="/nlab/show/functor">functor</a>.</p> <p>It is a theorem that a semigroup homomorphism between <a class="existingWikiWord" href="/nlab/show/groups">groups</a> must be a monoid homomorphism (and additionally must preserve <a class="existingWikiWord" href="/nlab/show/inverse+elements">inverse elements</a>, which is also necessary to be the <em>correct</em> definition of <strong>group homomorphism</strong>.)</p> <p>A <strong><a class="existingWikiWord" href="/nlab/show/ring">ring</a> homomorphism</strong> is a function between rings that is a homomorphism for both the additive group and the multiplicative monoid. Traditional <a class="existingWikiWord" href="/nlab/show/ring+theory">ring theory</a> sometimes actually uses rng homomorphisms even when the rngs in question are assumed to have identity elements, so be careful when reading old books.</p> <p>A <a class="existingWikiWord" href="/nlab/show/rig">rig</a> <strong>homomorphism</strong> is a function between rigs that is a homomorphism for both the additive monoid and the multiplicative monoid.</p> <p>A <a class="existingWikiWord" href="/nlab/show/semi-ring">semi-ring</a> <strong>homomorphism</strong> is a function between semi-rings that is a homomorphism for both the additive semigroup and the multiplicative monoid.</p> <h3 id="general">General</h3> <p>More generally, a <strong>homomorphism</strong> between sets equipped with any algebraic structure is a map preserving this structure. This can be made precise using <a class="existingWikiWord" href="/nlab/show/Lawvere+theories">Lawvere theories</a>, <a class="existingWikiWord" href="/nlab/show/monads">monads</a>, etc.</p> <p>Generalizing further, we may simply treat ‘homomorphism’ as a synonym for ‘<a class="existingWikiWord" href="/nlab/show/morphism">morphism</a>’ in any <a class="existingWikiWord" href="/nlab/show/category">category</a>, although there is a strong tendency to use ‘homomorphism’ in the case of ‘algebraic’ categories: for example, nobody seems to speak of a homomorphism between <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>s (<a class="existingWikiWord" href="/nlab/show/continuous+map">continuous map</a>s), or between <a class="existingWikiWord" href="/nlab/show/manifold">manifold</a>s. Here we put ‘algebraic’ in scare quotes to indicate the <em>field</em> of <a class="existingWikiWord" href="/nlab/show/algebra">algebra</a>; even morphisms in <a class="existingWikiWord" href="/nlab/show/algebraic+categories">algebraic categories</a> from other fields (such as the category of <a class="existingWikiWord" href="/nlab/show/compacta">compacta</a>) are not usually called homomorphisms.</p> <h2 id="examples">Examples</h2> <ul> <li> <p>an <a class="existingWikiWord" href="/nlab/show/action">action</a> homomorphism is an <a class="existingWikiWord" href="/nlab/show/equivariant+function">equivariant function</a></p> </li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/set">set</a> homomorphism is an <a class="existingWikiWord" href="/nlab/show/extensional+function">extensional function</a></p> </li> <li> <p>an <a class="existingWikiWord" href="/nlab/show/inequality+space">inequality space</a> homomorphism is a <a class="existingWikiWord" href="/nlab/show/strongly+extensional+function">strongly extensional function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homomorphism+of+L-infinity+algebras">homomorphism of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <msub><mi>L</mi> <mn>∞</mn></msub> </mrow> <annotation encoding="application/x-tex">L_\infty</annotation> </semantics> </math>-algebras</a></p> </li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p>homomorphisms are to <a class="existingWikiWord" href="/nlab/show/functions">functions</a> as <a class="existingWikiWord" href="/nlab/show/logical+relations">logical relations</a> are to <a class="existingWikiWord" href="/nlab/show/relations">relations</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/crossed+homomorphism">crossed homomorphism</a></p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on August 21, 2024 at 01:42:50. 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