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- ...> a subgroup of <math>H</math>. Then there exists a unique <math>G</math>-morphism <math>f : G/K \to G/S</math> for which <math>f(K) = a</math>; this mapping7 KB (1,332 words) - 17:45, 9 September 2008
- ...{\to} G</math> be two extensions of <math>G</math> by <math>F</math>. A ''morphism of extensions'' from <math>\mathcal{E}</math> to <math>\mathcal{E}'</math> ...m of <math>E</math> onto <math>E'</math>. In other words, every extension morphism is an extension isomorphism.5 KB (901 words) - 19:53, 27 May 2008
- ...nd <math>T</math>, respectively. An ''<math>\Omega</math>-[[homomorphism |morphism]]'' of <math>S</math> into <math>T</math> is a function <math>h: S \to T</m .../math>. A mapping <math>h: S \to T</math> is called a ''<math>\phi</math>-morphism'' if2 KB (321 words) - 08:24, 13 June 2008
- ** (identity) For and object <math>X</math>, there is an identity morphism <math>1_X:X\to X</math> such that for any <math>f:A\to B</math>: <cmath>1_B ...e. a morphism from <math>X</math> to <math>Y</math> would be replaced by a morphism from <math>Y</math> to <math>X</math>).5 KB (792 words) - 18:01, 7 April 2012
- * sends every morphism <math>f:X\to Y</math> of <math>\mathcal{C}</math> to a morphism <math>F(f):F(X)\to F(Y)</math> of <math>\mathcal{D}</math>. ...atisfying the same properties as above, except that <math>F(f)</math> is a morphism from <math>F(Y)</math> to <math>F(X)</math>, and instead of having <math>F(1 KB (248 words) - 21:42, 2 September 2008
- ...rphi_X:F(X)\to G(X)</math> in <math>\mathcal{D}</math> such that for every morphism <math>f:X\to Y</math> of <math>\mathcal{C}</math>, we have:<cmath>\varphi_Y1 KB (201 words) - 21:16, 2 September 2008
- Notice that, as before, each morphism <math>\varphi(g)</math> has an inverse <math>\varphi(g^{-1})</math>, and so ...a category with only one object, (say <math>A</math>), and in which every morphism is invertible (and is therefore an automorphism of <math>A</math>). In term5 KB (917 words) - 20:17, 7 September 2008
- ...ism <math>f^{op}:B\to A</math> in <math>\mathcal{C}^{op}</math> (and every morphism <math>g</math> in <math>\mathcal{C}^{op}</math> is equal to <math>f^{op}</m ...athcal{C}^{op}) = \text{Ob}(\mathcal{C})</math> we claim that the identity morphism on <math>A</math> is just <math>1_A^{op}</math>. Indeed, for any <math>f^{o2 KB (438 words) - 16:07, 8 September 2008
- Thus for every object <math>B</math> of <math>\mathcal{C}</math>, the morphism any morphism <math>f : B \to C</math> of <math>\mathcal{C}</math>, and any element4 KB (698 words) - 17:21, 6 November 2017
- Function composition is generalized to composition of [[morphism|morphisms]] in [[category theory]].750 bytes (116 words) - 22:17, 3 October 2020