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Traced Monoidal Category
In category theory, a traced monoidal category is a category with some extra structure which gives a reasonable notion of feedback. A traced symmetric monoidal category is a symmetric monoidal category C together with a family of functions :\mathrm^U_:\mathbf(X\otimes U,Y\otimes U)\to\mathbf(X,Y) called a ''trace'', satisfying the following conditions: * naturality in X: for every f:X\otimes U\to Y\otimes U and g:X'\to X, ::\mathrm^U_(f \circ (g\otimes \mathrm_U)) = \mathrm^U_(f) \circ g * naturality in Y: for every f:X\otimes U\to Y\otimes U and g:Y\to Y', ::\mathrm^U_((g\otimes \mathrm_U) \circ f) = g \circ \mathrm^U_(f) * dinaturality in U: for every f:X\otimes U\to Y\otimes U' and g:U'\to U ::\mathrm^U_((\mathrm_Y\otimes g) \circ f)=\mathrm^_(f \circ (\mathrm_X\otimes g)) * vanishing I: for every f:X \otimes I \to Y \otimes I, (with \rho_X \colon X\otimes I\cong X being the right unitor), ::\mathrm^I_(f)=\rho_Y \circ f \circ \rho_X^ * vanishing II: for every f:X\ot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, category theory is used in almost all areas of mathematics, and in some areas of computer science. In particular, many constructions of new mathematical objects from previous ones, that appear similarly in several contexts are conveniently expressed and unified in terms of categories. Examples include quotient spaces, direct products, completion, and duality. A category is formed by two sorts of objects: the objects of the category, and the morphisms, which relate two objects called the ''source'' and the ''target'' of the morphism. One often says that a morphism is an ''arrow'' that ''maps'' its source to its target. Morphisms can be ''composed'' if the target of the first morphism equals the source of the second one, and morphism com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Monoidal Category
In category theory, a branch of mathematics, a symmetric monoidal category is a monoidal category (i.e. a category in which a "tensor product" \otimes is defined) such that the tensor product is symmetric (i.e. A\otimes B is, in a certain strict sense, naturally isomorphic to B\otimes A for all objects A and B of the category). One of the prototypical examples of a symmetric monoidal category is the category of vector spaces over some fixed field ''k,'' using the ordinary tensor product of vector spaces. Definition A symmetric monoidal category is a monoidal category (''C'', ⊗, ''I'') such that, for every pair ''A'', ''B'' of objects in ''C'', there is an isomorphism s_: A \otimes B \to B \otimes A that is natural in both ''A'' and ''B'' and such that the following diagrams commute: *The unit coherence: *: *The associativity coherence: *: *The inverse law: *: In the diagrams above, ''a'', ''l'' , ''r'' are the associativity isomorphism, the left unit isomorphism, and the right ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trace Diagram Naturality 1
Trace may refer to: Arts and entertainment Music * ''Trace'' (Son Volt album), 1995 * ''Trace'' (Died Pretty album), 1993 * Trace (band), a Dutch progressive rock band * ''The Trace'' (album) Other uses in arts and entertainment * ''Trace'' (magazine), British hip-hop magazine * ''Trace'' (manhwa), a Korean internet cartoon * ''Trace'' (novel), a novel by Patricia Cornwell * ''The Trace'' (film), a 1994 Turkish film * ''The Trace'' (video game), 2015 video game * ''Sama'' (film), alternate title ''The Trace'', a 1988 Tunisian film * Trace, a fictional character in the game '' Metroid Prime Hunters'' * Trace, the protagonist of ''Axiom Verge'' * Trace, another name for Portgas D. Ace, a fictional character in the manga ''One Piece'' * TRACE, the main brand for a number of music channels such as Trace Urban Language * Trace (deconstruction), a concept in Derridian deconstruction * Trace (linguistics), a syntactic placeholder resulting from a transformation * TRACE (psych ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trace Diagram Naturality 2
Trace may refer to: Arts and entertainment Music * ''Trace'' (Son Volt album), 1995 * ''Trace'' (Died Pretty album), 1993 * Trace (band), a Dutch progressive rock band * ''The Trace'' (album) Other uses in arts and entertainment * ''Trace'' (magazine), British hip-hop magazine * ''Trace'' (manhwa), a Korean internet cartoon * ''Trace'' (novel), a novel by Patricia Cornwell * ''The Trace'' (film), a 1994 Turkish film * ''The Trace'' (video game), 2015 video game * ''Sama'' (film), alternate title ''The Trace'', a 1988 Tunisian film * Trace, a fictional character in the game '' Metroid Prime Hunters'' * Trace, the protagonist of ''Axiom Verge'' * Trace, another name for Portgas D. Ace, a fictional character in the manga ''One Piece'' * TRACE, the main brand for a number of music channels such as Trace Urban Language * Trace (deconstruction), a concept in Derridian deconstruction * Trace (linguistics), a syntactic placeholder resulting from a transformation * TRACE (psych ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trace Diagram Dinaturality
Trace may refer to: Arts and entertainment Music * ''Trace'' (Son Volt album), 1995 * ''Trace'' (Died Pretty album), 1993 * Trace (band), a Dutch progressive rock band * ''The Trace'' (album) Other uses in arts and entertainment * ''Trace'' (magazine), British hip-hop magazine * ''Trace'' (manhwa), a Korean internet cartoon * ''Trace'' (novel), a novel by Patricia Cornwell * ''The Trace'' (film), a 1994 Turkish film * ''The Trace'' (video game), 2015 video game * ''Sama'' (film), alternate title ''The Trace'', a 1988 Tunisian film * Trace, a fictional character in the game '' Metroid Prime Hunters'' * Trace, the protagonist of ''Axiom Verge'' * Trace, another name for Portgas D. Ace, a fictional character in the manga ''One Piece'' * TRACE, the main brand for a number of music channels such as Trace Urban Language * Trace (deconstruction), a concept in Derridian deconstruction * Trace (linguistics), a syntactic placeholder resulting from a transformation * TRACE (psych ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trace Diagram Vanishing
Trace may refer to: Arts and entertainment Music * ''Trace'' (Son Volt album), 1995 * ''Trace'' (Died Pretty album), 1993 * Trace (band), a Dutch progressive rock band * ''The Trace'' (album) Other uses in arts and entertainment * ''Trace'' (magazine), British hip-hop magazine * ''Trace'' (manhwa), a Korean internet cartoon * ''Trace'' (novel), a novel by Patricia Cornwell * ''The Trace'' (film), a 1994 Turkish film * ''The Trace'' (video game), 2015 video game * ''Sama'' (film), alternate title ''The Trace'', a 1988 Tunisian film * Trace, a fictional character in the game '' Metroid Prime Hunters'' * Trace, the protagonist of ''Axiom Verge'' * Trace, another name for Portgas D. Ace, a fictional character in the manga ''One Piece'' * TRACE, the main brand for a number of music channels such as Trace Urban Language * Trace (deconstruction), a concept in Derridian deconstruction * Trace (linguistics), a syntactic placeholder resulting from a transformation * TRACE (psych ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trace Diagram Associativity
Trace may refer to: Arts and entertainment Music * ''Trace'' (Son Volt album), 1995 * ''Trace'' (Died Pretty album), 1993 * Trace (band), a Dutch progressive rock band * ''The Trace'' (album) Other uses in arts and entertainment * ''Trace'' (magazine), British hip-hop magazine * ''Trace'' (manhwa), a Korean internet cartoon * ''Trace'' (novel), a novel by Patricia Cornwell * ''The Trace'' (film), a 1994 Turkish film * ''The Trace'' (video game), 2015 video game * ''Sama'' (film), alternate title ''The Trace'', a 1988 Tunisian film * Trace, a fictional character in the game '' Metroid Prime Hunters'' * Trace, the protagonist of ''Axiom Verge'' * Trace, another name for Portgas D. Ace, a fictional character in the manga ''One Piece'' * TRACE, the main brand for a number of music channels such as Trace Urban Language * Trace (deconstruction), a concept in Derridian deconstruction * Trace (linguistics), a syntactic placeholder resulting from a transformation * TRACE (psych ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trace Diagram Superposition
Trace may refer to: Arts and entertainment Music * ''Trace'' (Son Volt album), 1995 * ''Trace'' (Died Pretty album), 1993 * Trace (band), a Dutch progressive rock band * ''The Trace'' (album) Other uses in arts and entertainment * ''Trace'' (magazine), British hip-hop magazine * ''Trace'' (manhwa), a Korean internet cartoon * ''Trace'' (novel), a novel by Patricia Cornwell * ''The Trace'' (film), a 1994 Turkish film * ''The Trace'' (video game), 2015 video game * ''Sama'' (film), alternate title ''The Trace'', a 1988 Tunisian film * Trace, a fictional character in the game '' Metroid Prime Hunters'' * Trace, the protagonist of ''Axiom Verge'' * Trace, another name for Portgas D. Ace, a fictional character in the manga ''One Piece'' * TRACE, the main brand for a number of music channels such as Trace Urban Language * Trace (deconstruction), a concept in Derridian deconstruction * Trace (linguistics), a syntactic placeholder resulting from a transformation * TRACE (psych ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trace Diagram Yanking
Trace may refer to: Arts and entertainment Music * ''Trace'' (Son Volt album), 1995 * ''Trace'' (Died Pretty album), 1993 * Trace (band), a Dutch progressive rock band * ''The Trace'' (album) Other uses in arts and entertainment * ''Trace'' (magazine), British hip-hop magazine * ''Trace'' (manhwa), a Korean internet cartoon * ''Trace'' (novel), a novel by Patricia Cornwell * ''The Trace'' (film), a 1994 Turkish film * ''The Trace'' (video game), 2015 video game * ''Sama'' (film), alternate title ''The Trace'', a 1988 Tunisian film * Trace, a fictional character in the game '' Metroid Prime Hunters'' * Trace, the protagonist of ''Axiom Verge'' * Trace, another name for Portgas D. Ace, a fictional character in the manga ''One Piece'' * TRACE, the main brand for a number of music channels such as Trace Urban Language * Trace (deconstruction), a concept in Derridian deconstruction * Trace (linguistics), a syntactic placeholder resulting from a transformation * TRACE (psych ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compact Closed Category
In category theory, a branch of mathematics, compact closed categories are a general context for treating dual objects. The idea of a dual object generalizes the more familiar concept of the dual of a finite-dimensional vector space. So, the motivating example of a compact closed category is FdVect, the category having finite-dimensional vector spaces as objects and linear maps as morphisms, with tensor product as the monoidal structure. Another example is Rel, the category having sets as objects and relations as morphisms, with Cartesian monoidal structure. Symmetric compact closed category A symmetric monoidal category (\mathbf,\otimes,I) is compact closed if every object A \in \mathbf C has a dual object. If this holds, the dual object is unique up to canonical isomorphism, and is denoted A^*. In a bit more detail, an object A^* is called the dual of A if it is equipped with two morphisms called the unit \eta_A:I\to A^*\otimes A and the counit \varepsilon_A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
