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• 10:24, 9 April 2025 Leigh.Samphier talk contribs created page User:Leigh.Samphier/Topology/Adjunction Induces Unit of Adjunction (Created page with "{{Proofread}} == Theorem == Let $\mathbf {Set}$ be the category of sets. Let $\mathbf C$, $\mathbf D$ be locally small categories. Let $\tuple {F, G, \alpha}$ be an adjunction between $\mathbf C$ and $\mathbf D$. The '''unit of adjunction $\tuple {F, G, \alpha}$''' is the Definition:Natural Transformation|...")
• 10:12, 9 April 2025 Leigh.Samphier talk contribs created page User:Leigh.Samphier/Topology/Adjunction Induces Counit of Adjunction (Created page with "{{Proofread}} == Theorem == Let $\mathbf {Set}$ be the category of sets. Let $\mathbf C$, $\mathbf D$ be locally small categories. Let $\tuple {F, G, \alpha}$ be an adjunction between $\mathbf C$ and $\mathbf D$. Then: :there exists a counit of adjunction $\tuple {F, G, \alpha}$ That is, there exists a ...")
• 09:27, 9 April 2025 Leigh.Samphier talk contribs created page Definition talk:Natural Transformation/Covariant Functors (Change of Notation: new section)
• 12:12, 8 April 2025 Leigh.Samphier talk contribs created page User:Leigh.Samphier/Topology/Definition:Counit of Adjunction (Created page with "== Definition == Let $\mathbf {Set}$ be the category of sets. Let $\mathbf C$, $\mathbf D$ be locally small categories. Let $\tuple {F, G, \alpha}$ be an adjunction between $\mathbf C$ and $\mathbf D$. The '''unit of adjunction $\tuple {F, G, \alpha}$''' is the Definition:Natural Transformation|natural tran...")
• 11:35, 8 April 2025 Leigh.Samphier talk contribs created page User:Leigh.Samphier/CategoryTheory/Category CLat is Subcategory of Frm (Created page with "{{Proofread}} == Theorem == Let $\mathbf{Frm}$ denote the category of frames. Let $\mathbf{CLat}$ denote the category of complete lattices. Then: :$\mathbf{CLat}$ is a subcategory of $\mathbf{Frm}$ == Proof == {{qed}} Category:Subcategory Category:Category Theory")
• 10:59, 8 April 2025 Leigh.Samphier talk contribs created page User:Leigh.Samphier/Topology/Frame is Distributive Lattice (Created page with "== Theorem == Let $L= \struct{S, \preceq}$ be a frame. Then: :$\struct {S, \vee, \wedge, \preceq}$ is a distributive lattice where $\vee$ and $\wedge$ denote the join and meet operations on $S$, respectively. == Proof == {{qed}} Category:Frames Category:Ditributive Lattices")
• 10:40, 8 April 2025 Leigh.Samphier talk contribs created page User:Leigh.Samphier/Topology/Definition:Frame Isomorphism/Definition 4 (Created page with "== Definition == Let $L_1 = \struct {S_1, \preceq_1}$ and $L_2 = \struct {S_2, \preceq_2}$ be frames. <onlyinclude> Let $\phi: S_1 \to S_2$ be a mapping. We say $\phi: L_1 \to L_2$ is a '''frame isomorphism''' {{iff}} $\phi : L_1 \to L_2$ is a complete lattice isomorphism </o...")
• 10:37, 8 April 2025 Leigh.Samphier talk contribs created page User:Leigh.Samphier/Topology/Definition:Frame Isomorphism/Definition 3 (Created page with "== Definition == Let $L_1 = \struct {S_1, \preceq_1}$ and $L_2 = \struct {S_2, \preceq_2}$ be frames. <onlyinclude> Let $\phi: S_1 \to S_2$ be a mapping. We say $\phi: L_1 \to L_2$ is a '''frame isomorphism''' {{iff}}: :$\phi : \struct {S_1, \vee_1, \wedge_1, \preceq_1} \to \struct {S_2, \vee_2, \wedge_2, \preceq_2}$ is a Definition:Lattice Isom...")
• 09:27, 1 April 2025 Leigh.Samphier talk contribs created page User:Leigh.Samphier/Topology/Identity Mapping is Complete Lattice Homomorphism (Created page with "{{Proofread}} == Theorem == Let $L = \struct{A, \preceq}$ be a complete lattice. Let $\operatorname{id}_A$ denote the identity mapping on $A$. Then: :$\operatorname{id}_A$ is a complete lattice homomorphism of $L$ to $L$ == Proof == {{qed}} Category:Complete Lattice Homomorphisms")
• 09:21, 1 April 2025 Leigh.Samphier talk contribs created page User:Leigh.Samphier/Topology/Composite Complete Lattice Homomorphisms is Complete Lattice Homomorphism (Created page with "{{Proofread}} == Theorem == Let $L_1 = \struct{A_1, \preceq_1}$, $L_2 = \struct{A_2, \preceq_2}$ and $L_3 = \struct{A_3, \preceq_3}$ be complete lattices. Let $\phi_1: L_1 \to L_2$ and $\phi_2: L_2 \to L_3$ be complete lattice homomorphisms. Let $\phi_2 \circ \phi_1 : A_1 \to A_3$ be the composite mapping of $\phi_1$ and $\phi_2$...")
• 08:40, 1 April 2025 Leigh.Samphier talk contribs deleted page User:Leigh.Samphier/CategoryTheory/Category CLat is Subcategory of Lat (content was: "{{Proofread}} == Theorem == Let $\mathbf {CLat}$ denote the category of complete lattices. Let $\mathbf {Lat}$ denote the category of lattices. Then: :$\mathbf {CLat}$ is a subcategory of $\mathbf {Lat}$ == Pr...", and the only contributor was "Leigh.Samphier" (talk))
• 08:36, 1 April 2025 Leigh.Samphier talk contribs deleted page User:Leigh.Samphier/CategoryTheory/Category Frm is Subcategory of Lat (content was: "{{Proofread}} == Theorem == Let $\mathbf {Frm}$ denote the category of frames. Let $\mathbf {Lat}$ denote the category of lattices. Then: :$\mathbf {Frm}$ is a subcategory of $\mathbf {Lat}$ == Proof == Follows immediately from: * User:Leigh.Samphier/C...", and the only contributor was "Leigh.Samphier" (talk))
• 08:35, 1 April 2025 Leigh.Samphier talk contribs deleted page User:Leigh.Samphier/CategoryTheory/Category Frm is Subcategory of DLat (content was: "{{Proofread}} == Theorem == Let $\mathbf {Frm}$ denote the category of frames. Let $\mathbf {DLat}$ denote the category of distributive lattices. Then: :$\mathbf {Frm}$ is a subcategory of $\mathbf {DLat}$ == Proof == By definition of Defi...", and the only contributor was "Leigh.Samphier" (talk))
• 11:57, 30 March 2025 Leigh.Samphier talk contribs deleted redirect User:Leigh.Samphier/Topology/Equivalence of Definitions of Complete Lattice Isomorphism/Definition 1 Implies Definition 2 by overwriting (Deleted to make way for move from "User:Leigh.Samphier/Topology/Equivalence of Definitions of Complete Lattice Isomorphism/Definition 3 Implies Definition 2")
• 11:57, 30 March 2025 Leigh.Samphier talk contribs moved page User:Leigh.Samphier/Topology/Equivalence of Definitions of Complete Lattice Isomorphism/Definition 3 Implies Definition 2 to User:Leigh.Samphier/Topology/Equivalence of Definitions of Complete Lattice Isomorphism/Definition 1 Implies Definition 2 over redirect
• 11:55, 30 March 2025 Leigh.Samphier talk contribs deleted page User:Leigh.Samphier/Topology/Equivalence of Definitions of Complete Lattice Isomorphism/Definition 1 Implies Definition 3 (content was: "#REDIRECT User:Leigh.Samphier/Topology/Equivalence of Definitions of Complete Lattice Isomorphism/Definition 2 Implies Definition 3", and the only contributor was "Leigh.Samphier" (talk))
• 11:50, 30 March 2025 Leigh.Samphier talk contribs moved page User:Leigh.Samphier/Topology/Equivalence of Definitions of Complete Lattice Isomorphism/Definition 1 Implies Definition 3 to User:Leigh.Samphier/Topology/Equivalence of Definitions of Complete Lattice Isomorphism/Definition 2 Implies Definition 3
• 11:45, 30 March 2025 Leigh.Samphier talk contribs deleted page User:Leigh.Samphier/Topology/Equivalence of Definitions of Complete Lattice Isomorphism/Definition 2 Implies Definition 3 (content was: "#REDIRECT User:Leigh.Samphier/Topology/Equivalence of Definitions of Complete Lattice Isomorphism/Definition 1 Implies Definition 3", and the only contributor was "Leigh.Samphier" (talk))
• 11:41, 30 March 2025 Leigh.Samphier talk contribs moved page User:Leigh.Samphier/Topology/Equivalence of Definitions of Complete Lattice Isomorphism/Definition 2 Implies Definition 3 to User:Leigh.Samphier/Topology/Equivalence of Definitions of Complete Lattice Isomorphism/Definition 1 Implies Definition 3
• 11:37, 30 March 2025 Leigh.Samphier talk contribs moved page User:Leigh.Samphier/Topology/Equivalence of Definitions of Complete Lattice Isomorphism/Definition 1 Implies Definition 2 to User:Leigh.Samphier/Topology/Equivalence of Definitions of Complete Lattice Isomorphism/Definition 3 Implies Definition 2
• 08:25, 24 March 2025 Leigh.Samphier talk contribs deleted page User:Leigh.Samphier/Topology/Complete Lattice is a Frame (content was: "== Theorem == Let $L = \struct{S, \preceq}$ be a complete lattice. Then: :$L$ is a frame == Proof == By definition of complete lattice: :$\forall T \subseteq S: T$ admits both a supremum and an Definition:Infimum of S...", and the only contributor was "Leigh.Samphier" (talk))
• 08:23, 24 March 2025 Leigh.Samphier talk contribs created page User:Leigh.Samphier/Topology/Complete Lattice is a Frame (Created page with "== Theorem == Let $L = \struct{S, \preceq}$ be a complete lattice. Then: :$L$ is a frame == Proof == By definition of complete lattice: :$\forall T \subseteq S: T$ admits both a supremum and an infimum in $L$ Hence: :$\forall x, y \in S$, the join and Definition:Meet...")
• 11:09, 17 March 2025 Leigh.Samphier talk contribs created page User:Leigh.Samphier/Topology/Complete Lattice Homomorphism is Frame Homomorphism (Created page with "== Theorem == Let $L_1 = \struct {S_1, \preceq_1}$ and $L_2 = \struct {S_2, \preceq_2}$ be complete lattices. Let $\phi: L_1 \to L_2$ be a complete lattice homomorphsim between $L_1$ and $L_2$. Then: :$\phi$ is a frame homomorphism == Proof == By definition of User:Leigh.Samphier/Topology/Definition:Complete Lattice Homomorp...")
• 10:28, 17 March 2025 Leigh.Samphier talk contribs created page User:Leigh.Samphier/Topology/Complete Lattice is Lattice (Created page with "== Theorem == Let $L= \struct{S, \preceq}$ be a complete lattice. Then: :$L= \struct{S, \preceq}$ is a lattice == Proof == Follows from the definitions: :* complete lattice :* lattice {{qed}} Category:Complete Lattices Category:Lattices")
• 09:57, 16 March 2025 Leigh.Samphier talk contribs created page User:Leigh.Samphier/CategoryTheory/Category Frm is Subcategory of DLat (Created page with "{{Proofread}} == Theorem == Let $\mathbf {Frm}$ denote the category of frames. Let $\mathbf {Lat}$ denote the category of lattices. Then: :$\mathbf {Frm}$ is a subcategory of $\mathbf {Lat}$ == Proof == By definition of frame: :every frame is a complete lattice...")
• 11:28, 14 March 2025 Leigh.Samphier talk contribs created page User:Leigh.Samphier/CategoryTheory/Complete Lattice Isomorphism is Isomorphism in Category CLat (Created page with "{{Proofread}} == Theorem == Let $\mathbf{CLat}$ denote the category of complete lattices. Let $f : L_1 \to L_2$ be a morphism of $\mathbf{Lat}$. Then: :$f$ is an isomorphism of $\mathbf{CLat}$ {{iff}} $f$ is a complete lattice isomorphsm == Proo...")
• 11:10, 14 March 2025 Leigh.Samphier talk contribs created page User:Leigh.Samphier/CategoryTheory/Category of Complete Lattices is Category (Created page with "{{Proofread}} == Theorem == Let $\mathbf {CLat}$ denote the category of complete lattices. Then: :$\mathbf {CLat}$ is a metacategory == Proof == From User:Leigh.Samphier/CategoryTheory/Category CLat is Subcategory of Lat: :$\mathbf {CLat}$ is a subcategory of the category of lattices By defin...")
• 10:36, 14 March 2025 Leigh.Samphier talk contribs created page User:Leigh.Samphier/CategoryTheory/Category CLat is Subcategory of Lat (Created page with "{{Proofread}} == Theorem == Let $\mathbf {CLat}$ denote the category of complete lattices. Let $\mathbf {Lat}$ denote the category of lattices. Then: :$\mathbf {CLat}$ is a subcategory of $\mathbf {Lat}$ == Proof == By definition of distributive lattice: :a distributive lattice is a De...")
• 10:24, 14 March 2025 Leigh.Samphier talk contribs created page User:Leigh.Samphier/Topology/Inverse of Complete Lattice Isomorphism is Complete Lattice Isomorphism (Created page with "{{Proofread}} == Theorem == Let $L_1 = \struct {A_1, \preceq_1}$ and $L_2 = \struct {A_2, \preceq_2}$ be complete lattices. Let $\phi: L_1 \to L_2$ be a complete lattice isomorphism. Let $\phi^{-1} : A_2 \to A_1$ be the inverse of $\phi : A_1 \to A_2$. Then: :$\phi^{-1} : L_2 \to L_1$ is a User:Leigh.Samphier/Topology/Definition:...")
• 11:15, 13 March 2025 Leigh.Samphier talk contribs deleted page User:Leigh.Samphier/CategoryTheory/Definition:Category of Distributive Lattices (content was: "#REDIRECT Definition:Category of Distributive Lattices", and the only contributor was "Leigh.Samphier" (talk))
• 11:14, 13 March 2025 Leigh.Samphier talk contribs deleted page User:Leigh.Samphier/CategoryTheory/Composite Semilattice Homomorphisms is Semilattice Homomorphism (content was: "#REDIRECT Composite Semilattice Homomorphisms is Semilattice Homomorphism", and the only contributor was "Leigh.Samphier" (talk))
• 11:11, 13 March 2025 Leigh.Samphier talk contribs created page User:Leigh.Samphier/Topology/Complete Lattice Homomorphism is Lattice Homomorphism (Created page with "{{Proofread}} == Theorem == Let $L_1 = \struct {A_1, \vee_1, \wedge_1, \preceq_1}$ and $L_2 = \struct {A_2, \vee_2, \wedge_2, \preceq_2}$ be complete lattices. Let $\phi : L_1 \to L_2$ be a complete lattice homomorphism. Then: :$\phi : L_1 \to L_2$ is an Lattice Homomorphism == Proof == <onlyinclude> </onlyinclude>{{qed}} ...")
• 11:05, 13 March 2025 Leigh.Samphier talk contribs created page User:Leigh.Samphier/Topology/Equivalence of Definitions of Complete Lattice Isomorphism/Definition 3 Implies Definition 1 (Created page with "{{Proofread}} == Theorem == Let $L_1 = \struct {A_1, \vee_1, \wedge_1, \preceq_1}$ and $L_2 = \struct {A_2, \vee_2, \wedge_2, \preceq_2}$ be complete lattices. Let $\phi : L_1 \to L_2$ is an order isomorphism. Then: :$\phi : L_1 \to L_2$ is a lattice isomorphism == Proof == <onlyinclude> </onlyinclude>{{qed}} Category:Lattice Isomorphisms Category:Equivalence...")
• 11:03, 13 March 2025 Leigh.Samphier talk contribs created page User:Leigh.Samphier/Topology/Equivalence of Definitions of Complete Lattice Isomorphism/Definition 2 Implies Definition 3 (Created page with "{{Proofread}} == Theorem == Let $L_1 = \struct {A_1, \vee_1, \wedge_1, \preceq_1}$ and $L_2 = \struct {A_2, \vee_2, \wedge_2, \preceq_2}$ be complete lattices. Let $\phi : L_1 \to L_2$ is a bijective complete lattice homomorphism. Then: :$\phi : L_1 \to L_2$ is a order isomorphism == Proof == <onlyinclude...")
• 10:32, 13 March 2025 Leigh.Samphier talk contribs created page User:Leigh.Samphier/Topology/Equivalence of Definitions of Complete Lattice Isomorphism/Definition 1 Implies Definition 2 (Created page with "{{Proofread}} == Theorem == Let $L_1 = \struct {A_1, \vee_1, \wedge_1, \preceq_1}$ and $L_2 = \struct {A_2, \vee_2, \wedge_2, \preceq_2}$ be complete lattices. {{TFAE|def=Complete Lattice Isomorphism}} === Definition 1 === {{:User:Leigh.Samphier/Topology/Definition:Complete Lattice Isomorphism/Definition 1}} === User:Leigh.Samphier/Topology/Definit...")
• 10:13, 13 March 2025 Leigh.Samphier talk contribs created page User:Leigh.Samphier/Topology/Equivalence of Definitions of Complete Lattice Isomorphism (Created page with "{{Proofread}} == Theorem == <onlyinclude> Let $L_1 = \struct {A_1, \vee_1, \wedge_1, \preceq_1}$ and $L_2 = \struct {A_2, \vee_2, \wedge_2, \preceq_2}$ be complete lattices. {{TFAE|def=Complete Lattice Isomorphism}} === Definition 1 === {{:User:Leigh.Samphier/Topology/Definition:Complete Lattice Isomorphism/Definition 1}} === User:Leigh.Samphier/Top...")
• 09:41, 13 March 2025 Leigh.Samphier talk contribs created page User:Leigh.Samphier/Topology/Definition:Complete Lattice Isomorphism/Definition 3 (Created page with "== Definition == Let $L_1 = \struct {A_1, \preceq_1}$ and $L_2 = \struct {A_2, \preceq_2}$ be complete lattices. <onlyinclude> Let $\phi: A_1 \to A_2$ be a mapping. We say $\phi : L_1 \to L_2$ is a '''complete lattice isomorphism''' {{iff}} $\phi : L_1 \to L_2$ is a lattice isomorphism. </onlyinclude> == [...")
• 09:34, 13 March 2025 Leigh.Samphier talk contribs undeleted page User:Leigh.Samphier/Topology/Definition:Complete Lattice Isomorphism/Definition 2 (1 revision)
• 09:34, 13 March 2025 Leigh.Samphier talk contribs undeleted page User:Leigh.Samphier/Topology/Definition:Complete Lattice Isomorphism/Definition 1 (2 revisions)
• 09:29, 13 March 2025 Leigh.Samphier talk contribs deleted page User:Leigh.Samphier/Topology/Definition:Complete Lattice Isomorphism/Definition 2 (content was: "== Definition == Let $L_1 = \struct {A_1, \preceq_1}$ and $L_2 = \struct {A_2, \preceq_2}$ be complete lattices. <onlyinclude> Let $\phi: A_1 \to A_2$ be a mapping. We say $\phi : L_1 \to L_2$ is a '''User:Leigh.Samphier/Topology/Definition:Complete Lattice Isomorphism|complete lattice isomorphism...", and the only contributor was "Leigh.Samphier" (talk))
• 09:28, 13 March 2025 Leigh.Samphier talk contribs deleted page User:Leigh.Samphier/Topology/Definition:Complete Lattice Isomorphism/Definition 1 (content was: "== Definition == Let $L_1 = \struct {S_1, \preceq_1}$ and $L_2 = \struct {S_2, \preceq_2}$ be complete lattices. <onlyinclude> Let $\phi: L_1 \to L_2$ be a (complete lattice) homomorphism. We say $\phi: L_1 \to L_2$ is a '''User:Leigh.Samphier/Top...", and the only contributor was "Leigh.Samphier" (talk))
• 10:44, 12 March 2025 Leigh.Samphier talk contribs created page User:Leigh.Samphier/Topology/Definition:Complete Lattice Isomorphism/Definition 2 (Created page with "== Definition == Let $L_1 = \struct {A_1, \preceq_1}$ and $L_2 = \struct {A_2, \preceq_2}$ be complete lattices. <onlyinclude> Let $\phi: A_1 \to A_2$ be a mapping. We say $\phi : L_1 \to L_2$ is a '''complete lattice isomorphism''' {{iff}} $\phi : L_1 \to L_2$ is an order isomorphism. </onlyinclude> == De...")
• 10:40, 12 March 2025 Leigh.Samphier talk contribs created page User:Leigh.Samphier/Topology/Definition:Complete Lattice Isomorphism/Definition 1 (Created page with "== Definition == Let $L_1 = \struct {S_1, \preceq_1}$ and $L_2 = \struct {S_2, \preceq_2}$ be complete lattices. <onlyinclude> Let $\phi: L_1 \to L_2$ be a (complete lattice) homomorphism. We say $\phi: L_1 \to L_2$ is a '''complete lattice isomorphism''' {{iff}} $\phi : S_1 \to S_2$ is a Definition...")
• 10:29, 12 March 2025 Leigh.Samphier talk contribs created page User:Leigh.Samphier/Topology/Definition:Complete Lattice Isomorphism (Created page with "== Definition == <onlyinclude> Let $L_1 = \struct {A_1, \preceq_1}$ and $L_2 = \struct {A_2, \preceq_2}$ be complete lattices. === Definition 1 === {{:User:Leigh.Samphier/Topology/Definition:Lattice Isomorphism/Definition 1}} === Definition 2 === {{:User:Leigh.Samphier/Topology...")
• 10:04, 12 March 2025 Leigh.Samphier talk contribs created page Category:Category of Distributive Lattices (Created page with "{{SubjectCategoryNodef}} Category:Category Theory")
• 10:01, 12 March 2025 Leigh.Samphier talk contribs deleted page User:Leigh.Samphier/Topology/Equivalence of Definitions of Lattice Isomorphism (content was: "#REDIRECT Equivalence of Definitions of Lattice Isomorphism", and the only contributor was "Leigh.Samphier" (talk))
• 10:00, 12 March 2025 Leigh.Samphier talk contribs deleted page User:Leigh.Samphier/Topology/Inverse of Lattice Isomorphism is Lattice Isomorphism (content was: "#REDIRECT Inverse of Lattice Isomorphism is Lattice Isomorphism", and the only contributor was "Leigh.Samphier" (talk))
• 10:00, 12 March 2025 Leigh.Samphier talk contribs deleted page User:Leigh.Samphier/CategoryTheory/Category of Distributive Lattices is Category (content was: "#REDIRECT Category of Distributive Lattices is Category", and the only contributor was "Leigh.Samphier" (talk))
• 09:59, 12 March 2025 Leigh.Samphier talk contribs moved page User:Leigh.Samphier/CategoryTheory/Category of Distributive Lattices is Category to Category of Distributive Lattices is Category
• 09:58, 12 March 2025 Leigh.Samphier talk contribs deleted page User:Leigh.Samphier/CategoryTheory/Category DLat is Full Subcategory of Lat (content was: "#REDIRECT Category DLat is Full Subcategory of Lat", and the only contributor was "Leigh.Samphier" (talk))