---- > [!definition] Definition. ([[two-object slice category]]) > Let $\mathsf{C}$ be a [[category]] and $A,B$ two objects in $C$. The **two-object slice category** $\mathsf{C}_{A,B}$ is as follows. > $\text{Obj}(\mathsf{C}_{A,B})$ is defined to be diagrams > > ```tikz > \usepackage{tikz-cd} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBoBGAXVJADcBDAGwFcYkQAtEAX1PU1z5CKcqWLU6TVuwCCPPiAzY8BIqIBMEhizaIQAIR4SYUAObwioAGYAnCAFskZEDghJRkneyvzrdx4jOrkjqNNrSeqYgNIz0AEYwjAAKAirCIDZYpgAWOEbcQA > \begin{tikzcd} > & A \\ > Z \arrow[ru, "f"] \arrow[rd, "g"'] & \\ > & B > \end{tikzcd} > \end{document} > ``` > > in $\mathsf{C}$, and morphisms > ```tikz > \usepackage{tikz-cd} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBoBGAXVJADcBDAGwFcYkQAtAfXJAF9S6TLnyEU5UsWp0mrdgEF+gkBmx4CRCQCZpDFm0QgAQkqFrRRLRV2yDIUyuHqxyAMzWaeuYYeqRGlAAWDxl9dm4tXycLFABWSRsww0UBM38XeJ1PW3YTPmkYKABzeCJQADMAJwgAWyQyEBwIJAlQ7xByngcq2vqaJqQrNrsirppGegAjGEYABWiAkEqsIoALHBAaOFWsco3EAFpyVI7qusR3RubEQJOe8-irpAA2bKSOrki7s6RHgcQAOzjKYzebmRbLNYbN7tUZfSh8IA > \begin{tikzcd} > & A & & & & A \\ > Z_1 \arrow[ru, "f_1"] \arrow[rd, "g_1"', shift left] & & {} \arrow[r] & {} & Z_2 \arrow[ru, "f_2"] \arrow[rd, "g_2"'] & \\ > & B & & & & B > \end{tikzcd} > \end{document} > ``` > > > are *commutative* diagrams > > ```tikz > \usepackage{tikz-cd} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZAJgBoAGAXVJADcBDAGwFcYkQBBEAX1PU1z5CKcqQCM1Ok1bsAWgH0xPPiAzY8BImPGSGLNohALiy-uqFEyxXdIMgAQj0kwoAc3hFQAMwBOEALZI2iA4EEiiUvrsXoogNIz0AEYwjAAKAhrCIIwwXjhxIMlgUEgAzOS83n6BiMGhZTR6MoausfFJKenmmoY+WK4AFvk0RSWIALTllSC+AUE09Yhkkc0gADpr2K7+9AUJyWkZFoZYYNiwpjPVSMuLEU12MSbtB12CPSCn52zTszW3YUQpUatnYrWe2Q6h26WS+WAu3Eo3CAA > \begin{tikzcd} > & & A \\ > Z_1 \arrow[rru, "f_1", bend left] \arrow[rrd, "g_1"', bend right] \arrow[r, "\sigma" description] & Z_2 \arrow[ru, "f_2" description] \arrow[rd, "g_2" description] & \\ > & & B > \end{tikzcd} > \end{document} > ``` > > (i.e., we must have $f_{1}=f_{2} \sigma$ and $g_{1}=g_{2} \sigma$). > > The composition of morphisms > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZAJgBoAGAXVJADcBDAGwFcYkQBBEAX1PU1z5CKcqQCM1Ok1bsAWgH0xPPiAzY8BImPGSGLNohALiy-uqFEyxXdIMgAQqdUCNw5AGYdNPTMMAdPxwYAA8cYHowKG4nNUFNFAAWLyl9OXkTXjM4twBWZJ87BXcYlwsUADYKG1TDLkznc3jkSutvW3ZHbkkYKABzeCJQADMAJwgAWyRtEBwIJFEU3xAhxRAaRnoAIxhGAAVS+JBGGCGcNZBtyKR3cnrRiamaWeu2mpBe1fWtnf3G4RARlhegALM40S5QJAAWhudzGk0Q02eiDIizsAWwvXG9HOG22ewO-ywYGwsCc9wRqORCwK7BWJi++N+2XYxNJbDhDxRTzmiE8aPYHwZR2+BL+rJJWDJnIReRmvMqIqZhIl7POtP8gXozHJ8KQiuRAHZGT8VYY2VK2K8litijL9TykAAOE1ilnmyVk612D52lQUpByo3gmBXPk09qGem6rlB3kui6hyGIGELPGm8WGQEgsECwxCniUbhAA > \begin{tikzcd} > & & A & & & & A \\ > Z_1 \arrow[rru, "f_1", bend left] \arrow[rrd, "g_1"', bend right] \arrow[r, "\sigma" description] & Z_2 \arrow[ru, "f_2" description] \arrow[rd, "g_2" description] & & \text{and} & Z_2 \arrow[r, "\tau" description] \arrow[rru, "f_2", bend left] \arrow[rrd, "g_2"', bend right] & Z_3 \arrow[ru, "f_3" description] \arrow[rd, "g_3" description] & \\ > & & B & & & & B > \end{tikzcd} > \end{document} > ``` > > > is defined to be the diagram obtained by first 'concatenating' thus: > > ```tikz > \usepackage{tikz-cd} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBoBGAXVJADcBDAGwFcYkQAtAfXJAF9S6TLnyEU5CtTpNW7bgCZ+gkBmx4CReZJoMWbRJy4BmJULWiiR0sSm7ZBgIKmVw9WORX5tmfpAAhfikYKABzeCJQADMAJwgAWyQtEBwIJCsQRnoAIxhGAAVXCwMsMGxYEB0fdkjjZxj4xJoUpAAWGkyc-MKNYtKscsq9dhDagSjYhMQJZNTEJI7cgvMekBKytkH7EAAdbZx6ZjqJpGnmxHS7XxrFduzF7rFVvoGQHLAoNOIxkHrJ09m2hk7l1lo81v0NtIhgYRjdXjB3kgALRGL7KX5IMgzNI0N4fc5YhYgkQrcEvS7VHhHBqILFnQFEpYksHPSEUmFU3EI-EotHjGl02bTRkPdhktlVAy7bAhOL0QJ8IA > \begin{tikzcd} > & & & A \\ > Z_1 \arrow[rrru, "f_1" description, bend left] \arrow[rrrd, "g_1" description, bend right] \arrow[r, "\sigma" description] & Z_2 \arrow[r, "\tau" description] \arrow[rru, "f_2" description, bend left] \arrow[rrd, "g_2" description, bend right] & Z_3 \arrow[ru, "f_3" description] \arrow[rd, "g_3" description] & \\ > & & & B > \end{tikzcd} > \end{document} > ``` > > and then 'removing the center' > > ```tikz > \usepackage{tikz-cd} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBoBGAXVJADcBDAGwFcYkQAtAfXJAF9S6TLnyEUAJgrU6TVu24BmfoJAZseAkQWli0hizaIQAQWVD1oraXF7ZhkACF+0mFADm8IqABmAJwgAtkjkNDgQSJIgjPQARjCMAArCGmIgWGDYsCA0+nJG3lxKAj7+QYghIGFI2lGx8UkWmkbpmWw5duxuhWYgfoFIZJXhiJFxYFDVg9FxicmWzRlYWe0G7AW8xb2lA6HDNdP1c01pi8syq0ZdvDRjE4gAtArEm31lg1XlNAezjaktS21znkQAAdEHYNwBegAAjBOHozGcfCAA > \begin{tikzcd} > & & & A \\ > Z_1 \arrow[rrru, "f_1" description, bend left] \arrow[rrrd, "g_1" description, bend right] \arrow[rr, "\sigma \tau" description] & & Z_3 \arrow[ru, "f_3" description] \arrow[rd, "g_3" description] & \\ > & & & B > \end{tikzcd} > \end{document} > ``` > > where the result is commutative because $\mathsf{C}$ is a [[category]]. > > It is clear the identity morphisms exist and behave correctly with respect to composition, and that associativity holds. ^definition > [!intuition] > This is essentially 'just what you get' if you try to extend the definition of [[slice category]] $\mathsf{C}_{A}$ to, instead of considering objects to be morphisms $Z \to A$ entering $A$, consider objects to be *pairs* of morphisms $B \leftarrow Z \to A$. ^intuition ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` #reformatrevisebatch02