---- > [!definition]+ Definition. ([[fibered two-object coslice category]]) > Flipping most of the arrows in [[fibered two-object slice category]] gives an analog to [[two-object coslice category]], denoted $\mathsf{C}^{\alpha, \beta}$ in the fibered case. Explicitly, the objects of $\mathsf{C}^{\alpha, \beta}$ are commutative diagrams > > ```tikz > \usepackage{tikz-cd} > \begin{document} > % https://tikzcd.yichuanshen.de/#[](multi-object%20coslice%20category.md)ZARgBoAGAXVJADcBDAGwFcYkQBBEAX1PU1z5CKMgCZqdJq3YAhHnxAZseAkVGliEhizaIQAYXn9lQouQ1apukAC0eEmFADm8IqABmAJwgBbJOZAcCCR1SR12AB0IpjQAC3ojEC9fJDJA4MRQxnoAIxhGAAUBFWEQTywnWJwQGm1pPSi8nATeD28-RABmGiD-Wqt2d0Tkju701JpsvMLi0z1yyur+8L0ne24gA > \begin{tikzcd}[arrows=<-] > & A \arrow[rd, "\alpha"] & \\ > Z \arrow[ru, "f"] \arrow[rd, "g"'] & & C \\ > & B \arrow[ru, "\beta"'] & > \end{tikzcd} > \end{document} > ``` > and morphisms > > ```tikz > \usepackage{tikz-cd} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZARgBoAGAXVJADcBDAGwFcYkQBBEAX1PU1z5CKMgCZqdJq3YAhHnxAZseAkVGliEhizaIQAYXn9lQouQ1apukAC0A+sSOKBK4cgDMFmtul6nSwVUUABYvSR12fxdTFABWMJ9re1EokyDkADYKSwi9Ll5jQLcAdgSrdkMC5zS3LPFvcr05bgkYKABzeCJQADMAJwgAWyRzEBwIJHVw3xAAHVmmNAALeid+oaQyMYnEKcZ6ACMYRgAFaKCQPqx2pZwQBty52aOcVar14cRPbZGHmZ61gNPt9xpsaPsjqdzsJLtdbvdptZ2oCNohQj9ELF3kCkKUMQAOP7WeaLFYoz4AThooMQhJAEOOZxq7CuNzuRPY8xebwUHyQWQxeMS7AB2NRAppVPph0Z0JZcPZiPYyJa3CAA > \begin{tikzcd}[arrows=<-] > & A \arrow[rd, "\alpha"] & & & & & A \arrow[rd, "\alpha"] & \\ > Z_1 \arrow[ru, "f"] \arrow[rd, "g"'] & & C & {} & {} \arrow[l] & Z_2 \arrow[ru, "f"] \arrow[rd, "g"'] & & C \\ > & B \arrow[ru, "\beta"'] & & & & & B \arrow[ru, "\beta"'] & > \end{tikzcd} > \end{document> > ``` > > correspond to commutative diagrams > ```tikz > \usepackage{tikz-cd} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZARgBoAGAXVJADcBDAGwFcYkQBBEAX1PU1z5CKMgCZqdJq3YAhHnxAZseAkVGliEhizaIQAYXn9lQouQ1apukAC0A+sSOKBK4cgDMFmtul6nSwVUUABYvSR12fxdTFABWMJ9re1EokyDkADYKSwi9Ll5jQLcAdgSrdkMC5zS3LPFvcr05bgkYKABzeCJQADMAJwgAWyRzEBwIJHVw3xAAHVmmNAALeid+oaQyMYnEKcZ6ACMYRgAFaKCQPqx2pZwQBty52aOcVar14cRPbZGHmZ61gNPt9xpsaPsjqdzsJLtdbvdptZ2oCNohQj9ELF3kCkKUMQAOP7WeaLFYoz4AThooMQhJAEOOZxq7CuNzuRPY8xebwUHyQWQxeMS7AB2NRAppVPph0Z0JZcPZiPYyJa3CAA > \begin{tikzcd}[arrows=<-] > & & A \arrow[rd, "\alpha"] & \\ > Z_1 \arrow[rru, "f_1", bend left] \arrow[rrd, "g_1"', bend right] & Z_2 \arrow[l, "\sigma"] \arrow[ru, "f_2"] \arrow[rd, "g_2"'] & & C \\ > & & B \arrow[ru, "\beta"'] & > \end{tikzcd} > \end{document} > ``` > > Two morphisms compose analogously to in [[two-object slice category]]: first 'concatenate': > ```tikz > \usepackage{tikz-cd} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZARgBpiBdUkANwEMAbAVxiRAC0B9AJhAF9S6TLnyEU3clVqMWbLgGZ+gkBmx4CReaQAMU+s1aIQAQSVC1ozaW56ZhkACEzK4erHIALJOr7ZRgMLOqiIaKNre0gZynMT8UjBQAObwRKAAZgBOEAC2SBIgOBBIXpF+IAA65YxoABZ0zpk5SFoFRYglDHQARjAMAAqulkYZWIk1OCA+dmyVPTj1AulZuYhkrXnUnT39g6EgDDBpE1NRRmmciosgjStrhc2b3b0DFnsjY8el9okXDctI4XWiHyPTAUGagN89nOvCuNwB1HuiBaoPBiAAtPJAVtnrsxCB3uNJl82D9Ycp4YhAUi1lCZuV5kw-k1EABWRFtSHTIyVbCJbILCn-NkcjYgVEQk5lc6xOHC9lAlEwMFITHYp47V74wmfOlGH6yih8IA > \begin{tikzcd}[arrows=<-] > & & & A \arrow[rd, "\alpha"] & \\ > Z_1 \arrow[rrru, "f_1", bend left] \arrow[rrrd, "g_1"', bend right] & Z_2 \arrow[l, "\sigma"] \arrow[rru, "f_2", bend left] \arrow[rrd, "g_2"', bend right] & Z_3 \arrow[l, "\tau"] \arrow[ru, "f_3"] \arrow[rd, "g_3"'] & & C \\ > & & & B \arrow[ru, "\beta"'] & > \end{tikzcd} > \end{document> > ``` > and then 'remove the center': > ```tikz > \usepackage{tikz-cd} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZARgBoAGAXVJADcBDAGwFcYkQBBEAX1PU1z5CKMgCZqdJq3YAhHnxAZseAkVGliEhizaIQAYXn9lQouQ1apukAC0A+sSOKBK4cgDMFmtul6nSwVUUABYvSR12fxdTFABWMJ9re1EokyDkADYKSwi9Ll5jQLcAdgSrdkMC5zS3LPFvcr05bgkYKABzeCJQADMAJwgAWyRzEBwIJHVw3xAAHVmmNAALeid+oaQyMYnEKcZ6ACMYRgAFaKCQPqx2pZwQBty52aOcVar14cRPbZGHmZ61gNPt9xpsaPsjqdzsJLtdbvdptZ2oCNohQj9ELF3kCkKUMQAOP7WeaLFYoz4AThooMQhJAEOOZxq7CuNzuRPY8xebwUHyQWQxeMS7AB2NRAppVPph0Z0JZcPZiPYyJa3CAA > \begin{tikzcd}[arrows=<-] > & & & A \arrow[rd, "\alpha"] & \\ > Z_1 \arrow[rrru, "f_1", bend left] \arrow[rrrd, "g_1"', bend right] & & Z_3 \arrow[ll, "\tau \sigma"] \arrow[ru, "f_3"] \arrow[rd, "g_3"'] & & C \\ > & & & B \arrow[ru, "\beta"'] & > \end{tikzcd} > \end{document} > ``` > What identities are, and that they behave nicely with respect to composition, is clear. Associativity is immediate as well. ^definition ---- #### ---- #### 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