---- > [!definition]+ Definition. ([[lifting]]) > Given morphisms $f:X \to Y$ and $g: Z \to Y$ in a [[category]] $\mathsf{C}$, a **lifting of $f$ to $Z$** is a morphism $h$ satisfying $f=g \circ h$, i.e., the following diagram commutes: > ```tikz \usepackage{tikz-cd} \begin{document} % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBoBGAXVJADcBDAGwFcYkQANEAX1PU1z5CKchWp0mrdgE0efEBmx4CRUcXEMWbRCABaPcTCgBzeEVAAzAE4QAtkjIgcEJKIlb2FkDUb0ARjCMAAoCysIgVljGABY4cpY29ogATDTOrjSaUjrG8SDWdg5pLimZktog0QbcQA \begin{tikzcd} & Z \arrow[d, "g"] \\ X \arrow[r, "f"'] \arrow[ru, "h"] & Y \end{tikzcd} \end{document} >``` > > We say that $f$ **factors through $h$**. ^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 > ``` #reformatrevisebatch01 # Pre-LLM (Don't Edit (unless you are manually restoring from a maintenence required tag, in which case make sure to remove this heading)! Reformatting Occurred at 6/13/2024, 8:14:57 PM) --- tags: - definition Created: 2024-06-05 Modified: 2024-06-05 mathLink: auto aliases: - factors through - lift - lifts --- > [!metadata]- Created::[[2024-06-05]] Modified::[[2024-06-05]] [[Tags]]:: #definition \ Examples:: *[[Examples]]* Nonexamples:: *[[Nonexamples]]* Constructions:: *[[Constructions|Used in the construction of...]]* Specializations:: *[[Specializations]]* Generalizations:: *[[Generalizations]]* Justifications and Intuition:: *[[Justifications and Intuition]]* Properties:: *[[Properties]]* Sufficiencies:: *[[Sufficiencies]]* Equivalences:: *[[Equivalences]]* ---- > [!definition] Definition. ([[lifting]]) > Given morphisms $f:X \to Y$ and $g: Z \to Y$ in a [[category]] $\mathsf{C}$, a **lifting of $f$ to $Z$** is a morphism $h$ satisfying $f=g \circ h$, i.e., the following diagram commutes: > ```tikz \usepackage{tikz-cd} \begin{document} % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBoBGAXVJADcBDAGwFcYkQANEAX1PU1z5CKchWp0mrdgE0efEBmx4CRUcXEMWbRCABaPcTCgBzeEVAAzAE4QAtkjIgcEJKIlb2FkDUb0ARjCMAAoCysIgVljGABY4cpY29ogATDTOrjSaUjrG8SDWdg5pLimZktog0QbcQA \begin{tikzcd} & Z \arrow[d, "g"] \\ X \arrow[r, "f"'] \arrow[ru, "h"] & Y \end{tikzcd} \end{document} >``` > > We say that $f$ **factors through $h$**. ^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 > ``` #reformatrevisebatch01