Category theory MOC
Free category
Free categories are the free objects in , #m/def/cat
forming the left adjoint to the forgetful functor to the Underlying quiver
The free category is constructed by considering all composable words, called paths, as morphisms.
Universal property
If is a Small category with Underlying quiver
and is a quiver homomorphism then there exists a unique adjunct such that the following diagram commutes:
%0A%20%20%20%20..%20controls%20(%24(%5Ctikztostart)!%5Cpv%7Bpos%7D!(%5Ctikztotarget)!%5Cpv%7Bheight%7D!270%3A(%5Ctikztotarget)%24)%0A%20%20%20%20and%20(%24(%5Ctikztostart)!1-%5Cpv%7Bpos%7D!(%5Ctikztotarget)!%5Cpv%7Bheight%7D!270%3A(%5Ctikztotarget)%24)%0A%20%20%20%20..%20(%5Ctikztotarget)%5Ctikztonodes%7D%7D%2C%0A%20%20%20%20settings%2F.code%3D%7B%5Ctikzset%7Bquiver%2F.cd%2C%231%7D%0A%20%20%20%20%20%20%20%20%5Cdef%5Cpv%23%231%7B%5Cpgfkeysvalueof%7B%2Ftikz%2Fquiver%2F%23%231%7D%7D%7D%2C%0A%20%20%20%20quiver%2F.cd%2Cpos%2F.initial%3D0.35%2Cheight%2F.initial%3D0%7D%0A%25%20TikZ%20arrowhead%2Ftail%20styles.%0A%5Ctikzset%7Btail%20reversed%2F.code%3D%7B%5Cpgfsetarrowsstart%7Btikzcd%20to%7D%7D%7D%0A%5Ctikzset%7B2tail%2F.code%3D%7B%5Cpgfsetarrowsstart%7BImplies%5Breversed%5D%7D%7D%7D%0A%5Ctikzset%7B2tail%20reversed%2F.code%3D%7B%5Cpgfsetarrowsstart%7BImplies%7D%7D%7D%0A%25%20TikZ%20arrow%20styles.%0A%5Ctikzset%7Bno%20body%2F.style%3D%7B%2Ftikz%2Fdash%20pattern%3Don%200%20off%201mm%7D%7D%0A%25%20https%3A%2F%2Fq.uiver.app%2F%23q%3DWzAsNSxbMCwwLCIgQ1xcR2FtbWEiXSxbMCwyLCJcXG1hdGhzZiBEIl0sWzIsMCwiVUNcXEdhbW1hIl0sWzIsMiwiVVxcbWF0aHNmIEQiXSxbNCwwLCJcXEdhbW1hIl0sWzAsMSwiXFxleGlzdHMhIGciLDIseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XSxbNCwyLCJcXGV0YV9DIiwyXSxbNCwzLCJmIl0sWzIsMywiVWciLDJdXQ%3D%3D%0A%5Cbegin%7Btikzcd%7D%0A%09%7B%20C%5CGamma%7D%20%26%26%20%7BUC%5CGamma%7D%20%26%26%20%5CGamma%20%5C%5C%0A%09%5C%5C%0A%09%7B%5Cmathsf%20D%7D%20%26%26%20%7BU%5Cmathsf%20D%7D%0A%09%5Carrow%5B%22%7B%5Cexists!%20g%7D%22'%2C%20dashed%2C%20from%3D1-1%2C%20to%3D3-1%5D%0A%09%5Carrow%5B%22Ug%22'%2C%20from%3D1-3%2C%20to%3D3-3%5D%0A%09%5Carrow%5B%22%7B%5Ceta_C%7D%22'%2C%20from%3D1-5%2C%20to%3D1-3%5D%0A%09%5Carrow%5B%22f%22%2C%20from%3D1-5%2C%20to%3D3-3%5D%0A%5Cend%7Btikzcd%7D%0A#invert)
#state/tidy | #lang/en | #SemBr