nLab
imaginary element

The category of definable sets and definable functions for a fixed language $L$ (or more generally, for an $L$-theory $T$) does not have finite colimits. Given a 1-st order language $L$, and a theory $T$ in $L$, we say that $T$ admits/has elimination of imaginaries if it one can take the quotients of definable sets by equivalence relations.

• Bruno Poizat, Une théorie de Galois imaginaire, J. Symbolic Logic 48 (1984), no.4, 1151-1170, MR85e:03083, doi
• wikipedia imaginary element
• Anand Pillay, Some remarks on definable equivalence relations in O-minimal structures, J. Symbolic Logic 51 (1986), 709-714, MR87h:03046, doi
• Jan Holly, Definable operations on sets and elimination of imaginaries, Proc. Amer. Math. Soc. 117 (1993), no. 4, 1149–1157, MR93e:03052, doi, pdf
• Ehud Hrushovski, Groupoids, imaginaries and internal covers, arxiv/math.LO/0603413; On finite imaginaries, arxiv/0902.0842
• D. Haskell, E. Hrushovski, H.D.Macpherson, Definable sets in algebraically closed valued fields: elimination of imaginaries, J. reine und angewandte Mathematik 597 (2006)
• Saharon Shelah, Classification theory and the number of non-isomorphic models, Studies in Logic and the Foundations of Mathematics 92, North Holland, Amsterdam 1978