Is the category of convergence spaces cartesian-closed? Convergence spaces are a generalization of topological spaces; we denote the category of convergence spaces with continuous maps with ${\bf Conv}$. Is ${\bf Conv}$ cartesian-closed?
 A: There seemed to be several slightly different notions of convergence spaces that were considered extensively in the 70ies. So I apologize if the following references do not actually answer your question (because I am missing some subtle differences between the definitions used in the papers below and the definition you linked to). 
For the convergence spaces of Cook and Fischer, Theorem 5 in the following paper should prove that they are cartesian closed: 


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*H.R. Fischer and C.H. Cook. On Equicontinuity and Continuous Convergence. Math. Ann. 159 (1965), pp. 94-104. (available here)
For a slightly different notion of convergence, the $L\ast$-spaces, there is the following paper: 


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*G.A. Edgar. A Cartesian closed category for topology.
General Topology and Appl. 6 (1976), no. 1, 65–72. (available here)
There are also uniform convergence spaces, cartesian closed by the following paper: 


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*R.S. Lee. The category of uniform convergence spaces is Cartesian closed.
Bull. Austral. Math. Soc. 15 (1976), no. 3, 461–465, 
DOI: 10.1017/S0004972700022905. 

