{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,6,10]],"date-time":"2024-06-10T01:10:14Z","timestamp":1717981814757},"reference-count":16,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2015,5,7]],"date-time":"2015-05-07T00:00:00Z","timestamp":1430956800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["The Review of Symbolic Logic"],"published-print":{"date-parts":[[2015,12]]},"abstract":"Abstract<\/jats:title>If${\\cal C}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>is a category whose objects are themselves categories, and${\\cal C}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>has a rich enough structure, it is known that we can recover the internal structure of thecategories<\/jats:italic>in${\\cal C}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>entirely in terms of thearrows<\/jats:italic>in${\\cal C}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. In this sense, the internal structure of the categories in a rich enough category of categories is visible in the structure of the category of categories itself.<\/jats:p>In this paper, we demonstrate that this result follows as a matter of logic \u2013 given one starts from the right definitions. This is demonstrated by first producing an abstraction principle whose abstracts are functors, and then actually recovering the internal structure of the individual categories that intuitively stand at the sources and targets of these functors by examining the way these functors interact. The technique used in this construction will be useful elsewhere, and involves providing an abstract corresponding not to everyobject<\/jats:italic>of some given family, but to all the relevantmappings<\/jats:italic>of some family of objects.<\/jats:p>This construction should settle, in particular, questions about whether categories of categories qualify asautonomous<\/jats:italic>mathematical objects \u2013 categories of categories are perfectly acceptable autonomous objects and thus, in particular, suitable for foundational purposes.<\/jats:p>","DOI":"10.1017\/s1755020315000167","type":"journal-article","created":{"date-parts":[[2015,7,16]],"date-time":"2015-07-16T06:07:25Z","timestamp":1437026845000},"page":"705-721","source":"Crossref","is-referenced-by-count":1,"title":["ABSTRACTIONIST CATEGORIES OF CATEGORIES"],"prefix":"10.1017","volume":"8","author":[{"given":"SHAY ALLEN","family":"LOGAN","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2015,5,7]]},"reference":[{"key":"S1755020315000167_ref1","unstructured":"Bednarczyk M. A. , & Borzyszkowski A. M . (1998). Concrete (co) constructions in the category of small categories. Unpublished manuscript. http:\/\/citeseerx.ist.psu.edu\/viewdoc\/download?doi=10.1.1.116.5522&rep=rep1&type=pdf."},{"key":"S1755020315000167_ref6","doi-asserted-by":"publisher","DOI":"10.1093\/0199241279.003.0012"},{"key":"S1755020315000167_ref2","doi-asserted-by":"publisher","DOI":"10.2307\/421158"},{"key":"S1755020315000167_ref9","first-page":"64","article-title":"Pluralism and the foundations of mathematics","volume":"19","author":"Hellman","year":"2006","journal-title":"Scientific Pluralism"},{"key":"S1755020315000167_ref16","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1093\/oso\/9780198239208.003.0008","volume-title":"Language, Thought, and Logic, Essays in Honour of Michael Dummett.","author":"Wright","year":"1997"},{"key":"S1755020315000167_ref5","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780199246182.001.0001","volume-title":"The Limits of Abstraction","author":"Fine","year":"2002"},{"key":"S1755020315000167_ref10","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.50.5.869"},{"key":"S1755020315000167_ref8","volume-title":"Frege\u2019s Theorem","author":"Heck","year":"2011"},{"key":"S1755020315000167_ref11","doi-asserted-by":"publisher","DOI":"10.1093\/philmat\/nkr024"},{"key":"S1755020315000167_ref12","doi-asserted-by":"crossref","unstructured":"Logan S . (Forthcoming). Categories for the neologicist. 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