BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Toby St Clere Smithe (Topos Institute) DTSTART:20220322T170000Z DTEND:20220322T180000Z DTSTAMP:20260423T052624Z UID:Intercats/6 DESCRIPTION:Title: Categories by proxy and the limits of Para\nby Toby St Clere Smithe (Topos Institute) as part of Intercats: Seminar on Categorical Interaction \n\n\nAbstract\nThe notion of parameterization is of great importance in c ategorical cybernetics\, providing space for morphisms to be learnt\, or f or their choice to be 'externally' determined. At the same time\, the conc ept of 'randomness pushback' tells us that the randomness of a stochastic channel can also (in nice circumstances) be so externalized\, leaving inst ead a random choice of deterministic map. The usual perspective on paramet erization is an 'internal' one\, treating the parameter as a modification of a morphism's (co)domain. In general\, however\, this perspective is not wide enough to retain all the structure of the category at hand: an 'exte rnal' perspective seems mathematically\, as well as philosophically\, nece ssary. (In earlier work\, we attempted to provide such an external perspec tive using an enriched-categorical notion of parameterization\, but this i s similarly insufficient.)\n\nHere\, we describe an alternative perspectiv e\, considering an internal category parameterized by its 'external' unive rse. We build an indexed double category over the double category of spans in the universe\, with each base object representing a choice of 'paramet erizing context'. When the internal category has limits or a subobject cla ssifier\, so does its parameterization\; with appropriate quotienting\, so does the corresponding Grothendieck construction. By decorating the spans with (sub)distributions\, the same facts hold true even in the stochastic case\, suggesting semantics for notions of 'stochastic type' and 'stochas tic term'. In this setting\, we can reformulate Bayesian lenses as "Bayesi an dependent optics"\, treating generative models as such stochastic terms .\n LOCATION:https://researchseminars.org/talk/Intercats/6/ END:VEVENT END:VCALENDAR