CiteExport$(function(){PrimeFaces.cw("TieredMenu","widget_formSmash_upper_j_idt166",{id:"formSmash:upper:j_idt166",widgetVar:"widget_formSmash_upper_j_idt166",autoDisplay:true,overlay:true,my:"left top",at:"left bottom",trigger:"formSmash:upper:exportLink",triggerEvent:"click"});}); $(function(){PrimeFaces.cw("OverlayPanel","widget_formSmash_upper_j_idt167_j_idt169",{id:"formSmash:upper:j_idt167:j_idt169",widgetVar:"widget_formSmash_upper_j_idt167_j_idt169",target:"formSmash:upper:j_idt167:permLink",showEffect:"blind",hideEffect:"fade",my:"right top",at:"right bottom",showCloseIcon:true});});

On the local cartesian closure of exact completionsPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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(English)Manuscript (preprint) (Other academic)
##### Abstract [en]

##### Keywords [en]

Exact completion, local cartesian closure, weak limits, internal projective objects
##### National Category

Algebra and Logic
##### Research subject

Mathematics
##### Identifiers

URN: urn:nbn:se:su:diva-159377OAI: oai:DiVA.org:su-159377DiVA, id: diva2:1242506
#####

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt507",{id:"formSmash:j_idt507",widgetVar:"widget_formSmash_j_idt507",multiple:true});
#####

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt519",{id:"formSmash:j_idt519",widgetVar:"widget_formSmash_j_idt519",multiple:true});
#####

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt530",{id:"formSmash:j_idt530",widgetVar:"widget_formSmash_j_idt530",multiple:true}); Available from: 2018-08-28 Created: 2018-08-28 Last updated: 2019-03-29
##### In thesis

A characterisation of cartesian closure of exact completions as a property of the projective objects was given by Carboni and Rosolini. We show that the argument used to prove that characterisation is equivalent to the projectives being closed under binary products (equivalently, being internally projective). The property in question is the existence of weak simple products (a slight strengthening of weak exponentials) and the argument used relies on two claims: that weak simple products endow the internal logic with universal quantification, and that an exponential is the quotient of a weak exponential. We show that either these claims hold if and only if the projectives are internally projectives, which entails that Carboni and Rosolini's characterisation only applies to ex/lex completions. We then argue that this limitation depends on the universal property of weak simple products, and derive from this observation an alternative notion, which we call generalised weak simple product. We conclude by showing that existence of generalised weak simple products in the subcategory of projectives is equivalent to the cartesian closure of the exact category, thus obtaining a complete characterisation of (local) cartesian closure for exact completions of categories with weak finite limits.

1. Exact completion and type-theoretic structures$(function(){PrimeFaces.cw("OverlayPanel","overlay1264924",{id:"formSmash:j_idt827:0:j_idt831",widgetVar:"overlay1264924",target:"formSmash:j_idt827:0:parentLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

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CiteExport$(function(){PrimeFaces.cw("TieredMenu","widget_formSmash_lower_j_idt1339",{id:"formSmash:lower:j_idt1339",widgetVar:"widget_formSmash_lower_j_idt1339",autoDisplay:true,overlay:true,my:"left top",at:"left bottom",trigger:"formSmash:lower:exportLink",triggerEvent:"click"});}); $(function(){PrimeFaces.cw("OverlayPanel","widget_formSmash_lower_j_idt1340_j_idt1342",{id:"formSmash:lower:j_idt1340:j_idt1342",widgetVar:"widget_formSmash_lower_j_idt1340_j_idt1342",target:"formSmash:lower:j_idt1340:permLink",showEffect:"blind",hideEffect:"fade",my:"right top",at:"right bottom",showCloseIcon:true});});