Discussion of Homotopy Type Theory and Univalent Foundations
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From: "Martín Hötzel Escardó" <escardo.martin@gmail.com>
To: Homotopy Type Theory <HomotopyTypeTheory@googlegroups.com>
Subject: [HoTT] Question regarding terminology regarding injectivity of objects
Date: Fri, 8 Feb 2019 13:06:07 -0800 (PST)
Message-ID: <0cfb9acf-e5f9-4ee7-979e-f7d66b82a178@googlegroups.com> (raw)

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Question regarding terminology regarding injectivity of objects:

(1) An object D is called injective over an arrow j:X->Y if the 
"restriction map" 

    hom(Y,D) -> hom(X,D) 
        g   |-> g o j 

is a surjection. This is fairly standard terminology (where does it come 
from, by the way). 

(2) I am working with the situation where the restriction map is a **split** 
surjection. 

I thought of the terminology "D is split injective over j", but perhaps 
this is awkward. Is there a standard terminology for this notion. Or, 
failing that, a terminology that at least one person has already used in 
the literature or in the folklore. Or, failing that too, a good suggestion 
by any of you? 

(Before anybody says that there is no difference assuming choice, I remark 
that I am interested in the "infty-category of types (in a universe)" in 
HoTT/UF, where the homs are infty-groupoids and so the section has to be a 
morphism, not just a theoretical function.)

Thanks, 
Martin

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Discussion of Homotopy Type Theory and Univalent Foundations

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