From mboxrd@z Thu Jan 1 00:00:00 1970 X-Received: by 10.129.193.71 with SMTP id e7mr7645524ywl.185.1520285421544; Mon, 05 Mar 2018 13:30:21 -0800 (PST) X-BeenThere: homotopytypetheory@googlegroups.com Received: by 10.129.122.131 with SMTP id v125ls870479ywc.40.gmail; Mon, 05 Mar 2018 13:30:19 -0800 (PST) X-Received: by 10.129.78.84 with SMTP id c81mr7174987ywb.214.1520285419972; Mon, 05 Mar 2018 13:30:19 -0800 (PST) ARC-Seal: i=1; a=rsa-sha256; t=1520285419; cv=none; d=google.com; s=arc-20160816; b=XY0djyUrvGsmCpT0Z0hQtVV4GYTaBLY815iPO67bb9a/lSB4Gu3D3SyA4MZY5bh+W1 6QjmN1XfX7K07hbqmkLlEixbeaIadazOP+YrRyJBTpqSXh7C93Ll1T4zfguHpjaa6oS2 YcQUhlevMQEKqDQbIMkpXhHcXS9gBnvmpJeNTGCIYhBnotM3NT3YOfypK3Okmw4SPFb5 14Cy8VeCNXNBUb+nHsjnEzrVx76SzilCG1t094t+wHDbZ4+NNIzUhpxKW88YTV1t+VNA B3WT84GORd8b5fyLyEGd7urQfRZ/eWXUuGh1dnUKamp9VrJj5Y6Iufq/rX+4oYrjYGCG g6DA== ARC-Message-Signature: i=1; a=rsa-sha256; c=relaxed/relaxed; d=google.com; s=arc-20160816; h=to:subject:message-id:date:from:mime-version:dkim-signature :arc-authentication-results; bh=U21LMIRH6p8/kk29tpvzSY+vl/IdGZ8/hGH62kwk19c=; b=AbxAJLAyUC1URYYtZCa2LrnEk6iYEjwFHguoJKUpcQc7+QFXGdzP4/1WEONGvC6aMz frsJEZ806WiRPs4DJaldlfX/Lk34NfWhjH6rEo9AR5Yx7wAPjKebD4CiQAHyg3nP5kAq OTq0PWZ0ezGjYO8vrw28PFpJlAVzqU5X9qpY6qELCbqTXOfYgTutvzF9ogbAOAi7TUHO PSXuw0gKemtzMBAyTfE6SILyaqGj822qiuraGqZ/lBcLEiVQGt01G1fhRvztjAGazwiq kTbq26SzWaD4JCH1msby+I7n8eYM+ObOfx/dkC1/qlMHHBCnAd2ZQFhnOohtODUcg79C zOHA== ARC-Authentication-Results: i=1; gmr-mx.google.com; dkim=pass head...@sandiego-edu.20150623.gappssmtp.com header.s=20150623 header.b=kNOize1x; spf=neutral (google.com: 2607:f8b0:4002:c05::236 is neither permitted nor denied by domain of shu...@sandiego.edu) smtp.mailfrom=shu...@sandiego.edu Return-Path: Received: from mail-yw0-x236.google.com (mail-yw0-x236.google.com. [2607:f8b0:4002:c05::236]) by gmr-mx.google.com with ESMTPS id l13si795632ywm.0.2018.03.05.13.30.19 for (version=TLS1_2 cipher=ECDHE-RSA-AES128-GCM-SHA256 bits=128/128); Mon, 05 Mar 2018 13:30:19 -0800 (PST) Received-SPF: neutral (google.com: 2607:f8b0:4002:c05::236 is neither permitted nor denied by domain of shu...@sandiego.edu) client-ip=2607:f8b0:4002:c05::236; Authentication-Results: gmr-mx.google.com; dkim=pass head...@sandiego-edu.20150623.gappssmtp.com header.s=20150623 header.b=kNOize1x; spf=neutral (google.com: 2607:f8b0:4002:c05::236 is neither permitted nor denied by domain of shu...@sandiego.edu) smtp.mailfrom=shu...@sandiego.edu Received: by mail-yw0-x236.google.com with SMTP id x197so6192711ywg.11 for ; Mon, 05 Mar 2018 13:30:19 -0800 (PST) DKIM-Signature: v=1; a=rsa-sha256; c=relaxed/relaxed; d=sandiego-edu.20150623.gappssmtp.com; s=20150623; h=mime-version:from:date:message-id:subject:to; bh=U21LMIRH6p8/kk29tpvzSY+vl/IdGZ8/hGH62kwk19c=; b=kNOize1xzEwXI4d4rInUKo2AFxPvQZJDXBLAyqIb9GtvStnXVXDceYmH9Z2Mbeugk1 sGXv7JTvkyiX6xzcVn4tarWFe77kq5mx46ZMo9cMufD1FW4aelEb6sZiRb3NQNwsTAeq O8TkEXdWoxYYdg7lkaKoPwFFiT1Bhp4C+eh4OYE62U+M2jSnCu+wxzu+yrqHOHN/FV9K z5lJCCxW+9p1vpAa8Kl3j+NbCsamgWr9VU83Gso6yPujtpUCA96FoVSEsazo6VpjZn4N 3UAi8Md6EoVjxs659nl1FhtV7Cz3SVTVfsleTcqQFZN63BM3ksrcoxJuHY/qdEHz9LFV G+ug== X-Gm-Message-State: APf1xPCx7vhgVvCxDe+cLZOZdNZhw0OvqERQLP8EM/8HgNK/Yp1Q67NY XVMV3DD22NlXnugRtppbKecW25Mt X-Received: by 10.129.68.26 with SMTP id r26mr9883202ywa.371.1520285419564; Mon, 05 Mar 2018 13:30:19 -0800 (PST) Return-Path: Received: from mail-yw0-f177.google.com (mail-yw0-f177.google.com. [209.85.161.177]) by smtp.gmail.com with ESMTPSA id n2sm5380862ywl.41.2018.03.05.13.30.19 for (version=TLS1_2 cipher=ECDHE-RSA-AES128-GCM-SHA256 bits=128/128); Mon, 05 Mar 2018 13:30:19 -0800 (PST) Received: by mail-yw0-f177.google.com with SMTP id m123so3989738ywd.1 for ; Mon, 05 Mar 2018 13:30:19 -0800 (PST) X-Received: by 10.129.48.66 with SMTP id w63mr10237131yww.178.1520285418908; Mon, 05 Mar 2018 13:30:18 -0800 (PST) MIME-Version: 1.0 Received: by 10.223.174.130 with HTTP; Mon, 5 Mar 2018 13:29:56 -0800 (PST) From: Michael Shulman Date: Mon, 5 Mar 2018 13:29:56 -0800 X-Gmail-Original-Message-ID: Message-ID: Subject: Categories with 2-families To: "HomotopyT...@googlegroups.com" Content-Type: text/plain; charset="UTF-8" By a standard construction (described for example at https://ncatlab.org/nlab/show/categorical+model+of+dependent+types), categories with families are equivalent to categories with attributes, i.e. to comprehension categories that are discrete fibrations. However, it seems to me that the attributes-to-families construction can be applied to a non-discrete comprehension category as well, producing something that one might call a "category with 2-families" (or maybe "pseudo-families"?): a category C together with a representable morphism Tm -> Ty of pseudofunctors C^op -> Cat. Have such things been defined or used anywhere before?