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[2607:f8b0:4864:20::d29]) by gmr-mx.google.com with ESMTPS id x9si603712ita.0.2018.12.19.19.24.41 for (version=TLS1_2 cipher=ECDHE-RSA-AES128-GCM-SHA256 bits=128/128); Wed, 19 Dec 2018 19:24:41 -0800 (PST) Received-SPF: pass (google.com: domain of josephcmac@gmail.com designates 2607:f8b0:4864:20::d29 as permitted sender) client-ip=2607:f8b0:4864:20::d29; Received: by mail-io1-xd29.google.com with SMTP id s22so215200ioc.8 for ; Wed, 19 Dec 2018 19:24:41 -0800 (PST) X-Received: by 2002:a6b:f017:: with SMTP id w23mr1062758ioc.12.1545276280898; Wed, 19 Dec 2018 19:24:40 -0800 (PST) MIME-Version: 1.0 From: =?UTF-8?Q?Jos=C3=A9_Manuel_Rodriguez_Caballero?= Date: Wed, 19 Dec 2018 22:24:30 -0500 Message-ID: Subject: [HoTT] Topology is Combinatorics over Finite Fields To: HomotopyTypeTheory@googlegroups.com Content-Type: multipart/alternative; boundary="000000000000cd8aac057d6babf4" X-Original-Sender: josephcmac@gmail.com X-Original-Authentication-Results: gmr-mx.google.com; dkim=pass header.i=@gmail.com header.s=20161025 header.b=Q1EuciQ5; spf=pass (google.com: domain of josephcmac@gmail.com designates 2607:f8b0:4864:20::d29 as permitted sender) smtp.mailfrom=josephcmac@gmail.com; dmarc=pass (p=NONE sp=QUARANTINE dis=NONE) header.from=gmail.com Precedence: list Mailing-list: list HomotopyTypeTheory@googlegroups.com; contact HomotopyTypeTheory+owners@googlegroups.com List-ID: X-Google-Group-Id: 1041266174716 List-Post: , List-Help: , List-Archive: , --000000000000cd8aac057d6babf4 Content-Type: text/plain; charset="UTF-8" Content-Transfer-Encoding: quoted-printable Hello, According to Weil's conjectures (right now, theorems), some topological invariants have combinatorial interpretations related to finite fields. Could the slogan "Topology is Combinatorics over Finite Fields" be justified to some extent? I say that a type T is determined by a set S of algebraic equations over an arbitrary finite field if all the topological invariants of T can be interpreted in a natural way as the number of solutions of the system S.I say that a type T is determined by a set S of algebraic equations over an arbitrary finite field if all the topological invariants of T can be interpreted in a natural way as the number of solutions of the system S. Is any type in HoTT determined by a set of algebraic equations over an arbitrary finite field? I know that these questions may be rather ambiguous. I think that Weil's conjectures are just a particular case of a more general duality between combinatorics over a finite field and topology, but it is hard to find the right way to state the problem. Kind Regards, Jos=C3=A9 M. --=20 You received this message because you are subscribed to the Google Groups "= Homotopy Type Theory" group. To unsubscribe from this group and stop receiving emails from it, send an e= mail to HomotopyTypeTheory+unsubscribe@googlegroups.com. For more options, visit https://groups.google.com/d/optout. --000000000000cd8aac057d6babf4 Content-Type: text/html; charset="UTF-8" Content-Transfer-Encoding: quoted-printable
Hello,
=C2=A0 According to Weil's= conjectures (right now, theorems), some topological invariants have combin= atorial interpretations related to finite fields. Could the slogan "To= pology is Combinatorics over Finite Fields" be justified to some exten= t?

=C2=A0 I say that a type T is determined by a s= et S of algebraic equations over an arbitrary finite field if all the topol= ogical invariants of T can be interpreted in a natural way as the number of= solutions of the system S.I say that a type T is determined by a set S of = algebraic equations over an arbitrary finite field if all the topological i= nvariants of T can be interpreted in a natural way as the number of solutio= ns of the system S. Is any type in HoTT determined by a set of algebraic eq= uations over an arbitrary finite field?

=C2=A0 I k= now that these questions may be rather ambiguous. I think that Weil's c= onjectures are just a particular case of a more general duality between com= binatorics over a finite field and topology, but it is hard to find the rig= ht way to state the problem.

Kind Regards,
Jos=C3=A9 M.

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