### Rainer Vogt

#### Posted by Tom Leinster

I was sad to learn that Rainer Vogt died last month. He is probably
best-known to Café readers for his work on homotopy-algebraic
structures, especially his seminal 1973 book with Michael Boardman,
*Homotopy Invariant Algebraic Structures on Topological Spaces*.

It was Boardman and Vogt who first studied simplicial sets satisfying what
they called the “restricted Kan condition”, later renamed
*quasi-categories* by André Joyal, and today (thanks especially to
Jacob Lurie) the most deeply-explored incarnation of the notion of
$(\infty, 1)$-category. Their 1973 book also asked and answered
fundamental questions like this:

Given a topological group $X$ and a homotopy equivalence between $X$ and another space $Y$, what structure does $Y$ acquire?

Clearly $Y$ is some kind of “group up to homotopy” — but the details take some working out, and Boardman and Vogt did just that.

Martin Markl wrote a nice tribute to Vogt, which I reproduce with permission here:

Martin Markl writes:The first time I encountered the name Rainer Vogt was during my PhD study in Praha, when among randomly chosen books I was reading appeared a Russian translation of Boardman and Vogt’s “Homotopy invariant algebraic structures on topological spaces.” I did not expect that this kind of structures would turn to be the central theme of my professional career. I did not know who Rainer Vogt was either, I only realized that, along with Rainer Maria Rilke, he was the only person christened “Rainer” I knew.I met Rainer in person several years later, in 1998, when he delivered plenary talks at the 18th Winter School “Geometry and Physics” in Srni, a remote Bohemian village of Sumava Forest. It stricken me how he physically resembled my grandfather from the mother side. He obviously knew about my humble work on operads, and invited me to participate in the “Workshop on Operads” in Osnabrück, in June of the same year. I have been visiting Rainer regularly since.

Rainer was not only an excellent mathematician, but also a devoted amateur choir singer. Once I visited him shortly before Christmas. He brought me directly from the train station to a church in a neighboring village, where he sung in Bach’s Weihnachtsoratorium. I sat in the first row next to the priest who made frequent comments to me, not realizing that I do not understand a word. Another day he brought me to the house of his music teacher, where he rehearsed with some other people. I vividly remember his performing, in German, an aria from Smetana’s Bartered Bride, a kitschy comic opera which is considered a Czech national gem.

I learned about his serious illness during my stay at the Max-Planck-Institut für Matematik in Bonn in Winter 2014. Together with Michael Batanin and Clemens Berger, I visited him in a hospital in Osnabrück, and then once again shortly after his return home. He told me that listening to the record of Handel’s “Theodora” I brought to the hospital helped him greatly.

He was full of optimism, willing to fight his fate. I have been indirectly making inquiries about his health since, and was always assured that he was doing well. I believed I would meet him again in Bonn in January 2016. I was deeply shattered when I learned that Rainer died a couple of weeks ago.

Martin Markl, Praha, 26th of August 2015

A happy episode in my own mathematical life was an invitation from Rainer in 2000 to another workshop in operads, in his home university of Osnabrück, when I was a postdoc. That was one of the first mathematical invitations I received to anything anywhere, so it has a special place in my heart, and I have very pleasant memories of that week. In my conversations with him both then and by email in the years following, Rainer never spoke down to me or made me feel that he was a senior person and I was a young whippersnapper — something that I probably took for granted then, but now appreciate as a major virtue.

I don’t have the expertise to do full justice to Rainer Vogt’s work, but please feel free to add to what I’ve written in the comments.

## Re: Rainer Vogt

I never met Vogt, but his papers are underrated gems. I remember first stumbling on certain papers, that I had never even heard of, and being awestruck by their beauty and clarity. Young mathematicians can be well-served by reading Vogt; I think there are lots of half-forgotten ideas in those papers that are worth holding on to. Try “Convenient categories of topological spaces…” or “Modules of topological spaces…” as two particular examples. Thanks for remembering him here.