Segal's Original Definition of a Topological Category - MathOverflow

Segal's Original Definition of a Topological Category

2012-01-15T05:31:45Z

Nowadays we can associate to a topological space $X$ a category called the fundamental (or Poincare) $\infty$-groupoid given by taking $Sing(X)$.

There are many different categories that one can associate to a space $X$. For example, one could build the small category whose object set is the set of points with only the identity morphisms from a point to itself. It is claimed that the classifying space of this category returns the space: $BX=X$

The inspiration for these examples comes from three primary sources: Graeme Segal's famous 1968 paper Classifying Spaces and Spectral Sequences, Raoul Bott's Mexico notes (taken by Lawrence Conlon) Lectures on characteristic classes and foliations, and a 1995 pre-print called Morse Theory and Classifying Spaces by Ralph Cohen, G. Segal and John Jones.

In each of these papers there is a notion of a topological category. It is not just a category enriched in Top, since the set of objects can have non-discrete topology. Here is the definition that I can gleam from these articles:

A topological category consists of a pair of spaces $(Obj,Mor)$ with four continuous structure maps:

$i:Obj\to Mor$, which sends an object to the identity morphism
$s:Mor\to Obj$, which gives the source of an arrow
$t:Mor\to Obj$, which gives the target of an arrow
$\circ:Mor\times_{t,s}Mor\to Mor$, which is composition.

Were $i$ is a section of both $s$ and $t$, and all the axioms of a small category hold.

Is the appropriate modern terminology to describe this a Segal Space? What would Lurie call it? Based on reading Chris Schommer-Pries MO post and elsewhere this seems to be true. Would the modern definition of the above be a Segal Space where the Segal maps are identities? Also, why do we demand that the topology on objects be discrete for Segal Categories? Is there something wrong with allowing the object sets to have topologies?

Answer by Toby Bartels for Segal's Original Definition of a Topological Category

2012-01-15T06:55:53Z

I would call this an internal category in the category of topological spaces and continuous maps.

Answer by Peter May for Segal's Original Definition of a Topological Category

2012-01-15T16:13:53Z

Could I ask young people to use precise language? Calling a Kan complex an $\infty$-groupoid and asking what kind of category it is just jars. It feels so pointless (I'm toning down the language I'm tempted to use). As Tony pointed out, a topological category in the proposed sense is just a category internal to topological spaces. The notion of internal category is so familiar and elementary that it must long antecede any reference made in the question. (It seemed an old notion when I was using it in the early 1970s). A Segal space (original version) is a covariant functor from the category $\mathcal{F}$ of finite based spaces (the opposite of Segal's category $\Gamma$) to the category of based spaces. It is not a kind of category. Similarly, a Segal category is a functor from $\mathcal{F}$ to Cat. There is a forgetful functor from Segal spaces to simplicial spaces (simplicial objects in spaces). Parenthetically, the terms topological category and simplicial category are both ambiguous since, without clarification, they could mean either categories enriched in spaces or in simplicial sets, or they could mean categories internal to spaces or to simplicial sets. A category internal to simplicial sets is the same notion as a simplicial object in Cat, so the consistent meaning would be the internal one, but the more standard usage is that a simplicial category is a simplicially enriched category.

Answer by David Roberts for Segal's Original Definition of a Topological Category

2012-01-15T21:10:51Z

Topological categories were invented by Charles Ehresmann in the late 1950s, and can be seen in his 1959 paper I think called Catégories topologique et catégories differentiable. The usage 'topological category' for a Top-category is much newer.