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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?

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This isn't a Segal space because there is a further condition about homotopy pullbacks. I've called this a category object in Top before, and that (if I recall) was based on terminology from Tom Leinster's book on higher category theory. There is nothing wrong with allowing the object sets to have topologies, except that many straightforward facts become more difficult to prove (or false). Some places where these show up are in topological groupoids and topological stacks. –  Tyler Lawson Jan 15 '12 at 6:26
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Distinguish this from the unrelated notion of a topological concrete category; the adjective ‘concrete’ is often (and originally always) omitted. –  Toby Bartels Jan 15 '12 at 6:57
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I am amused by the use of the word "nowadays" since the simplicial singular complex has been used since Eilenberg's work, followed by Kan in the 1950s. Also this use fails to distinguish between a path space and the usual (strict) fundamental groupoid, so must lead to confusion. There is also the strict cubical homotopy groupoid of a filtered space, which is closely related to the standard relative homotopy groups, and is given an exposition in our recent book "Nonabelian algebraic topology", published by the EMS in 2011. The cubical methods have many advantages, explained there. –  Ronnie Brown Jan 15 '12 at 10:42
    
@Tyler, But couldn't we define X_2 to be the fibered pullback over s and t? Couldn't we do something similar for the higher X_n so that it is a pullback (on the nose)? –  Justin Curry Jan 15 '12 at 15:04
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adding to Ronnie's comment, the fact that the singular complex is Kan is 'classical', the fact that such things resemble infinity categories is 1970s, probably hidden in Boardman-Vogt, certainly well know to some people by 1980, although rarely in print. I mentioned it in several talks at category theory meetings around that time, and wondered why the search for lax infinity groupoids was taking up a lot of space as they were already well known. In fact much of the recent work reworks classical simplicial homotopy theory from the Moore seminar and it is well worth looking back at that. –  Tim Porter Jan 15 '12 at 15:15

3 Answers 3

up vote 16 down vote accepted

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

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One often places additional requirements, such as requiring $ s $ and $ t $ to be open maps; when working internal to the category of smooth (say real) manifolds and smooth maps, one often even requires them to be submersions. These are less easy to fit into a general notion applicable to any ambient category. –  Toby Bartels Jan 15 '12 at 7:00
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Toby beat me to this! This is just an internal category. Internal category makes sense in any category with finite limits. Another point is that the Cohen, Jones Segal paper was reported to have a gap in it and so was never finished, which is a pity because it was going in a very interesting direction. Finally it was John Jones (Warwick) and not Vaughan Jones, although both have Welsh origins, one directly the other slightly more distant!!! –  Tim Porter Jan 15 '12 at 7:26
    
@Toby: Concerning these additional requirements, this of course reminds of algebraic spaces. They are given by groupoids internal to schemes where $s,t$ are étale. @Tim: Even more generally, "internal category" makes sense in any ambient category $C$. It is an object $A$ together with a factorization of $\mathrm{Hom}(-,A) : C^{op} \to \mathrm{Set}$ through the forgetful functor $\mathrm{Cat} \to \mathrm{Set}$. For every concrete category $D \to \mathrm{Set}$, there is a notion of an object of $D$ internal to an ambient category. –  Martin Brandenburg Jan 15 '12 at 12:17
    
@Tim, Thanks for pointing out the Vaughan<->John mistake. –  Justin Curry Jan 15 '12 at 14:53
    
@Martin ...which is of course the way that Grothendieck handled all such constructions. –  Tim Porter Jan 15 '12 at 15:09

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.

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I agree completely with the first two sentences of Peter's answer (to say nothing about the rest). Name changes can lead to confusion and lack of contact with older work. And here a lack of attention to cases such as the strict case when there is something quite tricky to prove, e.g. the strict higher groupoids associated to a filtered space (I confess to a personal interest in these!). –  Ronnie Brown Jan 16 '12 at 10:31
    
Yet another meaning of ‘topological category’ is a category $C$ equipped with a faithful functor to $Set$ (or sometimes to something else) such that every (even large) sink from $C$ to a set has a final lift (nlab). This may be disambiguated as a ‘topological construct’ or as a ‘topological concrete category’; the reason for this terminology is examples from topology. (Also, my name is ‘Toby’, not ‘Tony’.) –  Toby Bartels Mar 23 '13 at 5:13
    
I apologize about the name, Toby; it gets confusing (I'm now collaborating with Tobias Barthel!). Mathematically, it seems bad enough to have two standard meanings of "topological category" without adding a third. –  Peter May Mar 23 '13 at 20:41

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.

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