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](http://ncatlab.org/nlab/show/Segal+space)? What would Lurie call it?** Based on reading [Chris Schommer-Pries MO post](https://mathoverflow.net/questions/29728/a-model-category-of-segal-spaces) 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?**



Segal's Original Definition of a Topological Category
I would call this an [internal category](http://ncatlab.org/nlab/show/internal+category) in the category of topological spaces and continuous maps.


