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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 Vaughan 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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# Segal's Original Definition of a Topological Category

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 Vaughan 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?