Timeline for Discreteness of topological category?
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Aug 15, 2016 at 2:42 | comment | added | PhysicsMath | @DanRamras: I see! Many thanks for making that crystal clear now! | |
| Aug 15, 2016 at 2:31 | comment | added | Dan Ramras | @PhysicsMath The notion of "discrete category" in the sense of the Wikipedia article you linked to is definitely not relevant here. Freed surely just meant "ordinary category" and used the term "discrete category" to emphasize the fact that in "ordinary" categories there is no topology on the objects/morphisms, i.e. they are "discrete sets". It's a slightly unfortunate coincidence of terminology. | |
| Aug 14, 2016 at 19:47 | comment | added | PhysicsMath | @DanielGrady: so that is not the same notion of "discrete category" as e.g. in the Wikipeida? | |
| Aug 14, 2016 at 15:11 | answer | added | Arun Debray | timeline score: 1 | |
| Aug 14, 2016 at 4:36 | comment | added | Todd Trimble | "Discrete" is a bit of a distraction here; you might as well say "ordinary category". In other words, as Daniel says, the data of a topological category consists of continuous maps (source and target maps $C_1 \to C_0$, a unit map $C_0 \to C_1$ which gives the identity assignment, and a composition map $C_1 \times_{C_0} C_1 \to C_1$). The equational axioms these data must satisfy are the same as those of ordinary category theory: associativity of composition, the identity axioms, etc. | |
| Aug 14, 2016 at 4:24 | comment | added | Daniel Grady | He defines a topological category as a pair $(C_1,C_0)$ with source and target maps and a degeneracy map. The requirement that these satisfy the algebraic relation of "discrete category" imply that this pair (and the maps between them) is actually a category internal to the category of topological spaces, where $C_1$ is the space of morphisms and $C_0$ is the space of objects. | |
| Aug 14, 2016 at 3:10 | comment | added | PhysicsMath | @Ubiquity: I don't understand either... | |
| Aug 14, 2016 at 3:04 | comment | added | user97187 | What does "algebraic relations of a discrete category" mean? For me, a topological category is a category enriched in topological spaces. | |
| Aug 14, 2016 at 2:34 | history | asked | PhysicsMath | CC BY-SA 3.0 |