Timeline for Why is an object not defined as identity morphism?
Current License: CC BY-SA 4.0
15 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 4, 2021 at 22:35 | comment | added | Tom Goodwillie | I always think of the Cheshire Cat when I think of a category without its objects. | |
| Apr 4, 2021 at 20:55 | answer | added | Ms. Molly Stewart-Gallus | timeline score: 0 | |
| Aug 30, 2020 at 8:11 | comment | added | smith K | @AndreasBlass I think this question is about to understand the concept of category theory not to understand the connection sets or groups in definition phase. The popularity due to wrong intuition does not help people. | |
| Jul 21, 2019 at 23:26 | comment | added | Mike Shulman | Actually I would argue that the homsets-definition (ncatlab.org/nlab/show/category#AFamilyOfCollectionsOfMorphisms) is the one that corresponds to how more people think of categories in practice. | |
| Jul 21, 2019 at 17:10 | answer | added | Gael Meigniez | timeline score: 2 | |
| Jul 20, 2019 at 0:45 | history | became hot network question | |||
| Jul 19, 2019 at 22:33 | vote | accept | ististyle | ||
| Jul 19, 2019 at 19:54 | comment | added | Simon Henry | Another area where the single sorted definition is more widespread is when working with strict $n$-categories or strict $\infty$-categories. Working with the single sorted definition allows to compose arrows of different dimension without writing iterated identities everywhere. It makes the manipulation of expressions a little more bearable, so lots of paper use it (where the "lots" need to be taken relatively as there is not that many paper on that topics anyway) | |
| Jul 19, 2019 at 18:48 | answer | added | Noah Schweber | timeline score: 16 | |
| Jul 19, 2019 at 18:20 | comment | added | Yemon Choi | My impression is that the single-sorted POV is slightly more widespread in the context of groupoids (considered as small categories where every morphism is iso) but I am not a specialist and would be happy to be corrected here by others. However, one often ends up introducing the "unit space" of the groupoid which is the "set of objects", so even then two sorts seem to emerge | |
| Jul 19, 2019 at 17:52 | comment | added | Jay Kangel | One can identify an object with its identity morphism. Then we have a category consisting of morphisms that can be composed. Usually the composition is thought of one function for the whole category. However, the composition function can be broken up into a performing compositions, one such place for each object. In higher order category theory there may be 2-morphisms between the usual morphisms. They may also be places or performing compositions. With the appropriate additional axioms higher order category theory fits into this scheme. | |
| Jul 19, 2019 at 17:45 | review | Close votes | |||
| Jul 25, 2019 at 3:05 | |||||
| Jul 19, 2019 at 16:40 | review | First posts | |||
| Jul 19, 2019 at 17:28 | |||||
| Jul 19, 2019 at 16:39 | comment | added | Andreas Blass | The two-sorted definition corresponds to how most people think of categories, and how they talk about categories. For example, we speak of the categories of sets (not of functions), of groups (not of group homomorphisms), of topological spaces (not of continuous maps), etc. (If I remember correctly, Ehresmann did write about the categories of functions, of homomorphisms, of continuous maps, etc., but that never caught on.) | |
| Jul 19, 2019 at 16:35 | history | asked | ististyle | CC BY-SA 4.0 |