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Mar 30 |
accepted | Reducing the simplices in the nerve of a category with an object with trivial endomorphism monoid |
Mar 25 |
comment |
Reducing the simplices in the nerve of a category with an object with trivial endomorphism monoid
Could you give me a reference for this? |
Mar 24 |
comment |
Reducing the simplices in the nerve of a category with an object with trivial endomorphism monoid
As pointed out by მამუკაჯიბლაძე, the arrow $b\rightarrow f$ closes the circle in the full subcategory where $a$ is not present, it is also a $S^1$. |
Mar 24 |
accepted | Pushout of categories along embeddings gives homotopy pushout? |
Mar 24 |
revised |
Reducing the simplices in the nerve of a category with an object with trivial endomorphism monoid
added 379 characters in body |
Mar 24 |
answered | Reducing the simplices in the nerve of a category with an object with trivial endomorphism monoid |
Mar 23 |
awarded | Self-Learner |
Mar 23 |
comment |
Reducing the simplices in the nerve of a category with an object with trivial endomorphism monoid
Sorry, I was not aware that the definition could cause so much confusion, but the idea is really simple: Take a simplex of the form $A_1\rightarrow A_2\rightarrow A_3\rightarrow ...\rightarrow A_k$ and look at it as a linear graph. Then delete all objects $A_i$ which are not equal to $X$. Then the number of connected components you get is the number of $X$-components. |
Mar 23 |
revised |
Reducing the simplices in the nerve of a category with an object with trivial endomorphism monoid
added 183 characters in body |
Mar 23 |
comment |
Pushout of categories along embeddings gives homotopy pushout?
Yes, one object category. You can take any monoid which is not contractible. |
Mar 23 |
answered | Pushout of categories along embeddings gives homotopy pushout? |
Mar 22 |
revised |
Pushout of categories along embeddings gives homotopy pushout?
edited title |
Mar 22 |
asked | Pushout of categories along embeddings gives homotopy pushout? |
Mar 20 |
revised |
Reducing the simplices in the nerve of a category with an object with trivial endomorphism monoid
deleted 335 characters in body |
Mar 20 |
revised |
Reducing the simplices in the nerve of a category with an object with trivial endomorphism monoid
added 336 characters in body |
Mar 19 |
comment |
Reducing the simplices in the nerve of a category with an object with trivial endomorphism monoid
A single $X$ is also a $X$-component, so your example is $X\rightarrow X\rightarrow Y\rightarrow X\rightarrow Y\rightarrow X$ and has thus three $X$-components. |
Mar 18 |
asked | Reducing the simplices in the nerve of a category with an object with trivial endomorphism monoid |
Mar 9 |
awarded | Nice Question |
Mar 8 |
accepted | Functors and coverings |
Mar 7 |
comment |
Functors and coverings
Thanks for your answer. The motivation for this question is the construction of the universal covering category as follows: Let $\pi_1(C)$ be the fundamental groupoid of $C$, i.e. the localization of $C$ on all arrows. Fix some object $X$ of $C$. Then let $UC$ be the slice catgory with respect to $X$ and the canonical $C\rightarrow\pi_1(C)$. Here rendiconti.dmi.units.it/volumi/25/25.pdf it is shown that it is simply conn. and I hoped I could easily see that it's also a covering of $C$. |