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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 $C$.