# Tag Info

Accepted

Accepted

### What are classifying spaces of algebraic categories like?

One famous way to make this interesting is Quillen's $Q$-construction, which involves taking the classifying space of $Q\mathcal C$ instead of $\mathcal C$, where $\mathcal C$ is a sufficiently nice ...
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### Let $K$ be a simplicial complex (that need not be finite). Prove $|K|$ is Hausdorff.

The space $|K|$ can be modelled in this way: $|K|$ is the set of maps $f$ from $V$ (the vertex set) to $\Bbb R_{\ge0}$ with the properties that $\{v\in V:f(v)>0\}$ is the vertex set of a simplex in ...
• 159k

### Simplicial sets which are not Kan complexes

Any finite, not contractible, simply connected simplicial set is not a Kan complex. It is a result of Serre that any space satisfying these properties has infinitely many nontrivial homotopy groups. ...
• 11.8k

### Simplicial sets which are not Kan complexes

For another wealth of examples: take the nerve of any category that is not a groupoid. Roughly speaking, filling outer horns corresponds to inverse morphisms.
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### Intuition degeneracy maps

Degeneracy maps are indeed totally unnecessary to describe the simplicial set you're interested in, and in fact it's possible to do simplicial homotopy theory with semisimplicial sets, which are ...
• 52.9k

### The difference between weak Kan complexes and Kan complexes

If I recall and understand your question correctly, horn fillers are not necessarily unique. However, for (weak) Kan complexes they are unique up to contractible choice. One way to think about ...
• 11.2k
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### How can I understand maps out of a limit or into a colimit?

You can't understand maps into colimits or out of limits in general, no. But there are various interesting special cases in which you can. For instance, a compact object in a category $C$ is an object ...
• 52.9k
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### Kan extension and representable functors

Yes, $F'$ will always be corepresentable if $F$ is. Let's say $i:\mathcal G\hookrightarrow\mathcal C$ is the fully faithful dense inclusion you mention, and $F:\mathcal G\to\mathbf{Set}$ some functor, ...
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### The coproduct of any indexed collection of quasicategories is a quasicategory.

If $h:\Lambda^n_j\to X$ is an inner horn, then the image of $h$ is connected, and thus is contained in some connected component of $X$, which is in turn contained in one of the $X_s$. This means that ...
• 335k
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### the difference between chains and cochains

Practically, the difference is that if there are infinitely many simplices, a cochain can take nonzero values on all of them, whereas a chain must be a finite linear combination of simplices. In other ...
• 438k

### Do we distinguish two singular simplices if they have different vertex orders?

This is an interesting question. You got two answers confirming that there is only one continuous map on $\Delta^n$ into $X$, and of course this is correct. The standard $n$-simplex $\Delta^n$ is a ...
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• 335k
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### Completing a delta set to a simplicial set

Using Friedman's notation, let $\hat{\Delta}$ denote the category of nonempty finite ordinals and strictly increasing maps between them, and $\Delta$ the category of nonempty ordinals and monotonic (i....
• 1,265
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### Why are simplicial sets contravariant functors and not covariant?

First, let's look at the Simplex Category $\Delta$. We can see that $\Delta$, rather than $\Delta^{op}$, is a natural category to look at, both because it has a nice combinatorial description (the ...
• 5,478
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### Discrete simplicial sets: equivalent definitions, request for a proof

2 clearly implies 1: every $x\in X_n$ for $n>0$ is in the image of any degeneracy map $X_{n-1}\to X_{n}$ since the latter is an isomorphism by assumption; hence $x$ is degenerate (by definition). ...
• 1,526
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### Understanding the exponential object in slice categories

You basically have it right. But naming objects of slices categories by their domain make you miss the obvious "simpler" way to see it. (It is simpler in the writting, not in the unpacking which is ...
• 11.7k