# Tag Info

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### Errata on Rezk's paper

It looks like I completely missed this. Here's what I guess happens: although the original 2.19 was wrong, there is a weaker version that is true (I'll just state it for simplicial sets): If $X$ ...
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### Why are quasi-categories better than simplicial categories?

As a preface, I think that this question should be viewed as analogous to "what are the advantages of ZFC over type theory" or vice versa. We're talking about foundations -- in principle, it ...
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### When is a topological space the homotopy colimit of an open covering?

It is true in complete generality that $X$ is the homotopy colimit of $C_U$ (and hence that the fat realization computes the homotopy colimit in this case). This is a special case of Lurie's version ...
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### About definition of homotopy colimit of Kan and Bousfield

You can define homotopy limits and colimits in pointed as well as unpointed spaces. It so happens that the two notions of homotopy limit coincide, basically because the forgetful functor from pointed ...
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### P.G.Goerss, J.F.Jardine, "Simplicial Homotopy Theory" prerequisites

As the commenters already argued, I would not regard this book as a self-contained introduction. For instance, from a brief browse through the introductory chapters: The reader is assumed to be ...
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### What is this symmetric simplex category, concretely?

$\Delta_+$ is the monoidal category generated from the associative operad, considered as a non-symmetric operad. Similarly, $(\Delta_+)_{{\rm sym}}$ is the symmetric monoidal category generated from ...
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### Teaching Steenrod Operations

Its nice to look also at Bott's early paper "On symmetric products and the Steenrod squares. " Ann. of Math. (2) 57, (1953). 579–590. He uses an early version of Smith theory. Depending on how you ...
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### Homotopy type of the semi-simplicial set of symmetric groups

It is contractible. To see this first observe that it is simply connected. It has 2 arcs $a = (1,2)$ and $b = (2,1)$, however the tirangle $(3,1,2)$ gives us the relation that $ab = b$ and ...
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### Weak complicial sets: Are the morphisms too strict?

Indeed there is no such coherence result: it is false already for $2$-categories (see for instance Lemma 2 of this paper of Steve Lack). The solution to your troubling corollary is that the "correct" ...
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### Is the hom-simplicial set in the hammock localization a nerve?

The nLab description is not correct. For each "shape" of zig-zag, there is a "hammock category" for it (not a groupoid, and the nLab page I am looking at never mentions groupoids here), whose ...
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### Where to find the proof that these two version of simplicial homotopy are equivalent?

Proposition 6.2 in Chapter 1 of "Simplicial objects in algebraic topology", by J.P. May.
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