At ncatlab.org/…/category+of+simplices, Proposition 2 says that for a simplicial set $X$, the category of non-degenerate simplices of $X$ is a *reflective* subcategory of the category of simplices of $X$. Is that true? Does someone have a reference for that statement? At the moment I can only see it in the case that $X$ has the property that every face of every non-degenerate simplex of $X$ is itself non-degenerate.

The article local model structure on simplicial presheaves states that for a site with enough points stalkwise weak equivalences of simplicial presheaves coincide with weak equivalences of simplicial presheaves in the Bousfield localization of componentwise weak equivalences with respect to all hypercovers (i.e., weak equivalences in a local model structure).

Is a proof of this statement written up somewhere? (The article cited above gives a reference to Jardine, which claims, but does not prove this statement.)

Also, is it possible to formulate an analog of this statement for sites that do not have enough points? (Presumably we would have to talk about sufficiently refined (hyper)covers instead of points.)

]]>Adopting the perspective on cohomology as described at cohomology, one understands cohomology as

$\mathrm{H}^0(X,A) = \pi_0 \mathrm{Hom}(X,A),$in the appropriate context of some $(\infty,1)$-topos. Then, perspectives which are perhaps more familiar are described as ways of computing this using presentations of the relevant $(\infty,1)$-topos.

Here’s my interpretation of two such perspectives, Čech cohomology and abelian sheaf cohomology, at a very naive level.

For Čech cohomology, I start with a topological space $X$, and the Čech nerve provides a cofibrant replacement for $X$ in an $(\infty,1)$-category of simplicial topological spaces (I insist on simplicial topological spaces and not simplicial sets), ignoring potential oversights which are corrected by using hypercovers. For abelian sheaf cohomology, starting with a sheaf of abelian groups $A$, a fibrant replacement is given, in an $(\infty,1)$-category of simplicial abelian sheaves, by an injective resolution (through the lens of the Dold–Kan correspondence).

In my mind, in both cases, one has gone from objects to simplicial objects, and I am left wondering how to fully justify this leap. For instance, in the Čech case, we have gone from the Quillen model structure for the $(\infty,1)$-topos $\mathrm{Top}$ to some (?) model structure on simplicial topological spaces. Why did we have to leave our original $(\infty,1)$-category, and what guarantees that this process allows the computation to give the correct answer? Can we always use this tool of “taking simplicial objects” to help in the computation of the cohomology, for any $(\infty,1)$-topos we start with?

]]>At simplicial category, three meanings for the phrase are mentioned: the simplex category, a simplicial object in Cat, or an sSet-enriched category. Then at enriched category#examples, the phrase is apparently used to describe a category internal to sSet. Is this term overloaded a fourth time?

By the way, what does it mean for a simplicial set to be discrete? All the cells above 0 are degenerate?

]]>Hi,

in homotopy type theory, higher identity types have a globular structure. Is there also a kind of intensional type theory in which higher identity types have the structure of a simplicial set?

I suppose that reflexivity terms should correspond to degenerate simplices. But then, an identity term $p:\mathrm{Id}_A(x,y)$ should have *two* reflexivity terms associated to it! Would this make any sense at all? If so, how could an inductive definition of such higher identity types look like?

A while ago, we had some brief discussion on potential higher structures in categorical probability theory. We now have a definition of a certain abstract categorical structure which captures these phenomena and also many other examples. See the current working document.

Differently from “ordinary” higher category theory, composition in our “compositories” has the property that composing an $n$-morphism with an $m$-morphism along a common $k$-morphism face results in an $(n+m-k)$-morphism. We believe that this kind of composition is a natural structure which arises in many situations. Think, for example, of the nerve of a category, in which sequences of composable morphisms can simply be concatenated. Other examples arise from mathematical structures which can be glued along pairwise intersection, even though the general sheaf condition fails; this applies e.g. to the presheaf of metrics on a set.

Compositories may also provide a potential answer to Urs’ question on hyperstructures and higher spans.

Now the questions are:

- has anyone looked at these kind of structures before?
- is there a simple way to reformulate them in ordinary (higher) categorical terms?
- can you think of other examples?
- would it make sense to develop “compositorial” analogues of category-theoretical concepts like limits, adjunctions, etc.?

Thanks for any feedback!

]]>Is the set of natural transformations between two simplicial (abelian) groups a simplicial group or a chain complex?

]]>Is there a well defined internal hom for cosimplicial objects? (sets, algebras, rings)

]]>Simple Question:

1.) How many morphisms $f: [n] \rightarrow [m]$ are there in $\Delta$ ?

```
( Or equivalent: What is $|\Delta [m]_n |$ ? )
```

2.) Is there a research field in combinatorics that is concerned with monotonic integer maps from $[n]$ to $[m]$ (Just in case I have another ’combinatorical’ question concerning $\Delta$)

]]>(http://ncatlab.org/nlab/show/Eilenberg-Zilber+map)

(http://ncatlab.org/nlab/show/Alexander-Whitney+map)

are given in the "standard simplicial dimension notation". However in the setting of abelian simplicial groups

and chain complexes we have frequently the situation where we work with augmented simplicial sets

and in that scenario there is the 'upshifted dimension counting' where we define the dimension of the augmented

simplex as zero instead of $(-1)$. (Explained for example in

http://ncatlab.org/nlab/show/simplex+category

My question is now how this affects the definition of the above maps and since I can't find anything on the web

I suggest to add such a augmented definition to the nLab entries on those topics.

If someone can post a link or something where this is worked out, I will change the entry if you people agree. ]]>

Given a simplicial abelian group, the alternating sum defines a derivation, making the simplicial abelian group itself into a chain complex.

The derivation is then an endomorphism on that chain complex. But the complex has another point of view because it is still a simplicial set, too and the question is:

Does the alternating-sum-derivation respect the simplicial structure, i. e. does it commute with the face and degeneracy maps? (Maybe not because of the square to zero rule, but anyway …)

If not, is there a derivation respecting the simplicial structure?

]]>Suppose we have the sequence of sets $\mathbb{R}$, $\mathbb{R}^2$, $\mathbb{R}^3$, … Is there a Kan simplicial structure on this sequence of sets, that is not $n$-coskeletal for some $n \in \mathbb{N}$?

To be more precise, is there a simplicial set (functor) $R$ with $R([n]) = \mathbb{R}^{n+1}$ that is not $n$-coskeletal for some $n \in \mathbb{N}$?

And very closely related: is there a simplicial set (functor) $R$ with $R([n]) = \mathbb{R}^{n}$ (with $R([0]))=\{0\}$), that is not $n$-coskeletal for some $n \in \mathbb{N}$ ?

]]>Suppose we have a simplicial set X and a m-truncated Kan simplicial set Y. Then how is it possible to construct $Hom_{Simpl}(X,Y)$ as a subset $H \subset Hom_{Set}(X_m,Y_m)$?

Since Severa used this in his work on the n-jet functor (for X the nerve of the pair groupoid over an arbitrary set), it should be possible. Nevertheless I can’t find an explicit construction including a proof that what he constructed is indeed $Hom_{Simpl}(X,Y)$.

By an explicit construction I mean something like: Let $f \in H$ be given, then the appropriate simplicial morphism $F$ is given by $F[n]= X_n \rightarrow Y_n$ as follows : ??? where the commutation with the face and degeneracy maps is seen as follows ??? … On the other side we that any simplicial morphism is given that way, because ???

….

So if someone could give me a proof (I think it will be an induction on something like $Hom_{Simpl}(Sk^n X,Y) \subset Hom_{Set}(X_n,Y_n)$ or $Hom_{Simpl}(Horn_j^n X,Y) \subset Hom_{Set}(X_n,Y_n)$ ) it would be great.

Likely this doesn’t work for arbitrary simplicial sets X, so another topic is to find the appropriate conditions on X .

Moreover this should be put into the nLab, too…

If nobody knows a proof it would be nice, if we could work it out together. At the end I will take the time to put in the nLab. Unfortunately my skills on simplicial sets are not good enough, to do it by myself.

]]>I’d like to learn about the Jardine-Joyal model structure on simplicial presheaves (and sheaves?) on a site. My understanding is that this was developed in a letter from Joyal to Grothendieck, which is cited pretty widely in the literature. Does anyone have a copy or know where I could find it? (Or, if anyone knows other sources for this model structure, that would also be helpful.)

]]>It is well known that a category can be defined as a certain simplicial set obtained by iterated fibred products which satisfies the internal horn filler condition; moreover, requiring horn filling for all horns (i.e., the Kan condition) one obtains the notion of groupoid. Then both the notions of category and groupoid should have an internalization in any category where one is able to arrange things in a way to have the required fibered products, and to state the horn filling condition.

This is what happens, e.g., when one defines Lie groupoids imposing that the source and target maps are submersions. Similarly one has a notion of Lie category, which by some reason seems to be less widely known of the more particular notion of Lie groupoid (maybe this is not surprising.. after all I suspect the notion of category is less known of that of group..). Another classical example are topological categories and groupoids.

Moving from categories and groupoids to oo-categories and oo-groupoids, one should have a similar internal simplicial object based notion of, e.g., Lie oo-groupoid. However, in the nLab the oo-sheaf point of view seems to be largely preferred to the internal Kan object point of view. Why is it so? is the oo-sheaf version just more general and powerful or there are problems with the internal version? I’m asking this since at internal infinity-groupoid it is said that a classical example of the internal Kan complex definition of oo-groupoid are Lie oo-groupoids, but then at Lie infinity-groupoid there is no trace of the internal Kan complex definition.

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