## ag.algebraic geometry – Distinguishing ample divisors by minimally intersecting curves on a projective simplicial toric variety

My question has an easily formulated generalization, which I will state first. Let $$sigma subseteq mathbf{R}^n$$ be a strongly convex polyhedral cone. For each minimally generating lattice point $$m in sigma^o cap mathbf{Z}^n$$ of the interior cone $$sigma^o subseteq sigma$$, let $$S(m) subseteq sigma^{vee} cap mathbf{Z}^n$$ denote the set of lattice points $$u$$ with $$langle u,m rangle = 1$$. My question is:

Does $$S(m) = S(m’)$$ imply that $$m = m’$$?

As a special case, assume that $$sigma$$ is the nef cone of a simplicial projective toric variety $$X_{Sigma}$$. Then my question seems to amount to the following:

If $$D_1$$ and $$D_2$$ are two ample divisors minimally generating in the ample cone, then does $$D_1 cdot C = 1 Leftrightarrow D_2 cdot C = 1$$ for all effective curves $$C$$ imply that $$D_1 = D_2$$?

This is the case I am most interested in.

## simplicial stuff – Computation on homotopy colimit cocomplete triangulated categories

I have a couple of questions about dealing with homotopy (co)limits cocomplete triangulated categories.

Question I:The first one concerns a comment by Peter Arndt in this discussion about derived categories regarding posibibility to calculate the homotopy colimit when working with nice enough category. Peter wrote:

I also find this a very enlightening view point, but just for the record: Ho(co)lims in cocomplete triangulated categories are MUCH easier to compute by completing the right map to an exact triangle than by going via a simplicial (or any other) enrichment…

Where I can look up the theoretical background telling me that doing the these steps we indeed obtain an object homotopic to homotopical (co)limit. In other words why this cooking recipe work?

Question 2: searching for an answer for my first question I found in this paper on Homotopy limits in triangulated categories by Bökstedt & Neeman an approach by so called ‘Totalization of a complex’.

The steps in the construction look quite similar to the step Peter described and the constructed object is also described as homotopical colimit.

Question: How close the construction in the paper is to that one in first question. The principal aspect that confuses me is that the construction in the paper (as well the paper) not explicitly working with simplicial enrichments of homs.

Is using simplicial enrichment a more ‘modern’ approach to obtain the same object? And how would it flow into the construction?

## Alexander duality for Homology sphere which is the Geometric realization of a finite simplicial complex

The Alexander duality Theorem is usually stated for a triangulable pair $$(mathbb S^n, Y)$$ where $$Y$$ is a subset of the standard sphere $$mathbb S^n$$. My question is: Does the duality also hold if we rather replace $$mathbb S^n$$ by a compact orientable Homology sphere (without boundary) (https://en.m.wikipedia.org/wiki/Homology_sphere) ? I’m mainly interested in the cases $$n=2$$ and $$3$$. I’m willing to assume that the Homology sphere is the Geometric realization of a finite abstract simplicial complex (https://en.m.wikipedia.org/wiki/Abstract_simplicial_complex)

Thanks

## at.algebraic topology: shows that a certain simplicial set has small cardinality at the level

My question refers to Section 2 of the article "The simple model of univalent foundations (after Voevodsky)" (https://arxiv.org/pdf/1211.2851.pdf).

Leave $$alpha$$ being a strongly inaccessible cardinal.

Leave $$f: X a Y$$ Be a simplicial set map.

1. We say that $$f$$ it is well organized if you are equipped with a good order of $$Y_x: = f_n ^ {- 1} (x)$$ for each simplex $$x in X_n$$.

2. We say that $$f$$ it is $$alpha$$-little Yes $$left lvert {Y_x} right rvert < alpha$$ for every simplex $$x$$.

Leave $$f: X a Y$$ Y $$g: Z a Y$$ be well ordered simplicial maps. A morphism $$f a g$$ is a simplicial map that preserves fiber $$h: X a Z$$
such that $$h_n: f_n ^ {- 1} (y) to g_n ^ {- 1} (y)$$ is to preserve the order for each natural number $$n$$ and every $$y in Y_n$$.

Define the functor $$mathcal {U} _ { alpha}: mathbf {sSet} ^ { text {op}} to mathbf {Set}$$ so that $$mathcal {U} _ { alpha} (X)$$ consists of all kinds of isomorphism of $$alpha$$-small neat Kan fibrations $$Y a X$$. Also, leave
$$mathrm {U} _ { alpha} = mathcal {U} _ { alpha} circ mathcal {Y} ^ { text {op}}: varDelta ^ { text {op}} to mathbf {Set}$$ where $$mathcal {Y}: varDelta to mathbf {sSet}$$ denotes the inlay of Yoneda.

The authors say that if $$beta < alpha$$ also inaccessible then the single map $$mathrm {U} _ { beta} a 1$$ it is $$alpha$$-small (p. 23, near the bottom). This is equivalent to saying that the set

$$left ( mathrm {U} _ { beta} right) _n = mathcal {U} _ { beta} ( Delta (n))$$
has cardinality $$< alpha$$ for each $$n$$.

However, I can't see an overall theoretical justification for this. I would be grateful if someone could provide one!

## Is every simplicial spectrum equivalent to a simplicial spectrum of an abelian group?

I wonder if every simplicial $$S ^ 1$$stable spectrum equivalent to a simplicial abelian group $$S ^ 1$$-spectrum? Looks like you could use the stable Dold-Kan in your heart and use the Postnikov towers?

## Examples of simplicial complexes in which each edge k is contained exactly in $d$ k + 1 \$ -edges

Are there any (other than the entire complex / 1 case)?
There is a name for this ($$k$$-edge-regular call him)?

Thank you.

## simplicial volume – Flow rate question

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## discrete geometry – triangulations with simplicial polyhera sign

Leave $$partial S$$ be the limit of a compact polyhedron $$S subset mathbb {R} ^ 3$$, is supposed to be generic, in the sense that each face of $$S$$ It is a triangle, and so that there are exactly two triangles on each edge. So $$partial S$$ it is a triangulated oriented surface, with sets of vertices $$V$$edges $$E$$ and triangles $$T$$.

Not all $$S$$ they can be traingulated, that is, divided into tetrahedra (i.e. full-size simplifiers), with vertices of $$V$$ — a simple example of such $$S$$ It is the polyhedron of Schönhardt. However, the latter may have triangle signs, that is, decompose in the difference of two tetrahedron junctions with vertices of $$V$$: simply take your convex closure (an octahedron, a convex polytope, therefore, it is possible to triangulate it using vertices of $$V$$ only), and the disjoint union of 3 tetrahedra.

The conjecture is that such signed triangulation is possible for any $$S$$ – I have a sketch of an induction test in $$| V |$$, but maybe it is well known?

It is known that the weakest forms of this conjecture, in terms of measures, which allow real weights of simplifications, or simply whole weights, remain in all dimensions, see https://arxiv.org/abs/1210.3193 and https: //arxiv.org /abs/1508.07594

## simplicial material: are the 2-Segal (complete) spaces equal to the categories of infinity enriched in Span?

The question is basically in the title. In more general terms, I would like to know if this, or any reasonable variant thereof, is true. Or maybe, to better understand the gap between 2-segment spaces and enriched in Span $$infty$$-categories.

### Background

The 2-segment spaces (see, for example, the role of Dyckerhoff and Kapranov-s) are generally informally described as $$infty$$-categories with multiple value composition. This notion seems to be more naturally encoded by enrichment over the $$infty$$-category of spaces of spaces and this interpretation is mentioned both in the nLab entry and in this question from MO by Tim Campion. Dyckerhoff and Kapranov try something of this kind for 2-Segal sets, but for 2-Segal spaces there are some constructions involved with algebras in the $$( infty, 2)$$-Category of BiSpans, I'm not sure how they relate to enrichment about Span.

It seems important to keep in mind that for 1-Segal spaces, which are "category objects" in spaces, we need to impose I complete it to get a custom model $$infty$$-categories. Therefore, it seems reasonable that the general spaces of 2 segments encode "multicomponent category objects" and only after imposing some condition of integrity, we could expect to enrich ourselves with Span $$infty$$-categories. In the document mentioned above, there is a definition of integrity for 2-segment spaces (combine def 9.3.2 for the integrity of (co) Segal fibrations and def 9.3.4 for Hall (co) Segal fibration associated with a 2 Segal space ) This may or may not be the relevant notion of integrity and I admit that I do not understand it well.

Another problem is the comparison of morphisms on Both Sides. As Tim Campion pointed out in his question, the morphisms of the 2-segment spaces can correspond to the enriched functors in Span that have real maps (and not just sections) between the corresponding mapping spaces.

## Withdrawal of motor homotopia calculated as homotopia withdrawal in the homotopy category of simplicial presets

In this test, why is the recoil of motor homotopy in the homotopy category of simplicity presets calculated? Does the preimage of a homotopia recoil under a right Quillen annex give a homotopia recoil?