ct.category theory – Enough sets of colimits in small categories

Leave $ C $ Be a small category, and consider the kind of diagrams. $ G: D a C $, with $ D $ A small category, which has colimits in. $ C $. This is an appropriate class even when $ C $ It is very small, eg. when $ D $ has a terminal object $ t $, any functor $ G: D a C $ has a colimit $ G (t) $, and there is an appropriate class of small categories with a terminal object.

However, these colimits feel somewhat "trivial"; in some cases, at least, we can find a small set of diagrams that "carry all the non-trivial information" on the colimit diagrams in $ C $. For example, yes $ C $ it is a poset, then it is enough to consider injective functors $ G $ (and we can also take $ D $ to be discrete too), and these form an essentially small set. For a non-posetal $ C $ we can not restrict ourselves to injective functors, since co-products are not idempotent, but there may be some other restriction that works. Note that according to Freyd's theorem, a small non-posetal category has a limit on the cardinality of the coproducts it can admit; but this does not answer the question itself, since a particular colimit can exist even if the coproducts that would be necessary to build it from co-products and co-factors do not.

Here are two ways to ask the precise question:

  1. Given a small category $ C $, there is a small set $ L $ of diagrams $ G: D a C $ With colimits such that for any diagram. $ G & # 39 ;: D & # 39; a C $ with a colimit, there is a $ (D, G) in L $ and a final functor $ F: D to D & # 39; $ such that $ G = G & # 39; circ F $?

  2. Given a small category $ C $, there is a small set $ L $ of diagrams $ G: D a C $ with colimits such that if a functor $ H: C a E $ Keep the colimits of all the diagrams in. $ L $, then it keeps all the colimits that exist in $ C $?

Any solution to question 1 is also a solution to question 2, but I'm not sure if the opposite is true. The mention of Freyd's previous theorem suggests that a solution might require a classical logic; I would find it more surprising if such a set existed for a small complete non-posetal category, although I do not immediately see an argument that I can not.

Of course, you can also ask similar questions for rich categories, internal categories, $ infty $-categories, and so on. Bonus points go to a response that applies more generally in such contexts.

Theory of elementary sets – Ordnance sum inequality – Test verification

Motto Given three ordinals $ alpha $, $ beta $Y $ gamma $, so
$$
alpha < beta to gamma + alpha < gamma + beta.
$$

Test
Dice $ alpha < beta $, so $ alpha subsetneq beta $, $ alpha $ is an adequate initial segment of $ beta $A) Yes
begin {meets *}
{0 } times gamma cup {1 } times alpha subsetneq
{0 } times gamma cup {1 } times beta,
end {meets *}

and the l.h.s. is an initial segment of r.h.s., so the claim is as follows:
$$
gamma + alpha
= operatorname {ord} left ( {0 } times gamma cup {1 } times alpha right) <
operatorname {ord} left ( {0 } times gamma cup {1 } times beta right)
= gamma + beta
$$

real analysis – Existence of open sets $ U, V $ of matrices, so that for every $ A in $ there is $ B in V $, so $ B ^ 4 = A $

Show that there are open non-empty sets U and V of $ n times n $ matrices about $ mathbb {R} $ such that for each matrix $ A in U $ there is exactly one matrix $ B in V $ such that $ B ^ 4 = A $.

I have tried to address this problem in several ways, using characteristic polynomials, Jordan's canonical form, and calculation data, but I did not get anything useful.

google sheets – merging data sets

On a Google sheet I have several tables that contain a list of dates.
I want to create a single list (ordered) that contains all the dates and the table from which they come.

Table foo:
1.1.19
3.1.19

Table bar:
2.1.19
4.1.19

Result:
1.1.19 foo
2.1.19 bar
3.1.19 foo
4.1.19 bar

Is this possible in some way with regular formulas or do I need to switch to custom functions here?

Magento 2 sets the creation date to order when creating

I followed the steps below to create an order in Magento 2. But I want to set my preference date when creating it.

How to create order programmatically in Magento 2?

algebraic topology: why does Spivak define the realization functor, from the fuzzy simplicity sets to the extended pseudometric spaces the way it does?

In Spivak's article on the metric realization of fuzzy simplicial sets, he sends a blur $ n $-simple of strength $ a $ to the set
$$
{(x_0, x_1, dots, x_n) in mathbb {R ^ {n + 1}} | x_0 + x_1 + dots + x_n = – lg (a) }
$$

and the realization of a general. $ X $ It is through colimits:
$$
Re (X): = mathrm {colim _ { Delta ^ n <{a} to X}} Re ( Delta ^ n <{a})
$$

It is a very concise self-published article, and no justification is offered.

Why $ lg $? Is that just a good trick to store the fuzziness, or is it preserving some good distance properties? Second, this is realization, so you should be able to see it. What does a fuzzy complex look like?

inequality: step in the test on independent sets in a graph without triangles

I am reading Tau and Vu's book, Additive Combinatorics, and I found a step in a test that I can not verify.

On page 252, in the last line of the proof of Theorem 6.4, it is stated that the following inequality is maintained for $ m geq 1 $ Y $ d geq 16 $:

$ frac {d} {2 ^ m + 1} + frac {m} {2} frac {2 ^ m} {2 ^ m + 1} geq frac { log_2 d} {4} $

I would appreciate a proof of this fact. I have tried several ways to reorganize inequality, and I have tried to fix one variable and then take the derivative with respect to the other, but I am not sure if this will work. Any help would be appreciated.

The result in the book is cited to be in an article by Shearer, which I found here:
https://www.sciencedirect.com/science/article/pii/0012365X8390273X, but I can not find this result in this document. I am getting lost?

Infinity categories – Some property conditions in simplicial sets.

Suppose that $ C a D $ It is a trivial Kan fibration, and $ D & # 39; a D $ It is an equivalence of simplicial sets. Leave $ C & # 39; $ be your kick It is true that $ C & # 39; a D & # 39; $ Is it an equivalence?

Recall that a trivial Kan fibration is an equivalence.
Then it is enough to prove that $ C & # 39; a C $ it is an equivalence, that is to say, that the equivalences extracted along the fibrations are equivalences. In fact, for 2-3 property would follow that $ C & # 39; a C a D $, $ D & # 39; a D $ equivalences $ Rightarrow $ $ C & # 39; a D & # 39; $ equivalence.

Thank you!

abstract algebra – Why are these matrices representatives of sets for $ Gamma_0 (2) $ in $ SL_2 ( mathbb {Z}) $?

Consider the group $$ Gamma_0 (2) = lbrace begin {pmatrix}
a & b \
CD
end {pmatrix} in SL_2 ( mathbb {Z}): c equiv 0 (mod 2) rbrace $$

Y
$ I = begin {pmatrix}
1 & 0 \
0 and 1
end {pmatrix} $
,
$ S = begin {pmatrix}
0 & -1 \
1 and 0
end {pmatrix} $
, $ T ^ {- 1} S = begin {pmatrix}
-eleven \
1 and 0
end {pmatrix} $
, with $ T = begin {pmatrix}
eleven \
0 and 1
end {pmatrix} $

Now I want to show that these matrices are coset representatives of $ SL_2 ( mathbb {Z}) / Gamma_0 (2) $

I know there are three cosets from $ | SL_2 ( mathbb {Z}) / Gamma_0 (2) | = 3 $

$ SL_2 ( mathbb {Z}) / Gamma_0 (2) = lbrace I, S, T ^ {- 1} S rbrace $

I think $ " supset" $ It is clear why these matrices are in $ SL_2 ( mathbb {Z}) $ Y $ I in Gamma_0 (2) $ .

For the other inclusion I have no idea.

Thanks for the help .

elementary sets theory: sets $ A, B $ disjoint $ implies overline {A} cap B = A cap overline {B} = emptyset $?

If a set $ X subset mathbb {R} ^ n $ is disconnected / separated, then it can be written as the union of two relatively open subsets, for example $ A $ Y $ B $; that is to say $$ X = A cup B. $$ I keep reading that the previous statement implies the following about closures of $ A $ Y $ B $, Written as $ overline {A} $ Y $ overline {B} $: $$ overline {A} cap B = A cap overline {B} = emptyset. $$ I do not understand why the fact that $ A $ Y $ B $ disunity would imply that each of $ A $ Y $ B $ They are also unconnected with the closure of the other.