See also

- Wikipedia, Swiss-cheese operad in Operad.

Added further clarifications:

The relevance of regular cardinals for these concepts was already pointed out by Gabriel and Ulmer in their original treatise on locally presentable categories, where on page 2 we read:

Sei $\alpha$ eine Kardinalzahl, wobei $3 \le \alpha \lt \infty$. Eine geordnete Menge $(N,\le)$ heisst

$\alpha$-gerichtet, wenn es für jede Familie $(\nu_i)_{i\in I}$ in $N$ mit $Kard(I) \lt \alpha$ ein $\mu$ gibt derart, dass $\nu_i\le\mu$. Sei $\beta$ die kleinste reguläre Kardinalzahl $\ge \alpha$. Dann ist jede $\alpha$-gerichtete Menge auch $\beta$-gerichtet. Wir setzen deshalb im folgenden zusatzlich voraus, dass $\alpha$ regulär ist (vgl. §0).

where the meaning of $\alpha\lt\infty$ is explained on page 13:

]]>Ausserdem bezeichnen wir mit $\infty$ die kleinste Kardinalzahl, die nicht mehr zu $U$ gehört (die also in unserer Sprache keine Menge ist).

Dieser Arbeit liegt die Mengenlehre von Zermelo-Fraenkel und ein fest gewähltes

Universum $U$ zugrunde. Wir setzen dabei voraus, dass $U$ die Menge $\mathbf{N}$ der natürlichen Zahlen enthält.

I’ve added a small historical correction.

]]>The assumption $\alpha\lt\infty$ in the quote looks a bit confusing to me. If $\infty$ means $\aleph_0$ then the smallest cardinal $\geq\alpha$ is necessarily $\infty$. So I assume $\infty$ means “the size of the universe”?

That would actually be consistent with the use of “$\infty$-filtered” in Adamek/Lawvere/Rosicky’s “Continuous categories revisited”.

]]>Created:

An operad is a monoid in the monoidal category of symmetric sequences equipped with the substitution product.

A **module over an operad** is just a right module over this monoid.

Right modules are very different from left modules, the latter are essentially algebras over an operad.

V. A. Smirnov. ON THE COCHAIN COMPLEX OF TOPOLOGICAL SPACES. Mathematics of the USSR-Sbornik 43:1 (1982), 133–144. doi.

Martin Markl,

*Models for operads*, Comm. Algebra 24 (1996), no. 4, 1471–1500. arXiv:hep-th/9411208v1.

Created:

The Swiss cheese operad is an analogue of the little disks operad, where disks are replaced by half-disks, which contain both ordinary disks in their interior, as well as half-disks positioned at the flat boundary.

This structure can be organized into an operad in the category of modules over the little disks operad.

Alexander Voronov,

*The Swiss-Cheese Operad*, arXiv:math/9807037.Najib Idrissi,

*Swiss-Cheese operad and Drinfeld center*, arXiv:1507.06844.

Corrected version:

Regular cardinals $\lambda$ are used in the definitions of $\lambda$-filtered colimits, $\lambda$-presentable objects, $\lambda$-accessible categories, locally $\lambda$-presentable categories, $\lambda$-ind-completion, and many notions derived from these, e.g., $\lambda$-combinatorial model categories.

Then notions make sense for all cardinals, not necessarily regular. However, the relevant concepts reduce to those for regular cardinals.

The relevance of regular cardinals for these concepts was already pointed out by Gabriel and Ulmer in their original treatise on locally presentable categories, where on page 2 we read:

Sei $\alpha$ eine Kardinalzahl, wobei $3 \le \alpha \lt \infty$. Eine geordnete Menge $(N,\le)$ heisst

$\alpha$-gerichtet, wenn es für jede Familie $(\nu_i)_{i\in I}$ in $N$ mit $Kard(I) \lt \alpha$ ein $\mu$ gibt derart, dass $\nu_i\le\mu$. Sei $\beta$ die kleinste reguläre Kardinalzahl $\ge \alpha$. Dann ist jede $\alpha$-gerichtete Menge auch $\beta$-gerichtet. Wir setzen deshalb im folgenden zusatzlich voraus, dass $\alpha$ regulär ist (vgl. §0).

If $\lambda$ is not a regular cardinal, then a category has $\lambda$-filtered colimits if and only if it has $\lambda^+$-filtered colimits, and $\lambda^+$ is always a regular cardinal (assuming the axiom of choice). In this case, a category is locally $\lambda$-presentable if and only if it is locally $\lambda^+$-presentable.

]]>

Oh, wait, now I see. Lupercio & Uribe introduce that Prop. 7.2.2 only in v2, after they must have seen it in BCMMS. Will adjust the wording in the entry once more…

]]>Added pointer to

- Ernesto Lupercio, Bernardo Uribe,
*Gerbes over Orbifolds and Twisted K-theory*, Comm. Math. Phys. 245(3): 449-489. (arXiv:math/0105039, doi:10.1007/s00220-003-1035-x)

I have expanded publication data but also commentary on the three original references (along the lines indicated in the parallel thread here).

]]>I have added pointer to

- Ernesto Lupercio, Bernardo Uribe, Section 7.2 of:
*Gerbes over Orbifolds and Twisted K-theory*, Comm. Math. Phys. 245(3): 449-489. (arXiv:math/0105039, doi:10.1007/s00220-003-1035-x)

Their Prop. 7.2.2 is verbatim the characterization that BCMMS made the definition of “bundle gerbe module” a month and a half later (except that LU focus on open covers instead of more general surjective submersions, but that’s not an actual restriction and in any case not the core of the definition).

Also added pointer to

- Marco Mackaay,
*A note on the holonomy of connections in twisted bundles*, Cahiers de Topologie et Géométrie Différentielle Catégoriques, Tome 44 (2003) no. 1, pp. 39-62. (arXiv:math/0106019, numdam:CTGDC_2003__44_1_39_0)

which essentially recovers Lupercio & Uribe’s Def. 7.2.1.

From the arXiv timestamps I gather that it must have been an intense couple of weeks for all these auhtors in spring 2001. But Lupercio & Uribe came out first, by a fair margin. And in equivariant generality, right away…

]]>added (here) a description of the twisted Chern character as a twisted characteristic form of twisted connections on twisted virtual vector bundles, according to BCMMS 2002, Prop. 9.1

]]>I agree this is a counterexample.

I was also confused by the restriction to regular cardinals in these definitions. It is explained in the book by Gabriel and Ulmer where presentability is first introduced, but it seems later references never comment on this. On page 2 in the introduction they write (translated from German):

]]>Let $\alpha$ be a cardinal, with $3\leq \alpha\lt\infty$ [sic]. A poset $(N,\leq)$ is called

$\alpha$-filteredif for every family $(\nu_i)_{i\in I}$ in $N$ with $card(I)\lt\alpha$ there exists $\mu\in N$ such that $\nu_i\leq \mu$. Let $\beta$ be the smallest regular cardinal $\geq \alpha$. Then every $\alpha$-filtered poset is also $\beta$-filtered. We therefore assume in the sequel that $\alpha$ is regular.

Re #17: I must say that the definition of a locally λ-presentable category always confused me by its unmotivated introduction of regular cardinals.

According to #14, if a locally λ-presentable category is locally λ^+-presentable for any nonregular cardinal λ, then it would seem that Exercise 1.b(2) must be false: if μ<ν are regular cardinals then locally μ-presentable categories are always locally ν-presentable, and there are locally ν-presentable categories that are not μ-presentable.

Substituting μ=cf(λ) and ν=λ^+ produces a counterexample.

]]>Note that this second part is the statement of Exercise 1.b(3) in *Locally presentable and accessible categories* (page 59). But that doesn’t necessarily mean it’s correct, because the immediately preceding Exercise 1.b(2) is definitely wrong (I have verified this with Jiri Rosicky). There is a more extensive and corrected treatment of presentability for non-regular cardinals in section 3 of Internal sizes in μ-abstract elementary classes; perhaps it contains the answer to this question.

Added related concepts.

]]>Re #14: The first part was indeed nonsense.

What is a counterexample to the second part? (A category is locally λ-presentable if and only if it is locally cof(λ)-presentable.)

]]>Correction:

If $\lambda$ is not a regular cardinal, then a category has $\lambda$-filtered colimits if and only if it has $\lambda^+$-filtered colimits, and $\lambda^+$ is always a regular cardinal (assuming the axiom of choice).

]]>A category has $\lambda$-filtered colimits if and only if it has $cof(\lambda)$-filtered colimits.

A category is locally $\lambda$-presentable if and only if it is locally $cof(\lambda)$-presentable.

There are arbitrarily large cardinals of cofinality $\omega$, so if this were true then any category with $\lambda$-filtered colimits would have $\omega$-filtered colimits. I think one should replace $cof(\lambda)$ by $\lambda^+$ in these statements (the successor cardinal of $\lambda$, which is always regular). Indeed, if $cof(\lambda)\lt\lambda$, then every set of cardinality $\lt\lambda^+$ is a union of $\lt\lambda$ subsets of cardinality $\lt\lambda$, so every $\lambda$-filtered poset is automatically $\lambda^+$-filtered.

]]>Added:

Regular cardinals $\lambda$ are used in the definitions of $\lambda$-filtered colimits, $\lambda$-presentable objects, $\lambda$-accessible categories, locally $\lambda$-presentable categories, $\lambda$-ind-completion, and many notions derived from these, e.g., $\lambda$-combinatorial model categories.

Then notions make sense for all cardinals, not necessarily regular. However, the relevant concepts reduce to those for regular cardinals.

Recall that the cofinality $cof(\lambda)$ of a cardinal $\lambda$ is the smallest cardinal $\mu$ such that $\lambda$ is a sum of $\mu$ cardinals smaller than $\lambda$.

A cardinal $\lambda$ is regular if and only if $\lambda=cof(\lambda)$.

A category has $\lambda$-filtered colimits if and only if it has $cof(\lambda)$-filtered colimits.

A category is locally $\lambda$-presentable if and only if it is locally $cof(\lambda)$-presentable.

]]>

added some references

]]>The link to your webpage doesn’t work.

]]>There seems to be new forum spam at ‘HomePage’ spammed by someone called dgroyals1.

]]>I have added a remark (here) that the space of rational maps $\mathbb{C}P^1 \to \mathbb{C}P^n$ that is considered in Segal’s theorem is also considered in Gromov-Witten theory (after compactification and quotienting), as is nicely explicit in Bertram 02, p. 9.

This confluence looks like it ought to have drawn attention, but I don’t find literature in this direction.

]]>