topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
fiber space, space attachment
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
continuous metric space valued function on compact metric space is uniformly continuous
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
Theorems
Analysis Theorems
symmetric monoidal (∞,1)-category of spectra
Pyknotic sets (from the ancient Greek πυκνός, meaning thick, dense, or compact) aim to provide a convenient setting in the framework of homotopical algebra for working with algebraic objects that have some sort of topology on them.
A pyknotic set is a sheaf of sets on the site of compact Hausdorff topological spaces whose underlying set has rank smaller than the smallest strongly inaccessible cardinal. The Grothendieck topology on the latter category is generated by finite jointly effective epimorphic families of maps.
Pyknotic sets form a coherent topos, whereas the closely related category of condensed sets does not form a topos (BarHai 19, Sec. 0.3), only an infinitary pretopos.
This construction generalizes to pyknotic objects for any (∞,1)-category.
Compactly generated topological spaces embed fully faithfully into pyknotic sets, in a manner that preserves limits.
One key point, however, is that the relationship between compactly generated topological spaces and pyknotic sets is dual to the relationship between compactly generated topological spaces and general topological spaces: in topological spaces, compactly generated topological spaces are stable under colimits but not limits; in pyknotic sets, compactly generated topological spaces are stable under limits but not colimits (BarHai 19, 0.2.3).
The concept of pyknotic structure can be compared to that of cohesive structure:
one of the main peculiarities of the theory of pyknotic structures … is also one of its advantages: the forgetful functor is not faithful. (BarHai 19, 0.2.4)
The global sections functor is given by evaluation at the one-point compactum. This has a left adjoint sending a set, $X$, to the discrete pyknotic set attached to $X$, and a right adjoint to the indiscrete pyknotic set attached to $X$. The left adjoint, the discrete functor, does not preserve limits and so does not possess a further left adjoint. The topos of pyknotic sets is thus not cohesive (BarHai 19, 2.2.14).
For example locally compact abelian groups, normed rings, and complete locally convex topological vector spaces are pyknotic objects.
Pyknotic sets can be described as sheaves on several different sites.
Take $\mathcal{C}$ to be the Kleisli category for the ultrafilter monad: objects are sets and morphisms are functions between the sets of ultrafilters borne by those sets. Pyknotic sets are functors $\mathcal{C}^{op}\to Set$ that carry finite coproducts in $\mathcal{C}$ to products in $Set$. So they can be understood as models for a large Lawvere theory.
Take $\mathcal{C}$ to be the category of compact Hausdorff spaces (i.e., compacta) but where the sheaf condition is somewhat more elaborate, requiring in addition that $F$ carries the coequalizer displaying an epimorphism $K \to L$ as the cokernel of its kernel pair to monomorphism.
Take $\mathcal{C} \subset Top$ to be the subcategory of tiny compact hausdorff spaces that are extremally disconnected, called Stonean spaces. A pyknotic set carries finite coproducts to products in $Set$.
In the first of a series of talks (BarHaiMSRI), Barwick motivates the concept of pyknotic set as follows:
The category of topological abelian groups, $AbTop$, is not abelian. This can be seen by taking an abelian group and imposing two topologies, one finer than the other. Both the kernel and cokernel of the continuous map which is the identity on elements are $0$. This is an indication that $AbTop$ does not have enough objects. To rectify this, we can modify the category to allow ‘pyknotic’ structures on $0$, which can act as a cokernel here.
In Pontrjagin duality, the pairing and double dual maps are not necessarily continuous outside of locally compact abelian groups. This occurs because the category of topological spaces does not have an internal hom. This is an indication of too many objects, and restriction to compactly generated spaces may be imposed.
…the indiscrete topological space $\{0,1\}$, viewed as a sheaf on the site of compacta, is pyknotic but not condensed (relative to any universe). By allowing the presence of such pathological objects into the category of pyknotic sets, we guarantee that it is a topos, which is not true for the category of condensed sets. (BarHai 19, sec 0.3)
[Barwick and Haine] assume the existence of universes, fixing in particular a “tiny” and a “small” universe, and look at sheaves on tiny profinite sets with values in small sets; they term these pyknotic sets. In our language, placing ourselves in the small universe, this would be $\kappa$-condensed sets for the first strongly inaccessible cardinal $\kappa$ they consider (the one giving rise to the tiny universe). (Scholze 19, p. 7)
quasi-topological space (roughly, the concrete pyknotic sets)
Clark Barwick, Peter Haine, Pyknotic objects, I. Basic notions, (arXiv:1904.09966)
Clark Barwick, Peter Haine, MSRI: Pyknotic/Condensed Seminar,
Peter Scholze, Lectures on Condensed Mathematics, (pdf)
Sebastian Wolf, The pro-étale topos as a category of pyknotic presheaves, arxiv/2012.10502
Last revised on January 5, 2023 at 02:49:30. See the history of this page for a list of all contributions to it.