Skip to main content
enabled the --> enabled to
Source Link
Matthieu Romagny
  • 4.5k
  • 1
  • 31
  • 37

Fréchet defined metric spaces in 1906 and Hausdorff defined topological spaces in 1914. This enabled theto study the topological properties of spaces without first triangulating them.

Fréchet defined metric spaces in 1906 and Hausdorff defined topological spaces in 1914. This enabled the study the topological properties of spaces without first triangulating them.

Fréchet defined metric spaces in 1906 and Hausdorff defined topological spaces in 1914. This enabled to study the topological properties of spaces without first triangulating them.

Post Made Community Wiki by S. Carnahan
added 995 characters in body
Source Link
Dmitri Pavlov
  • 37.8k
  • 4
  • 97
  • 183

Around 1945, Leray introduced sheaves and spectral sequences. The relevant theory was further developed by Cartan, Serre, and others.

Eilenberg and MacLane introduced categories, functors, and natural transformations in 1945. Ever since then, category theory played an increasingly important role in homotopy theory, to the point where we are now often unable to cleanly separate them.

Around 1953, Cartan and Eilenberg completed their book on homological algebra (published in 1956).

Kan (advised by Eilenberg) systematically developed simplicial homotopy theory (and briefly also cubical homotopy theory) starting from around 1955. He introduced combinatorial homotopy groups, the Dold–Kan correspondence, adjoint functors, limits and colimits, Kan extensions, etc.

In 1971, Gabriel and Ulmer published their systematic account of locally presentable categories.

Segal introduced Γ-spaces around 1972. At the same time, May introduced operads, also in connection with infinite loop spaces.

Boardman and Vogt introduced quasicategories in 1973.

In 1977, Sullivan published his work on rational homotopy theory in the language of commutative differential graded algebras, complementing the previous work by Quillen.

Around 1979, Bousfield introduced what is now known as Bousfield localizations.

In 1983, Grothendieck introduced what is now known as Grothendieck homotopy theory, as well as derivators.

In 1989, Makkai and Paré published a systematic account of accessible categories.

In 1995, Baez and Dolan formulated the cobordism and tangle hypotheses, which perhaps qualifies as the first noticeable conjecture about (∞,n)-categories for arbitrary n.

In the late 1990s, Voevodsky introduced and developed motivic homotopy theory (including some joint work with Morel).

Around the late 1990s, Smith introduced combinatorial model categories and proved what is now known as the Smith recognition theorem and established the existence of left Bousfield localizations of left proper combinatorial model categories.

Monoidal model categories were systematically studied by Schwede and Shipley starting from the late 1990s1997.

In the mid-2000s2006 (based on a 2003 preprint), Lurie's Higher Topos Theory came out, first as an online draft, which was later published.

Eilenberg and MacLane introduced categories, functors, and natural transformations in 1945. Ever since then, category theory played an increasingly important role in homotopy theory, to the point where we are now often unable to cleanly separate them.

Kan (advised by Eilenberg) systematically developed simplicial homotopy theory (and briefly also cubical homotopy theory) starting from around 1955. He introduced combinatorial homotopy groups, the Dold–Kan correspondence, adjoint functors, limits and colimits, Kan extensions, etc.

Segal introduced Γ-spaces around 1972.

Boardman and Vogt introduced quasicategories in 1973.

Around 1979, Bousfield introduced what is now known as Bousfield localizations.

In 1995, Baez and Dolan formulated the cobordism and tangle hypotheses, which perhaps qualifies as the first noticeable conjecture about (∞,n)-categories for arbitrary n.

Around the late 1990s, Smith introduced combinatorial model categories and proved what is now known as the Smith recognition theorem and established the existence of left Bousfield localizations of left proper combinatorial model categories.

Monoidal model categories were systematically studied by Schwede and Shipley starting from the late 1990s.

In the mid-2000s, Lurie's Higher Topos Theory came out, first as an online draft, which was later published.

Around 1945, Leray introduced sheaves and spectral sequences. The relevant theory was further developed by Cartan, Serre, and others.

Eilenberg and MacLane introduced categories, functors, and natural transformations in 1945. Ever since then, category theory played an increasingly important role in homotopy theory, to the point where we are now often unable to cleanly separate them.

Around 1953, Cartan and Eilenberg completed their book on homological algebra (published in 1956).

Kan (advised by Eilenberg) systematically developed simplicial homotopy theory (and briefly also cubical homotopy theory) starting from around 1955. He introduced combinatorial homotopy groups, the Dold–Kan correspondence, adjoint functors, limits and colimits, Kan extensions, etc.

In 1971, Gabriel and Ulmer published their systematic account of locally presentable categories.

Segal introduced Γ-spaces around 1972. At the same time, May introduced operads, also in connection with infinite loop spaces.

Boardman and Vogt introduced quasicategories in 1973.

In 1977, Sullivan published his work on rational homotopy theory in the language of commutative differential graded algebras, complementing the previous work by Quillen.

Around 1979, Bousfield introduced what is now known as Bousfield localizations.

In 1983, Grothendieck introduced what is now known as Grothendieck homotopy theory, as well as derivators.

In 1989, Makkai and Paré published a systematic account of accessible categories.

In 1995, Baez and Dolan formulated the cobordism and tangle hypotheses, which perhaps qualifies as the first noticeable conjecture about (∞,n)-categories for arbitrary n.

In the late 1990s, Voevodsky introduced and developed motivic homotopy theory (including some joint work with Morel).

Around the late 1990s, Smith introduced combinatorial model categories and proved what is now known as the Smith recognition theorem and established the existence of left Bousfield localizations of left proper combinatorial model categories.

Monoidal model categories were systematically studied by Schwede and Shipley starting from 1997.

In 2006 (based on a 2003 preprint), Lurie's Higher Topos Theory came out, first as an online draft, which was later published.

Added links to papers
Source Link
Timothy Chow
  • 82.7k
  • 26
  • 363
  • 587

Segal introduced Γ-spacesΓ-spaces around 1972.

Brown studied the homotopy theory of sheaves of spaces and spectrasheaves of spaces and spectra in 1972.

Boardman and Vogt introduced quasicategoriesquasicategories in 1973.

Dwyer and Kan introduced and developed the theory of simplicial localizationssimplicial localizations starting from around 1979.

Around 1979, Bousfield introduced what is now known as Bousfield localizationsBousfield localizations.

In 1980s, Joyal established what is now known as the Joyal model structureJoyal model structure on simplicial sets.

In mid-1980s, Segal (following Witten) introduced what is now known as functorial field theoryfunctorial field theory, later studied by Atiyah, Kontsevich, Freed, Lawrence, and many others.

In 1985, Jardine gave an account of simplicial presheavessimplicial presheaves.

Around 1986, Lewis, May, Steinberger, McClure introduced genuine equivariant spectraequivariant spectra.

In 1995, Baez and Dolan formulated the cobordism and tangle hypothesescobordism and tangle hypotheses, which perhaps qualifies as the first noticeable conjecture about (∞,n)-categories for arbitrary n.

In 1997, Elmendorf, Kriz, Mandell, May published the first ever account of a symmetric monoidal category of spectrasymmetric monoidal category of spectra.

In 1998, Hovey, Shipley, Smith published an account of symmetric spectrasymmetric spectra.

In 1998, Rezk introduced complete Segal spacescomplete Segal spaces.

Around the late 1990s, Smith introduced combinatorial model categories and proved what is now known as the Smith recognition theoremSmith recognition theorem and established the existence of left Bousfield localizations of left proper combinatorial model categories.

Monoidal model categoriesMonoidal model categories were systematically studied by Schwede and Shipley starting from the late 1990s.

In the mid-2000s, Lurie's Higher Topos TheoryHigher Topos Theory came out, first as an online draft, which was later published.

Segal introduced Γ-spaces around 1972.

Brown studied the homotopy theory of sheaves of spaces and spectra in 1972.

Boardman and Vogt introduced quasicategories in 1973.

Dwyer and Kan introduced and developed the theory of simplicial localizations starting from around 1979.

Around 1979, Bousfield introduced what is now known as Bousfield localizations.

In 1980s, Joyal established what is now known as the Joyal model structure on simplicial sets.

In mid-1980s, Segal (following Witten) introduced what is now known as functorial field theory, later studied by Atiyah, Kontsevich, Freed, Lawrence, and many others.

In 1985, Jardine gave an account of simplicial presheaves.

Around 1986, Lewis, May, Steinberger, McClure introduced genuine equivariant spectra.

In 1995, Baez and Dolan formulated the cobordism and tangle hypotheses, which perhaps qualifies as the first noticeable conjecture about (∞,n)-categories for arbitrary n.

In 1997, Elmendorf, Kriz, Mandell, May published the first ever account of a symmetric monoidal category of spectra.

In 1998, Hovey, Shipley, Smith published an account of symmetric spectra.

In 1998, Rezk introduced complete Segal spaces.

Around the late 1990s, Smith introduced combinatorial model categories and proved what is now known as the Smith recognition theorem and established the existence of left Bousfield localizations of left proper combinatorial model categories.

Monoidal model categories were systematically studied by Schwede and Shipley starting from the late 1990s.

In the mid-2000s, Lurie's Higher Topos Theory came out, first as an online draft, which was later published.

Segal introduced Γ-spaces around 1972.

Brown studied the homotopy theory of sheaves of spaces and spectra in 1972.

Boardman and Vogt introduced quasicategories in 1973.

Dwyer and Kan introduced and developed the theory of simplicial localizations starting from around 1979.

Around 1979, Bousfield introduced what is now known as Bousfield localizations.

In 1980s, Joyal established what is now known as the Joyal model structure on simplicial sets.

In mid-1980s, Segal (following Witten) introduced what is now known as functorial field theory, later studied by Atiyah, Kontsevich, Freed, Lawrence, and many others.

In 1985, Jardine gave an account of simplicial presheaves.

Around 1986, Lewis, May, Steinberger, McClure introduced genuine equivariant spectra.

In 1995, Baez and Dolan formulated the cobordism and tangle hypotheses, which perhaps qualifies as the first noticeable conjecture about (∞,n)-categories for arbitrary n.

In 1997, Elmendorf, Kriz, Mandell, May published the first ever account of a symmetric monoidal category of spectra.

In 1998, Hovey, Shipley, Smith published an account of symmetric spectra.

In 1998, Rezk introduced complete Segal spaces.

Around the late 1990s, Smith introduced combinatorial model categories and proved what is now known as the Smith recognition theorem and established the existence of left Bousfield localizations of left proper combinatorial model categories.

Monoidal model categories were systematically studied by Schwede and Shipley starting from the late 1990s.

In the mid-2000s, Lurie's Higher Topos Theory came out, first as an online draft, which was later published.

Source Link
Dmitri Pavlov
  • 37.8k
  • 4
  • 97
  • 183
Loading