chars int64 40 24k | content_type stringclasses 9
values | id stringlengths 13 64 | source stringclasses 5
values | subset stringclasses 31
values | text stringlengths 40 24k |
|---|---|---|---|---|---|
2,763 | lean_file_chunk | b734775640977c44469bad8d2229e4f6172f87386fb25cb19e0edd7fed4d64a4 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Emily Riehl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Emily Riehl
-/
module
public import Mathlib.CategoryTheory.Functor.Currying
public import Mathlib.CategoryTheory.Monoidal.Cartesian.Cat
public import Mathlib.CategoryTheory.Monoidal.Clos... |
3,459 | lean_file_chunk | dd551936fe1a5379c90e4fcc993bbbc386d81af0a75240d20a7029d5815a7f3f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Nicolas Rolland. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolas Rolland
-/
module
public import Mathlib.CategoryTheory.Category.Cat
public import Mathlib.CategoryTheory.Adjunction.Basic
public import Mathlib.CategoryTheory.ConnectedCompon... |
4,002 | lean_file_chunk | dbcf1a8d8c81e9fde37e472a01fcf7c0ed76ff240812f30a1dcb4dd26d2f71d8 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
module
public import Mathlib.CategoryTheory.SingleObj
public import Mathlib.CategoryTheory.Limits.Shapes.Products
/-!
# Category of groupoids
This file contains ... |
5,806 | lean_file_chunk | 8fb2dd3e2fa5113d88d0c5b389d99567a9c7bfb91cafc12ff768a5cd2cc03d94 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
module
public import Mathlib.CategoryTheory.SmallObject.Iteration.Basic
/-!
# The functor from `Set.Iic j` deduced from a cocone
Given a functor `F : Set.Iio j ⥤ C` and `c :... |
5,714 | lean_file_chunk | 7ec01049991a05e16878334660a74a4366a8a83b5c8a836eb110d6142514cefd | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
module
public import Mathlib.CategoryTheory.SmallObject.Iteration.Basic
public import Mathlib.CategoryTheory.SmallObject.Iteration.ExtendToSucc
public import Mathlib.CategoryT... |
17,733 | lean_file_chunk | 2bfc859bad653eb3dfa14b7cc9c07a6d88f2845c295f39538bfc01dd86d363a8 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
module
public import Mathlib.CategoryTheory.Category.Preorder
public import Mathlib.CategoryTheory.Limits.FunctorCategory.Basic
public import Mathlib.CategoryTheory.Limits.Sha... |
7,510 | lean_file_chunk | 2c18430454c0b3f763afdc163daf3d4217eee27226f288d1c96454bbe82f1bcc | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
module
public import Mathlib.CategoryTheory.SmallObject.Iteration.Basic
/-!
# Extension of a functor from `Set.Iic j` to `Set.Iic (Order.succ j)`
Given a linearly ordered ty... |
1,390 | lean_file_chunk | 0b990102f9ba6152ef82778a8c83034c8b469bdb9f60df5e7f0c075b09b84c92 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
module
public import Mathlib.CategoryTheory.Limits.FunctorCategory.Basic
public import Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts
public import Mathlib.... |
1,956 | lean_file_chunk | 27ed334a0449c48d0a18f4e71a803719d26aed35807a9e86cce445f9408a94fe | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
module
public import Mathlib.CategoryTheory.Limits.Constructions.EpiMono
public import Mathlib.CategoryTheory.Limits.FunctorCategory.Basic
/-!
# Monomorphisms and epimorphism... |
965 | lean_file_chunk | 3fa145f9abfd319b2ff1e528c607c9d28e8e1dd8823e0f48a53434b58d49d66d | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
module
public import Mathlib.CategoryTheory.Limits.FunctorCategory.Basic
public import Mathlib.CategoryTheory.Limits.Filtered
/-!
# Functor categories have filtered c... |
23,992 | lean_file_chunk | 71a8380bcbd5a76996a061add39526a4463851e0e7343c19dd8bc7cc9d7a78c0 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
module
public import Mathlib.CategoryTheory.Functor.Currying
public import Mathlib.CategoryTheory.Limits.Preserves.Limits
/-!
# (Co)limits in functor categories.
We sh... |
800 | lean_file_chunk | ab55c707fc0913caa0c0ddb255099696ed6b1cc67c0b35b37b6d98831732e81d | lean_source_file | pkg_mathlib_source_file_chunks | /-- `colimitFlipIsoCompColim` is natural with respect to diagrams. -/
@[simps!]
def colimCompFlipIsoWhiskerColim [HasColimitsOfShape J C] :
flipFunctor K J C ⋙ colim ≅ (whiskeringRight _ _ _).obj colim :=
NatIso.ofComponents colimitFlipIsoCompColim fun {F G} η ↦ by
ext k
apply colimit_obj_ext
intro j
... |
2,321 | lean_file_chunk | 632edaa25688e62de01dbb585015891f051754374cee837250c8f622d50effbc | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 Leopold Mayer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leopold Mayer
-/
module
public import Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts
public import Mathlib.CategoryTheory.Limits.FunctorCategory.Basic
/-!
# Biproducts in funct... |
471 | lean_file_chunk | b6636ce785ea9fbcb66e810501f0772b3a68391abd9ee8606d13e4a9b0d89236 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
module -- shake: keep-all
public import Mathlib.CategoryTheory.Category.Init
public import Mathlib.Data.Finset.Attr
public import Mathlib.Tactic.Common
public import Mat... |
4,930 | lean_file_chunk | f855f9fe9fd963f5a3359ed2b82f8b2c3f7272075f0dabd43f2d38732efe112e | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Reid Barton
-/
module
public import Mathlib.CategoryTheory.Limits.Types.Limits
/-!
# Cones and limits
In this file, we give the natural isomorphism between cones on `F` ... |
13,831 | lean_file_chunk | 683b17e79cdb5ea39c61da2700d8143b4f4d0760516f4a5e1ede094f3ba2b7cb | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
module
public import Mathlib.CategoryTheory.Limits.Types.Pullbacks
public import Mathlib.CategoryTheory.Limits.Shapes.Pullback.IsPullback.Basic
/-!
# Pushouts in `Type`
We d... |
10,227 | lean_file_chunk | 8d16111121129d99121813a47a9a634e64430d910b4b80d54f8add9f018b3b87 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
module
public import Mathlib.CategoryTheory.Limits.Shapes.Products
public import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
public import Mathlib.CategoryTheory... |
7,319 | lean_file_chunk | 4d01f56b4caeece5e76c84d48bb160e0dca49f02bce9e9bc7535786de63cd7dd | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Reid Barton
-/
module
public import Mathlib.CategoryTheory.Limits.Types.Colimits
public import Mathlib.CategoryTheory.Filtered.Basic
/-!
# Filtered colimits in the catego... |
12,706 | lean_file_chunk | 4cbb909a695aebe9873e5ab4546e3257f7c2874d9d1be0fafa45ec5cc2d69f2d | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
module
public import Mathlib.CategoryTheory.Functor.Basic
public import Mathlib.CategoryTheory.Types.Basic
/-!
# The colimit type of a functor to types
Given a category `J` ... |
3,204 | lean_file_chunk | 93dc1f8b900eadcc577f8553ca61c743cc858ea63195956fc50b5146a8529aa8 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
module
public import Mathlib.CategoryTheory.Limits.Shapes.Equalizers
public import Mathlib.CategoryTheory.Limits.Types.Limits
public import Mathlib.Tactic.CategoryTheory... |
4,293 | lean_file_chunk | c7e65832bea7426b3b1867d288577a7262c9708b57bf825901c3b9f4258cce84 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
module
public import Mathlib.CategoryTheory.Filtered.Basic
public import Mathlib.CategoryTheory.Limits.Types.ColimitType
/-!
# Filtered colimits of types
In this file, given... |
13,555 | lean_file_chunk | 53e663fd73c2d0166e1deaa719dde8bcd6eccf8e06d66790b848014bdbb53bb3 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
module
public import Mathlib.CategoryTheory.Limits.Shapes.Products
public import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
public import Mathlib.CategoryTheory... |
9,428 | lean_file_chunk | 56bff4f26eed2cb25cf8a480cea1618e753bbd145371d20e7dcc63e01651b65d | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Reid Barton, Joël Riou
-/
module
public import Mathlib.Logic.UnivLE
public import Mathlib.CategoryTheory.Limits.HasLimits
public import Mathlib.CategoryTheory.Limits.Types... |
3,125 | lean_file_chunk | 6d4244a303d50a758d1cf0ba01da1d5a627e83a8c5bd1378f590cb43a63cfa7b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
module
public import Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer
public import Mathlib.CategoryTheory.Limits.Types.Limits
/-!
# Multiequalizers in Type
Given `J : Mu... |
10,523 | lean_file_chunk | 4ae2f402c91849b3af48afdbcff5e5a2bed8cd4f352795e1f5652ece4da17658 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Reid Barton
-/
module
public import Mathlib.Logic.UnivLE
public import Mathlib.CategoryTheory.Limits.HasLimits
/-!
# Limits in the category of types.
We show that the ca... |
3,814 | lean_file_chunk | 532ec2620affc4e6625d0bfcf1015e341c5dbe6793cb3bd147b40183855c369e | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Reid Barton
-/
module
public import Mathlib.CategoryTheory.Limits.Types.Limits
public import Mathlib.CategoryTheory.Limits.Shapes.Images
/-!
# Images in the category of t... |
5,333 | lean_file_chunk | 6a5dfad319a4f27539fa5c84bff87577ce0e8253bc8fbb79aa5629440d6592a8 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
module
public import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Multiequalizer
public import Mathlib.CategoryTheory.Limits.Shapes.MultiequalizerPullback
public import Math... |
8,183 | lean_file_chunk | c96675c42f2a6612a1f3797ce011d3638e9ba00c4edd40cdaf3ce9b853e794bd | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Christian Merten
-/
module
public import Mathlib.CategoryTheory.Limits.Shapes.Pullback.IsPullback.Defs
public import Mathlib.CategoryTheory.Limits.Types.Limits
/-!
# Pull... |
3,003 | lean_file_chunk | be5285836741108aa763881f1e1a3b5064a4178430d70be72ed360ad7fba2233 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
module
public import Mathlib.CategoryTheory.Limits.Shapes.Equalizers
public import Mathlib.CategoryTheory.Limits.Types.Colimits
public import Mathlib.Logic.Function.Coeq... |
4,163 | lean_file_chunk | 49ee83f327dd1a3503d3f26061215ca841e26d5d84eb108179d2ae9d8765c670 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
module
public import Mathlib.AlgebraicTopology.ExtraDegeneracy
public import Mathlib.CategoryTheory.Limits.FormalCoproducts.Cech
public import Mathlib.CategoryTheory.Limits.Co... |
17,085 | lean_file_chunk | 9eac8a763e78caf6ad3bcf724facb5ca3cf3b780a39d93f75a05ead86d3b150f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou, Kenny Lau
-/
module
public import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Products
public import Mathlib.CategoryTheory.Limits.Shapes.Opposites.Products
public import Mat... |
6,682 | lean_file_chunk | c6c67c653feef71deb8ca337c1e4f1b72ff7ca455d59e01f5c5ffd9c00555217 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
module
public import Mathlib.AlgebraicTopology.SimplicialObject.Basic
public import Mathlib.CategoryTheory.Limits.FormalCoproducts.Basic
/-!
# The Cech object for formal copr... |
10,765 | lean_file_chunk | 5023bc909ea11155a9cef7bf9a73b14a2e6fc9a0a5202e97df2e90eb1526dbdd | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
module
public import Mathlib.CategoryTheory.Limits.Preserves.Finite
public import Mathlib.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts
public import Mathlib.Cat... |
3,083 | lean_file_chunk | 8bdb5d44213ab348ee41d2353e494bffc61f63d1a12015cc7dbe90b88845600b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
module
public import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
public import Mathlib.CategoryTheory.Limits.Shapes.Pullback.Mono
public import Mathlib.CategoryT... |
10,294 | lean_file_chunk | b3c3da6f18ab5be1f537368c8f156b4f64b65e1c33db93614b90c8b097caeafc | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
module
public import Mathlib.CategoryTheory.Filtered.Basic
public import Mathlib.CategoryTheory.Limits.HasLimits
/-!
# Limits of eventually constant functors
If `F : J ⥤ C` ... |
11,766 | lean_file_chunk | 5a993d89e8be8560acec2feff7be1d2a3ac98c43ff95ab9b23ce361ead353a08 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
module
public import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
public import Mathlib.CategoryTheory.Limits.Shapes.Opposites.Filtered
pu... |
23,976 | lean_file_chunk | 68d751c03c7d4f16cee17dbb4889adca14c978c003fa91a94c8ce68ebdbef862 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Kim Morrison
-/
module
public import Mathlib.CategoryTheory.Limits.Constructions.BinaryProducts
public import Mathlib.CategoryTheory.Limits.Constructions.Equalizers
public... |
3,360 | lean_file_chunk | 3165d7fbc1b1eb0ba0ca188388f49bb10072b0fdcaaea9b0d1823d522cd0fe4f | lean_source_file | pkg_mathlib_source_file_chunks | /-- If a functor creates coequalizers and coproducts, it creates colimits.
We additionally require the rather strong condition that the functor reflects isomorphisms. It is
unclear whether the statement remains true without this condition. There are various definitions of
"creating colimits" in the literature, and whe... |
9,639 | lean_file_chunk | f83df1ea24e43655d24afa2b8af224eba4b7dd6ebb22f7082556698516c980af | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Andrew Yang
-/
module
public import Mathlib.CategoryTheory.Limits.Shapes.Equalizers
public import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
public import Mathlib... |
1,856 | lean_file_chunk | 4b5af29ec56be16aa87b212f95ad2279f84fcf0eb8a58856d19dceb5ce33d04c | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
module
public import Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks
public import Mathlib.CategoryTheory.Limits.Shapes.Products
public import Mathlib.CategoryTheory.Limits... |
7,377 | lean_file_chunk | 757abdbb47757ab9eddf426de9a2728b4acc7e421aacbdb8cba32608e1616935 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
module
public import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
public import Mathlib.CategoryTheory.Limits.Constructions.BinaryProducts
/-!
# Limits involving ... |
2,397 | lean_file_chunk | 4275e027a8f9f44773d001d8c7b68c27bbabb969967602f0c037cde600ddfd9f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
module
public import Mathlib.CategoryTheory.Limits.Shapes.WideEqualizers
public import Mathlib.CategoryTheory.Limits.Shapes.Products
public import Mathlib.CategoryTheory... |
11,461 | lean_file_chunk | 32133307bd68cfe55c608303c35db116bd490d39404d781cf9f80f24e117b150 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Andrew Yang
-/
module
public import Mathlib.CategoryTheory.Limits.Shapes.Terminal
public import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
public import Mathlib.C... |
4,094 | lean_file_chunk | 70c6795a94e6e3765bc7048b8cf7a19ed9fa4719feac6b8b7c2fa3ae3cfe00e2 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
module
public import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
public import Mathlib.CategoryTheory.Limits.Shapes.Equalizers
public import Mathlib.CategoryTh... |
8,586 | lean_file_chunk | 337db8a443d35b5a456a574d1d278cd84bcbd376cdfd6b7ce8d58ce1a93f6f76 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
module
public import Mathlib.CategoryTheory.Limits.FinallySmall
public import Mathlib.CategoryTheory.Limits.Presheaf
public import Mathlib.CategoryTheory.Filtered.Smal... |
14,870 | lean_file_chunk | dc4da2ac80de812e49488b6fdf594290e55823ad9bda9aea86f40c3cf538dcb3 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
module
public import Mathlib.CategoryTheory.Limits.Constructions.Filtered
public import Mathlib.CategoryTheory.Limits.FullSubcategory
public import Mathlib.CategoryThe... |
3,362 | lean_file_chunk | 996b6841b63548048030c94e86888f1e591bbb589c5961c09024e2b05a6db1e1 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
module
public import Mathlib.CategoryTheory.Limits.Preserves.Ulift
public import Mathlib.CategoryTheory.Limits.IndYoneda
public import Mathlib.CategoryTheory.Limits.In... |
2,021 | lean_file_chunk | 883885e47e9bbf8d3b5fe4d30402fd1c332856ac7c411d1bacc75aeff972ac7b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
module
public import Mathlib.CategoryTheory.Limits.FilteredColimitCommutesProduct
public import Mathlib.CategoryTheory.Limits.Indization.FilteredColimits
/-!
# Ind-ob... |
3,209 | lean_file_chunk | e41e6c9b2b85cf3653ed7b99c708af67b9d4941cc290d17a45a22b32a931f135 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
module
public import Mathlib.CategoryTheory.Limits.Indization.FilteredColimits
public import Mathlib.CategoryTheory.Limits.Indization.ParallelPair
public import Mathli... |
8,456 | lean_file_chunk | 938ebcdfe993210b2a6aac3bdb7fafdfebba8549090fa5d567f630c03a9d0606 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
module
public import Mathlib.CategoryTheory.Comma.Presheaf.Colimit
public import Mathlib.CategoryTheory.Limits.Filtered
public import Mathlib.CategoryTheory.Limits.Fil... |
7,233 | lean_file_chunk | fda7230978334fa412c0810555f9d983478ea73aee1e2bd175be1bf688c1b4d9 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
module
public import Mathlib.CategoryTheory.Comma.Final
public import Mathlib.CategoryTheory.Limits.Indization.IndObject
/-!
# Parallel pairs of natural transformatio... |
23,966 | lean_file_chunk | b94fa5681e0b815accac9b40f308b6b266c07cec1a359a58bb55d52449c21acf | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Jakob von Raumer
-/
module
public import Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts
public import Mathlib.CategoryTheory.Limits.Shapes.Kernels
/-!
# Biproducts a... |
20,971 | lean_file_chunk | c1b3679d710fb4cecdf8081365896186b220d985ea202ac1c65b28c24e28d9fd | lean_source_file | pkg_mathlib_source_file_chunks | Limits.IsColimit.ι_map (biproduct.isColimit f) (biproduct.bicone g).toCocone
(Discrete.natTrans fun j => p j.as) (Discrete.mk j)
@[reassoc (attr := simp)]
theorem biproduct.map_desc {f g : J → C} [HasBiproduct f] [HasBiproduct g] (p : ∀ j, f j ⟶ g j)
{P : C} (k : ∀ j, g j ⟶ P) :
biproduct.map p ≫ biproduct... |
9,613 | lean_file_chunk | e9ff4dee40f9c39d8cb6d3dbdf6c6a089dd6e5af9ddb7582cf33cc08ec32c66f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Jack McKoen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jack McKoen
-/
module
public import Mathlib.CategoryTheory.Limits.FunctorCategory.Basic
public import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
public import Mathlib.CategoryTh... |
20,173 | lean_file_chunk | 9dba2ce3b939a6857eb76f5875097d33b4b1bc1fc81e8eeaea2e1b2da087e0a8 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Bhavik Mehta
-/
module
public import Mathlib.CategoryTheory.PEmpty
public import Mathlib.CategoryTheory.Limits.IsLimit
public import Mathlib.CategoryTheory.EpiMono
public ... |
8,721 | lean_file_chunk | 6b10bb4f2dab88c4a6bb6836471e32f88dff42edeb758ae090adc6b0db354817 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
module
public import Mathlib.CategoryTheory.Limits.Shapes.Terminal
public import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
/-!
# Strict initial objects
This ... |
23,949 | lean_file_chunk | 4ec9b296e2ec2d2eda3cb7c98a2828d2dcc11a35f6ea6efe09da2c6c0755c46a | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Robin Carlier
-/
module
public import Mathlib.CategoryTheory.Limits.ConeCategory
public import Mathlib.CategoryTheory.Limits.Final
public import Mathlib.CategoryTheory.Lim... |
1,035 | lean_file_chunk | ea91bb0095ef225de819b5754ba00fc1ab92e58f8b807f4db707a22775ed0bb0 | lean_source_file | pkg_mathlib_source_file_chunks | lemma ι_colimitOfIsReflexivePairIsoCoequalizer_hom :
colimit.ι (ofIsReflexivePair f g) zero ≫ colimitOfIsReflexivePairIsoCoequalizer.hom =
coequalizer.π f g := @ι_reflexiveCoequalizerIsoCoequalizer_hom _ _ _ h
@[reassoc (attr := simp)]
lemma π_colimitOfIsReflexivePairIsoCoequalizer_inv :
coequalizer.π f ... |
9,707 | lean_file_chunk | 7fcaea71d1a918e475d8252459bf277a359f5166c649fd392cff460046ef1807 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
module
public import Mathlib.CategoryTheory.Functor.OfSequence
public import Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproducts
public import Mathlib.Category... |
7,653 | lean_file_chunk | a80bacae5da57e30292248131c542b6a63ba4c717ca8069f68bbd12d4532d791 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
module
public import Mathlib.CategoryTheory.Balanced
public import Mathlib.CategoryTheory.LiftingProperties.Basic
/-!
# Strong epimorphisms
In this file, we define s... |
2,769 | lean_file_chunk | 3e2301df9b13ac70dd015a243719fa4522fd69a00e77151fe1f238add6cf13aa | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
module
public import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
public import Mathlib.CategoryTheory.Limits.Shapes.Products
/-!
# A product as a binary... |
22,393 | lean_file_chunk | c21b6542df3e4d85b65d1e8cf77a060c1a85d47839e7a19d4955e721fea386e7 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.CategoryTheory.Comma.Over.Pullback
public import Mathlib.CategoryTheory.Limits.Shapes.KernelPair
public import Mathlib.CategoryTheory.Limits.S... |
23,959 | lean_file_chunk | a7f5e889e90fa5571d6ded4459fb512acb082bddf6e6d06589ea0ef37b0083c9 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Bhavik Mehta
-/
module
public import Mathlib.CategoryTheory.Limits.HasLimits
public import Mathlib.CategoryTheory.Discrete.Basic
/-!
# Categorical (co)products
This file... |
16,093 | lean_file_chunk | 9aa187fb64ebe8d179981fd00ab1f96ab3e60fd5c75dcef7f0584650c70b301a | lean_source_file | pkg_mathlib_source_file_chunks | { cone := Fan.mk (∏ᶜ fun i => ∏ᶜ g i) (fun X => Pi.π (fun i => ∏ᶜ g i) X.1 ≫ Pi.π (g X.1) X.2)
isLimit := mkFanLimit _ (fun s => Pi.lift fun b => Pi.lift fun c => s.proj ⟨b, c⟩)
(by simp)
(by intro s (m : _ ⟶ (∏ᶜ fun i ↦ ∏ᶜ g i)) w; aesop (add norm simp Sigma.forall)) }
set_option backward.isDefE... |
23,958 | lean_file_chunk | 55c531ef0b55aeb3f25964ef8627a151b4a4943c5ac16ae46eed6acd7ca880ba | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Bhavik Mehta
-/
module
public import Mathlib.CategoryTheory.EffectiveEpi.Basic
public import Mathlib.CategoryTheory.Limits.Shapes.Opposites.Equalizers
public import Mathli... |
2,640 | lean_file_chunk | 979367c6e20ad31eabd769fe7ec23664da9b5c06e5857528f96c99a1359a64da | lean_source_file | pkg_mathlib_source_file_chunks | isLimit := Cofork.isColimitOfπEquivIsLimitOp _ _ hf.w _ rfl hf.isColimit
/-- A regular epimorphism in `Cᵒᵖ` induces a regular monomorphism in `C`. -/
noncomputable def RegularEpi.unop {X Y : Cᵒᵖ} {f : X ⟶ Y} (hf : RegularEpi f) :
RegularMono f.unop where
Z := hf.W.unop
left := hf.left.unop
right := hf.right.... |
14,685 | lean_file_chunk | bb6971ade114ad5ce72a451242b5b116927c73adeb9583f7dcc8ae996c39cd86 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Bhavik Mehta
-/
module
public import Mathlib.CategoryTheory.Limits.Shapes.IsTerminal
public import Mathlib.CategoryTheory.Limits.HasLimits
/-!
# Initial and terminal obje... |
5,911 | lean_file_chunk | 2fccb97e33fce4b9abfe8dbfc4d54fe76564633fda784c8fcdf17d209742a0c9 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
module
public import Mathlib.CategoryTheory.Limits.Shapes.Equalizers
/-!
# Split coequalizers
We define what it means for a triple of morphisms `f g : X ⟶ Y`, `π : Y ⟶... |
23,960 | lean_file_chunk | 6c2df5c87a11f04bb52a891193ea2174b2a06298c8b2b0911aa99310c9da8aa1 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
module
public import Mathlib.Algebra.Notation.Pi.Basic
public import Mathlib.CategoryTheory.Limits.Shapes.Products
public import Mathlib.CategoryTheory.Limits.Shapes.Ima... |
4,006 | lean_file_chunk | 6886ef5b6e634b9073b3509e34cfe6e5dd527aebd5b60101bd5f692f52c79bc0 | lean_source_file | pkg_mathlib_source_file_chunks | Limits.Sigma.desc (Function.update (fun _ ↦ 0) b (𝟙 _))
set_option backward.isDefEq.respectTransparency false in
@[reassoc (attr := simp), grind =]
lemma Sigma.ι_π_eq_id (b : β) : Sigma.ι f b ≫ Sigma.π f b = 𝟙 _ := by
simp [Sigma.π]
set_option backward.isDefEq.respectTransparency false in
@[reassoc, grind =]
lemm... |
23,972 | lean_file_chunk | 725a1e4f10f36c44b986bdf59fee3d61d3af9447e3f449fffb80bd0bb238e108 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2018 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Markus Himmel
-/
module
public import Mathlib.CategoryTheory.EpiMono
public import Mathlib.CategoryTheory.Limits.HasLimits
public import Mathlib.CategoryTheory.Limits.Shap... |
23,975 | lean_file_chunk | a9cc0345744373f807a43c39a0cca717b99268d97579747bd2e5a0d2ef9cc964 | lean_source_file | pkg_mathlib_source_file_chunks | naturality := by rintro _ _ (_ | _ | _) <;> simp [t.condition] }
/-- This is a helper construction that can be useful when verifying that a category has all
coequalizers. Given `F : WalkingParallelPair ⥤ C`, which is really the same as
`parallelPair (F.map left) (F.map right)`, and a cofork on `F.map left` and `F.map ... |
8,417 | lean_file_chunk | 552b5ec63f530861365fce53768d706a5daae906aac997ae539f8be4d9960404 | lean_source_file | pkg_mathlib_source_file_chunks | noncomputable def coequalizerComparison [HasCoequalizer f g] [HasCoequalizer (G.map f) (G.map g)] :
coequalizer (G.map f) (G.map g) ⟶ G.obj (coequalizer f g) :=
coequalizer.desc (G.map (coequalizer.π _ _))
(by simp only [← G.map_comp]; rw [coequalizer.condition])
@[reassoc (attr := simp)]
theorem ι_comp_coeq... |
23,979 | lean_file_chunk | 7a0de3cb24659b179ea9c9763d2cdd9fea83f30bade8a1ca81111c82b32eb54c | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Jakob von Raumer
-/
module
public import Mathlib.CategoryTheory.Limits.HasLimits
public import Mathlib.CategoryTheory.Thin
/-!
# Wide pullbacks
We define the category `W... |
807 | lean_file_chunk | 1cb48df0c54560fda9488e66bd0541f5821e542d67d5dc38ae03e75cf5da0258 | lean_source_file | pkg_mathlib_source_file_chunks | functor := widePullbackShapeUnop J
inverse := widePushoutShapeOp J
unitIso := (widePullbackShapeOpUnop J).symm
counitIso := widePushoutShapeUnopOp J
/-- If a category has wide pushouts on a higher universe level it also has wide pushouts
on a lower universe level. -/
theorem hasWidePushouts_shrink [HasWidePushou... |
10,468 | lean_file_chunk | 9b33210c2135194db0dbe4c203c527b59ddaf83cdcdd03e19f7df9e52269438f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
module
public import Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer
/-!
# Ends and coends
In this file, given a functor `F : Jᵒᵖ ⥤ J ⥤ C`, we define its end `end_ F`,
w... |
23,994 | lean_file_chunk | f818bd286ff94a5bb8a370162044c34651753ff4e8fcee64f6817f1446630c0d | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Jakob von Raumer
-/
module
public import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
public import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
/-!
# Binary bi... |
17,242 | lean_file_chunk | 204653e5bb090d08e0715f4c0ad8fb319e35d2b84231114d8301895477330d7c | lean_source_file | pkg_mathlib_source_file_chunks | IsSplitMono.mk' { retraction := biprod.fst }
instance biprod.inr_mono {X Y : C} [HasBinaryBiproduct X Y] :
IsSplitMono (biprod.inr : Y ⟶ X ⊞ Y) :=
IsSplitMono.mk' { retraction := biprod.snd }
instance biprod.fst_epi {X Y : C} [HasBinaryBiproduct X Y] : IsSplitEpi (biprod.fst : X ⊞ Y ⟶ X) :=
IsSplitEpi.mk' { s... |
23,951 | lean_file_chunk | 3881a3ddf5910e22ee00e2d268b6fc934ed62342f78919131e25a91f1dc301a5 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Markus Himmel
-/
module
public import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
/-!
# Kernels and cokernels
In a category with zero morphisms, the kernel of a ... |
23,990 | lean_file_chunk | f142592e7c219dda83ca6b632b13f1d28eda0b8e4673e775e3b18d78baaa3f19 | lean_source_file | pkg_mathlib_source_file_chunks | /-- Every cokernel cofork `s` is isomorphic (actually, equal) to `cofork.ofπ (cofork.π s) _`. -/
def isoOfπ (s : Cofork f 0) : s ≅ Cofork.ofπ (Cofork.π s) (Cofork.condition s) :=
Cocone.ext (Iso.refl _) fun j => by cases j <;> cat_disch
/-- If `π = π'`, then `CokernelCofork.of_π π _` and `CokernelCofork.of_π π' _` a... |
4,868 | lean_file_chunk | 1f77b094704f1add190b06e7f3ea76debf1e20d5544837661d6324f6b95f2b5b | lean_source_file | pkg_mathlib_source_file_chunks | section Comparison
variable {D : Type u₂} [Category.{v₂} D] [HasZeroMorphisms D]
variable (G : C ⥤ D) [Functor.PreservesZeroMorphisms G]
/-- The comparison morphism for the kernel of `f`.
This is an isomorphism iff `G` preserves the kernel of `f`; see
`Mathlib/CategoryTheory/Limits/Preserves/Shapes/Kernels.lean`
-/
d... |
9,506 | lean_file_chunk | 504d0b43e6e14f56947d5e8ec99d62f60094222f16f973629f43b334da84abb2 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
module
public import Mathlib.CategoryTheory.Limits.Shapes.Equalizers
public import Mathlib.CategoryTheory.Limits.Shapes.RegularMono
public import Mathlib.CategoryTheory.... |
10,051 | lean_file_chunk | 7f0f08172a69c875490d7ae552992574922e49fa475fc1599fea5d2455eb50ee | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Johan Commelin
-/
module
public import Mathlib.CategoryTheory.Limits.Shapes.Terminal
/-!
# Zero objects
A category "has a zero object" if it has an object which is both ... |
2,885 | lean_file_chunk | 4550e8c38aece651ffd6e6e23ee6b83d6a7e188f5978228e26ec18a6ade3c80a | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
module
public import Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer
public import Mathlib.CategoryTheory.Limits.Shapes.Pullback.IsPullback.Defs
/-!
# Multicoequalizers t... |
3,836 | lean_file_chunk | a99d66a0986baf301b70c540e87f364c368f224843b909a5890044e38215584f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten
-/
module
public import Mathlib.CategoryTheory.Limits.Types.Colimits
public import Mathlib.CategoryTheory.Limits.Types.Limits
public import Mathlib.CategoryTheory.S... |
10,704 | lean_file_chunk | 63d3f940d6676f50179ff47235a803f646cfa7c495138fd7e018bdee0d1f71f6 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
module
public import Mathlib.CategoryTheory.FinCategory.AsType
public import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
public import Mathlib.CategoryTheory.Lim... |
11,044 | lean_file_chunk | 163532f45ae24f8808eba499d91f8e30809547c8ad4fee039f4a6063b17361ac | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
module
public import Mathlib.CategoryTheory.Grothendieck
public import Mathlib.CategoryTheory.Limits.HasLimits
/-!
# (Co)limits on the (strict) Grothendieck Con... |
927 | lean_file_chunk | 4343c4fbd7dcf7d7189a02dfddbe4f88b9f1777b8a6266f7947cc84997f3a9ee | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.CategoryTheory.IsConnected
public import Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks
/-!
# Connected shapes
In this file we prove th... |
5,858 | lean_file_chunk | 24368b919193369b89d0ef796b06b05ffb0f2a2f44f2046fc8f2a2d78009b8b5 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Jack McKoen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jack McKoen
-/
module
public import Mathlib.CategoryTheory.Limits.Shapes.Equalizers
/-!
# Split Equalizers
We define what it means for a triple of morphisms `f g : X ⟶ Y`, `ι : W ⟶ X` ... |
23,992 | lean_file_chunk | d298c69ee17a25fca89780feb2ebe69be991d7e0cbb11bd9f644f94204dcab74 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Joël Riou
-/
module
public import Mathlib.CategoryTheory.Limits.Shapes.Products
public import Mathlib.CategoryTheory.Limits.Shapes.Equalizers
public import Mathlib.CategoryThe... |
21,320 | lean_file_chunk | 6ba572f5f0234beb06b2ea345e01c92ed946528d5d13603eb042e6973bae52cf | lean_source_file | pkg_mathlib_source_file_chunks | (a : Fork (I.fstPiMapOfIsLimit c hd) (I.sndPiMapOfIsLimit c hd)) (i) :
(ofPiFork a).π.app (WalkingMulticospan.right i) =
a.ι ≫ I.fstPiMapOfIsLimit c hd ≫ d.proj _ :=
rfl
end Multifork
namespace MulticospanIndex
variable {J : MulticospanShape.{w, w'}} (I : MulticospanIndex J C)
variable {c : Fan I.left} (... |
3,426 | lean_file_chunk | a15060cd66ad3a546c1537625809eaa7293694e5a834bde41c857ea154659e77 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten
-/
module
public import Mathlib.CategoryTheory.Limits.Shapes.Products
public import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
/-!
# Constructors for comb... |
3,106 | lean_file_chunk | 54e8649c0cf80106b99cc103297fd4bd72745c75272c2104195bc8a033eb4722 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
module
public import Mathlib.CategoryTheory.Limits.Shapes.FiniteLimits
public import Mathlib.CategoryTheory.Limits.Shapes.Products
/-!
# Categories with finite (co)prod... |
1,397 | lean_file_chunk | 81a8dd87d35892928fcdba284069186391e3d3fc060ceb173fc83320ec087ae6 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
module
public import Mathlib.CategoryTheory.Adjunction.Limits
public import Mathlib.CategoryTheory.Limits.Shapes.Terminal
/-!
# Transporting existence of specific lim... |
23,983 | lean_file_chunk | 4b3a5cb703ef7381a576545bf80d5d9668d66b7cbe442b688b1703dfeb8e4a98 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
module
public import Mathlib.CategoryTheory.Limits.HasLimits
public import Mathlib.CategoryTheory.Limits.Shapes.Equalizers
/-!
# Wide equalizers and wide coequalizers
... |
4,612 | lean_file_chunk | 571f95e48cd333112d8089ff38b5557839684cf6ff2292182f020a31a551c7fa | lean_source_file | pkg_mathlib_source_file_chunks | abbrev wideCoequalizer : C :=
colimit (parallelFamily f)
/-- If a wideCoequalizer of `f` exists, we can access the corresponding projection by
saying `wideCoequalizer.π f`. -/
abbrev wideCoequalizer.π : Y ⟶ wideCoequalizer f :=
colimit.ι (parallelFamily f) one
/-- An arbitrary choice of coequalizer cocone for... |
23,953 | lean_file_chunk | fc8b9330f1962c34e9ebd2c9187d419bc8885317647531d908e335ccf05f3f0c | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Markus Himmel
-/
module
public import Mathlib.CategoryTheory.Limits.Shapes.Equalizers
public import Mathlib.CategoryTheory.Limits.Shapes.Pullback.Mono
public import Mathli... |
13,351 | lean_file_chunk | d948d9334049b91e473b27908e46679b9e069ea894d9a92c3498cc5f3fce08e8 | lean_source_file | pkg_mathlib_source_file_chunks | theorem ImageMap.factor_map {f g : Arrow C} [HasImage f.hom] [HasImage g.hom] (sq : f ⟶ g)
(m : ImageMap sq) : factorThruImage f.hom ≫ m.map = sq.left ≫ factorThruImage g.hom :=
(cancel_mono (image.ι g.hom)).1 <| by simp
set_option backward.isDefEq.respectTransparency false in
/-- To give an image map for a comm... |
23,998 | lean_file_chunk | 2472f3513bce861ec4e99de0e0c0461c302bda8a58d707227f8944e32e4c96d1 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Bhavik Mehta
-/
module
public import Mathlib.CategoryTheory.Comma.Over.Basic
public import Mathlib.CategoryTheory.Discrete.Basic
public import Mathlib.CategoryTheory.EpiMo... |
23,960 | lean_file_chunk | 7641c5c78c8b241fef328de63f40664854c9dca11d6ab9988364eb0688f853ff | lean_source_file | pkg_mathlib_source_file_chunks | /-- If the products `W ⨯ X` and `Y ⨯ Z` exist, then every pair of morphisms `f : W ⟶ Y` and
`g : X ⟶ Z` induces a morphism `prod.map f g : W ⨯ X ⟶ Y ⨯ Z`. -/
noncomputable def prod.map {W X Y Z : C} [HasBinaryProduct W X] [HasBinaryProduct Y Z]
(f : W ⟶ Y) (g : X ⟶ Z) : W ⨯ X ⟶ Y ⨯ Z :=
limMap (mapPair f g)
/-- ... |
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