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Abstract
We study the 2-category of categories with finite limits, Lex. We characterise descent, effective descent and chain descent morphisms. These classes of morphisms do not coincide in Lex. We also study relations between these and other naturally arising classes of conservative morphisms. We define, in a semantical way, a new false quotient-strongly conservative factorisation in Lex. We prove that the iteration of the descent construction eventually "stops" at this factorisation. This gives a syntactic description of the factorisation.
CONTENTS
0. Introduction.........................................................................................5
1. Basic notions......................................................................................6
1.1. Effective descent morphisms..........................................................7
1.2. Left exact categories......................................................................9
1.3. Factorisations in Lex.....................................................................11
1.4. Descent theorem for exact categories..........................................16
2. The exact completion of the left exact categories.............................17
3. A characterisation of the descent category......................................24
4. A characterisation of the effective descent morphisms in Lex..........30
5. Some further results.........................................................................34
6. The false quotient-strongly conservative factorisation in Lex...........36
7. Conservative morphisms in Lex........................................................44
References...........................................................................................51
1991 Mathematics Subject Classification: Primary 03G30, 18C10; Secondary 03C40, 18D05.
Zawadowski Marek W.. Lax descent theorems for left exact categories. 1995. <http://eudml.org/doc/275823>.
@book{ZawadowskiMarekW1995, abstract = {
Abstract
We study the 2-category of categories with finite limits, Lex. We characterise descent, effective descent and chain descent morphisms. These classes of morphisms do not coincide in Lex. We also study relations between these and other naturally arising classes of conservative morphisms. We define, in a semantical way, a new false quotient-strongly conservative factorisation in Lex. We prove that the iteration of the descent construction eventually "stops" at this factorisation. This gives a syntactic description of the factorisation.
CONTENTS
0. Introduction.........................................................................................5
1. Basic notions......................................................................................6
1.1. Effective descent morphisms..........................................................7
1.2. Left exact categories......................................................................9
1.3. Factorisations in Lex.....................................................................11
1.4. Descent theorem for exact categories..........................................16
2. The exact completion of the left exact categories.............................17
3. A characterisation of the descent category......................................24
4. A characterisation of the effective descent morphisms in Lex..........30
5. Some further results.........................................................................34
6. The false quotient-strongly conservative factorisation in Lex...........36
7. Conservative morphisms in Lex........................................................44
References...........................................................................................51
1991 Mathematics Subject Classification: Primary 03G30, 18C10; Secondary 03C40, 18D05.}, author = {Zawadowski Marek W.}, keywords = {descent; Beck triplability theorem; left exact categories; algebraic theories; Beth definability theorem}, language = {eng}, title = {Lax descent theorems for left exact categories}, url = {http://eudml.org/doc/275823}, year = {1995}, }
TY - BOOK AU - Zawadowski Marek W. TI - Lax descent theorems for left exact categories PY - 1995 AB -
Abstract
We study the 2-category of categories with finite limits, Lex. We characterise descent, effective descent and chain descent morphisms. These classes of morphisms do not coincide in Lex. We also study relations between these and other naturally arising classes of conservative morphisms. We define, in a semantical way, a new false quotient-strongly conservative factorisation in Lex. We prove that the iteration of the descent construction eventually "stops" at this factorisation. This gives a syntactic description of the factorisation.
CONTENTS
0. Introduction.........................................................................................5
1. Basic notions......................................................................................6
1.1. Effective descent morphisms..........................................................7
1.2. Left exact categories......................................................................9
1.3. Factorisations in Lex.....................................................................11
1.4. Descent theorem for exact categories..........................................16
2. The exact completion of the left exact categories.............................17
3. A characterisation of the descent category......................................24
4. A characterisation of the effective descent morphisms in Lex..........30
5. Some further results.........................................................................34
6. The false quotient-strongly conservative factorisation in Lex...........36
7. Conservative morphisms in Lex........................................................44
References...........................................................................................51
1991 Mathematics Subject Classification: Primary 03G30, 18C10; Secondary 03C40, 18D05. LA - eng KW - descent; Beck triplability theorem; left exact categories; algebraic theories; Beth definability theorem UR - http://eudml.org/doc/275823 ER -
