Exact colimits and fixed points

Isbell, John; Mitchell, Barry

doi:10.1090/S0002-9947-1976-0404377-3

Exact colimits and fixed points
HTML articles powered by AMS MathViewer

by John Isbell and Barry Mitchell PDF

Trans. Amer. Math. Soc. 220 (1976), 289-298 Request permission

Abstract:

In this paper we shall give details of some work sketched in [6] on the exactness of the functor colim: ${\text {Ab}}^\mathcal {\text {C}} \to {\text {Ab}}$. We shall also investigate the connection between this work and a paper of J. Adámek and J. Reiterman [1] characterizing those categories $\mathcal {\text {C}}$ with the property that every endomorphism of an indecomposable functor $\mathcal {\text {C}} \to$ Sets has a fixed point. Exactness of colim implies the fixed point property, and in some cases (such as when $\mathcal {\text {C}}$ has only finitely many objects) both conditions turn out to be equivalent to the components of $\mathcal {\text {C}}$ being filtered. We do not expect that the two conditions are equivalent in general, although we have no example. However the category of finite ordinals and order preserving injections is an example of a connected, nonfiltered category relative to which colim is exact. This was conjectured by Mitchell, and is proved by Isbell in [5].

References

J. Adámek and J. Reiterman, Fixed points in representations of categories, Trans. Amer. Math. Soc. 211 (1975), 239–247. MR 376799, DOI 10.1090/S0002-9947-1975-0376799-X

Eléments de mathématiques

Algèbre commutative

36

30

P. M. Cohn, On the free product of associative rings, Math. Z. 71 (1959), 380–398. MR 106918, DOI 10.1007/BF01181410
John R. Isbell, A note on exact colimits, Canad. Math. Bull. 11 (1968), 569–572. MR 238926, DOI 10.4153/CMB-1968-068-7
John Isbell, Exact colimits. I, Ann. of Math. (2) 100 (1974), 633–637. MR 352200, DOI 10.2307/1970960
John Isbell and Barry Mitchell, Exact colimits, Bull. Amer. Math. Soc. 79 (1973), 994–996. MR 318255, DOI 10.1090/S0002-9904-1973-13296-3
F. William Lawvere, Functorial semantics of algebraic theories, Proc. Nat. Acad. Sci. U.S.A. 50 (1963), 869–872. MR 158921, DOI 10.1073/pnas.50.5.869
Barry Mitchell, Rings with several objects, Advances in Math. 8 (1972), 1–161. MR 294454, DOI 10.1016/0001-8708(72)90002-3
Ulrich Oberst, Homology of categories and exactness of direct limits, Math. Z. 107 (1968), 87–115. MR 244347, DOI 10.1007/BF01111023