Let Top and Diff be the categories of topological and diffeological spaces,respectively. By using an adjunction between Top and Diff we show that the full subcategory NG of Top consisting of numerically generated spaces is complete, cocomplete and cartesian closed. In fact, NG can be embedded into Diff as a cartesian closed full subcategory. It follows then that the category NG0 of numerically generated pointed spaces is complete,cocomplete and monoidally closed with respect to the smash product. These features of NG0 are used to establish a simple but flexible method for constructing generalized homology and cohomology theories by using the notion of enriched bifunctors.
Abscisic acid (ABA) plays an important role in plant growth, development, and stress responses. ABA regulates many aspects of plant growth and development, including seed maturation, dormancy, germination, the transition from vegetative to reproductive growth, leaf senescence and responses to environmental stresses, such as drought, high salinity and cold. It is also known that mitogen-activated protein kinase (MAPK) cascades function in ABA signaling. Recently, we and another group have identified the ABA-inducible MAP3Ks MAP3K17 and MAP3K18 as the upstream MAP3Ks of MKK3, implicating the MAP3K17/18-MKK3-MPK1/2/7/14 cascade in ABA signaling. It has also been reported that overexpression of MAP3K18 in Arabidopsis causes an early leaf senescence phenotype, ABA hypersensitive stomata closing, and drought tolerance. In this study, we generated transgenic plants overexpressing MAP3K17 (35S:MAP3K17) and its kinase-inactive form (35S:MAP3K17KN). The bolting of 35S:MAP3K17 was earlier than WT, and the fresh weights of the seedlings were smaller, whereas 35S:MAP3K17KN showed the opposite phenotype. These results indicate that the transition from vegetative to reproductive growth can be regulated by overexpression of MAP3K17 and its kinase-inactive form. Moreover, 35S:MAP3K17 showed lower sensitivity to ABA during post-germinated growth, whereas 35S:MAP3K17 KN showed the opposite phenotype, suggesting the negative roles of MAP3K17 in the response to ABA. Our work provides the possibility to regulate plant growth and development by the genetic manipulation of ABA-induced MAPK cascades, leading to improved crop growth and productivity.