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Hybrid caches consisting of both SRAM and emerging Non-Volatile Random Access Memory (eNVRAM) bitcells increase cache capacity and reduce power consumption by taking advantage of eNVRAM's small area footprint and low leakage energy. However, they also inherit eNVRAM's drawbacks, including long write latency and limited endurance. To mitigate these drawbacks, many works propose heuristic strategies to allocate memory blocks into SRAM or eNVRAM arrays at runtime based on block content or access pattern. In contrast, this work presents a HW/SW Stack for Hybrid Caches (SHyCache), consisting of a hybrid cache architecture and supporting programming model, reminiscent of those that enable GP-GPU acceleration, in which application variables can be allocated explicitly to the eNVRAM cache, eliminating the need for heuristics and reducing cache access time, power consumption, and area overhead while maintaining maximal cache utilization efficiency and ease of programming. SHyCache improves performance for applications such as neural networks, which contain large numbers of invariant weight values with high read/write access ratios that can be explicitly allocated to the eNVRAM array. We simulate SHyCache on the gem5-X architectural simulator and demonstrate its utility by benchmarking a range of cache hierarchy variations using three neural networks, namely, Inception v4, ResNet-50, and SqueezeNet 1.0. We demonstrate a design space that can be exploited to optimize performance, power consumption, or endurance, depending on the expected use case of the architecture, while demonstrating maximum performance gains of 1.7/1.4/1.3x and power consumption reductions of 5.1/5.2/5.4x, for Inception/ResNet/SqueezeNet, respectively.